BIG IDEAS MATH Integrated Math 1: Solving Linear Equations; Student Edition 2016 [Student ed.] 1680331124, 9781680331127

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1 1.1 1.2 1.3 1.4 1.5

Solving Linear Equations Solving Simple Equations Solving Multi-Step Equations Solving Equations with Variables on Both Sides Solving Absolute Value Equations Rewriting Equations and Formulas

Density of Pyrite (p. 41)

SEE the Big Idea Cheerleading Competition (p. 29)

Boat (p. 22)

Biking (p. 14)

Average Speed (p. 6)

Maintaining Mathematical Proficiency Adding and Subtracting Integers Example 1

(Grade 7)

Evaluate 4 + (−12). ∣ −12 ∣ > ∣ 4 ∣. So, subtract ∣ 4 ∣ from ∣ −12 ∣.

4 + (−12) = −8 Use the sign of −12.

Example 2

Evaluate −7 − (−16). −7 − (−16) = −7 + 16

Add the opposite of −16.

=9

Add.

Add or subtract. 1. −5 + (−2)

2. 0 + (−13)

3. −6 + 14

4. 19 − (−13)

5. −1 − 6

6. −5 − (−7)

7. 17 + 5

8. 8 + (−3)

9. 11 − 15

Multiplying and Dividing Integers Example 3

(Grade 7)

⋅

Evaluate −3 (−5). The integers have the same sign.

⋅

−3 (−5) = 15 The product is positive.

Example 4

Evaluate 15 ÷ (−3). The integers have different signs.

15 ÷ (−3) = −5 The quotient is negative.

Multiply or divide.

⋅

⋅

10. −3 (8)

11. −7 (−9)

13. −24 ÷ (−6)

14. —

15. 12 ÷ (−3)

17. 36 ÷ 6

18. −3(−4)

⋅

16. 6 8

−16 2

12. 4 (−7)

19. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers,

(c) multiplying integers, and (d) dividing integers. Give an example of each.

Dynamic Solutions available at BigIdeasMath.com

1

Mathematical Practices

Mathematically proficient students carefully specify units of measure.

Specifying Units of Measure

Core Concept Operations and Unit Analysis Addition and Subtraction

When you add or subtract quantities, they must have the same units of measure. The sum or difference will have the same unit of measure. Example

Perimeter of rectangle = (3 ft) + (5 ft) + (3 ft) + (5 ft)

3 ft

= 16 feet

5 ft

When you add feet, you get feet.

Multiplication and Division

When you multiply or divide quantities, the product or quotient will have a different unit of measure. Example

Area of rectangle = (3 ft) × (5 ft) = 15 square feet

When you multiply feet, you get feet squared, or square feet.

Specifying Units of Measure You work 8 hours and earn $72. What is your hourly wage?

SOLUTION dollars per hour

dollars per hour

Hourly wage = $72 ÷ 8 h ($ per h) = $9 per hour

The units on each side of the equation balance. Both are specified in dollars per hour.

Your hourly wage is $9 per hour.

Monitoring Progress Solve the problem and specify the units of measure. 1. The population of the United States was about 280 million in 2000 and about

310 million in 2010. What was the annual rate of change in population from 2000 to 2010? 2. You drive 240 miles and use 8 gallons of gasoline. What was your car’s gas mileage

(in miles per gallon)? 3. A bathtub is in the shape of a rectangular prism. Its dimensions are 5 feet by 3 feet by

18 inches. The bathtub is three-fourths full of water and drains at a rate of 1 cubic foot per minute. About how long does it take for all the water to drain?

2

Chapter 1

Solving Linear Equations

1.1

Solving Simple Equations Essential Question

How can you use simple equations to solve

real-life problems?

Measuring Angles Work with a partner. Use a protractor to measure the angles of each quadrilateral. Copy and complete the table to organize your results. (The notation m∠ A denotes the measure of angle A.) How precise are your measurements? a.

A

b.

B

B

c.

A

A

B C

D

UNDERSTANDING MATHEMATICAL TERMS

Quadrilateral

A conjecture is an a. unproven statement about a general b. mathematical concept. c. After the statement is proven, it is called a rule or a theorem. You will learn more about these terms in Chapter 9.

D

m∠A (degrees)

m∠B (degrees)

D

C

m∠C (degrees)

C

m∠D (degrees)

m∠A + m∠B + m∠C + m∠D

Making a Conjecture Work with a partner. Use the completed table in Exploration 1 to write a conjecture about the sum of the angle measures of a quadrilateral. Draw three quadrilaterals that are different from those in Exploration 1 and use them to justify your conjecture.

Applying Your Conjecture

CONNECTIONS TO GEOMETRY You will learn more about angle measures of quadrilaterals in a future course.

Work with a partner. Use the conjecture you wrote in Exploration 2 to write an equation for each quadrilateral. Then solve the equation to find the value of x. Use a protractor to check the reasonableness of your answer. a.

b.

85°

c. 78°

100°

30°

x° 90°

80°

x°

72°

x° 60°

90°

Communicate Your Answer 4. How can you use simple equations to solve real-life problems? 5. Draw your own quadrilateral and cut it out. Tear off the four corners of

the quadrilateral and rearrange them to affirm the conjecture you wrote in Exploration 2. Explain how this affirms the conjecture. Section 1.1

Solving Simple Equations

3

Lesson

1.1

What You Will Learn Solve linear equations using addition and subtraction. Solve linear equations using multiplication and division. Use linear equations to solve real-life problems.

Core Vocabul Vocabulary larry conjecture, p. 3 rule, p. 3 theorem, p. 3 equation, p. 4 linear equation in one variable, p. 4 solution, p. 4 inverse operations, p. 4 equivalent equations, p. 4 Previous expression

Solving Linear Equations by Adding or Subtracting An equation is a statement that two expressions are equal. A linear equation in one variable is an equation that can be written in the form ax + b = 0, where a and b are constants and a ≠ 0. When you solve an equation, you use properties of real numbers to find a solution, which is a value that makes the equation true. Inverse operations are two operations that undo each other, such as addition and subtraction. When you perform the same inverse operation on each side of an equation, you produce an equivalent equation. Equivalent equations are equations that have the same solution(s).

Core Concept Addition, Subtraction, and Substitution Properties of Equality

CONNECTIONS TO GEOMETRY

Let a, b, and c be real numbers.

Segment lengths and angle measures are real numbers. You will use these properties later in the book to write logical arguments about geometric figures.

Addition Property of Equality

If a = b, then a + c = b + c.

Subtraction Property of Equality

If a = b, then a − c = b − c.

Substitution Property of Equality If a = b, then a can be substituted for b

(or b for a) in any equation or expression.

Solving Equations by Addition or Subtraction Solve each equation. Justify each step. Check your answer. a. x − 3 = −5

b. 0.9 = y + 2.8

SOLUTION a. x − 3 = −5 +3

Addition Property of Equality

Write the equation.

+3

Check x − 3 = −5 ? − 2 − 3 = −5

Add 3 to each side.

x = −2

Simplify.

−5 = −5

The solution is x = −2.

b. Subtraction Property of Equality

0.9 = y + 2.8 − 2.8

− 2.8

−1.9 = y

Write the equation.

Check

Subtract 2.8 from each side.

0.9 = y + 2.8 ? 0.9 = −1.9 + 2.8

Simplify.

The solution is y = −1.9.

Monitoring Progress



0.9 = 0.9



Help in English and Spanish at BigIdeasMath.com

Solve the equation. Justify each step. Check your solution. 1. n + 3 = −7

4

Chapter 1

Solving Linear Equations

1

2

2. g − —3 = −—3

3. −6.5 = p + 3.9

Solving Linear Equations by Multiplying or Dividing Just as addition and subtraction are inverse operations, multiplication and division are also inverse operations.

Core Concept Multiplication and Division Properties of Equality Let a, b, and c be real numbers. Multiplication Property of Equality If a = b, then a

⋅ c = b ⋅ c, c ≠ 0.

a b If a = b, then — = — , c ≠ 0. c c

Division Property of Equality

Solving Equations by Multiplication or Division Solve each equation. Justify each step. Check your answer. n a. −— = −3 5

b. π x = −2π

c. 1.3z = 5.2

SOLUTION n −— = −3 5

a. Multiplication Property of Equality

−5

⋅( )

Write the equation.

n − — = −5 (− 3) 5

⋅

Check

n −— = −3 5 15 ? −— = −3 5 −3 = −3

Multiply each side by − 5.

n = 15

Simplify.

The solution is n = 15.

b. π x = −2π Division Property of Equality

πx π

Write the equation.

−2π π

—=—

Divide each side by π.

x = −2



Check π x = −2π ? π (−2) = −2π

Simplify.

− 2π = −2π



The solution is x = −2.

c. 1.3z = 5.2 Division Property of Equality

1.3z 5.2 —=— 1.3 1.3 z=4

Write the equation.

Check 1.3z = 5.2 ? 1.3(4) = 5.2

Divide each side by 1.3. Simplify.

5.2 = 5.2

The solution is z = 4.

Monitoring Progress



Help in English and Spanish at BigIdeasMath.com

Solve the equation. Justify each step. Check your solution. y 3

4. — = −6

5. 9π = π x

Section 1.1

6. 0.05w = 1.4

Solving Simple Equations

5

Solving Real-Life Problems MODELING WITH MATHEMATICS Mathematically proficient students routinely check that their solutions make sense in the context of a real-life problem.

Core Concept Four-Step Approach to Problem Solving 1.

Understand the Problem What is the unknown? What information is being given? What is being asked?

2.

Make a Plan This plan might involve one or more of the problem-solving strategies shown on the next page.

3.

Solve the Problem Carry out your plan. Check that each step is correct.

4.

Look Back Examine your solution. Check that your solution makes sense in the original statement of the problem.

Modeling with Mathematics In the 2012 Olympics, Usain Bolt won the 200-meter dash with a time of 19.32 seconds. Write and solve an equation to find his average speed to the nearest hundredth of a meter per second.

REMEMBER

SOLUTION

The formula that relates distance d, rate or speed r, and time t is d = rt.

1. Understand the Problem You know the winning time and the distance of the race. You are asked to find the average speed to the nearest hundredth of a meter per second. 2. Make a Plan Use the Distance Formula to write an equation that represents the problem. Then solve the equation. 3. Solve the Problem

⋅ 200 = r ⋅ 19.32 d=r t

REMEMBER The symbol ≈ means “approximately equal to.”

200 19.32

19.32r 19.32

Write the Distance Formula. Substitute 200 for d and 19.32 for t.

—=—

Divide each side by 19.32.

10.35 ≈ r

Simplify.

Bolt’s average speed was about 10.35 meters per second. 4. Look Back Round Bolt’s average speed to 10 meters per second. At this speed, it would take 200 m 10 m/sec

— = 20 seconds

to run 200 meters. Because 20 is close to 19.32, your solution is reasonable.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

7. Suppose Usain Bolt ran 400 meters at the same average speed that he ran the

200 meters. How long would it take him to run 400 meters? Round your answer to the nearest hundredth of a second.

6

Chapter 1

Solving Linear Equations

Core Concept Common Problem-Solving Strategies Use a verbal model.

Guess, check, and revise.

Draw a diagram.

Sketch a graph or number line.

Write an equation.

Make a table.

Look for a pattern.

Make a list.

Work backward.

Break the problem into parts.

Modeling with Mathematics On January 22, 1943, the temperature in Spearfish, South Dakota, fell from 54°F at 9:00 a.m. to − 4°F at 9:27 a.m. How many degrees did the temperature fall?

SOLUTION 1. Understand the Problem You know the temperature before and after the temperature fell. You are asked to find how many degrees the temperature fell. 2. Make a Plan Use a verbal model to write an equation that represents the problem. Then solve the equation. 3. Solve the Problem Words

Number of degrees Temperature Temperature = − the temperature fell at 9:27 a.m. at 9:00 a.m.

Variable

Let T be the number of degrees the temperature fell.

Equation

−4

=

−

54

−4 = 54 − T

T Write the equation.

−4 − 54 = 54 − 54 − T −58 = − T

Subtract 54 from each side. Simplify.

58 = T

Divide each side by − 1.

The temperature fell 58°F.

REMEMBER The distance between two points on a number line is always positive.

4. Look Back The temperature fell from 54 degrees above 0 to 4 degrees below 0. You can use a number line to check that your solution is reasonable. 58 −8 −4

0

4

8 12 16 20 24 28 32 36 40 44 48 52 56 60

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

8. You thought the balance in your checking account was $68. When your bank

statement arrives, you realize that you forgot to record a check. The bank statement lists your balance as $26. Write and solve an equation to find the amount of the check that you forgot to record.

Section 1.1

Solving Simple Equations

7

Exercises

1.1

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY Which of the operations +, −, ×, and ÷ are inverses of each other? 2. VOCABULARY Are the equations − 2x = 10 and −5x = 25 equivalent? Explain. 3. WRITING Which property of equality allows you to check a solution of an equation? Explain. 4. WHICH ONE DOESN’T BELONG? Which expression does not belong with the other three? Explain

your reasoning. x 8=— 2

3=x÷4

x 3

—=9

x−6=5

Monitoring Progress and Modeling with Mathematics In Exercises 5–14, solve the equation. Justify each step. Check your solution. (See Example 1.)

USING TOOLS The sum of the angle measures of a

5. x + 5 = 8

6. m + 9 = 2

quadrilateral is 360°. In Exercises 17–20, write and solve an equation to find the value of x. Use a protractor to check the reasonableness of your answer.

7. y − 4 = 3

8. s − 2 = 1

17.

9. w + 3 = −4

10. n − 6 = −7

11. −14 = p − 11

12. 0 = 4 + q

13. r + (−8) = 10

x°

amusement park ticket costs $12.95 less than the original price p. Write and solve an equation to find the original price.

x° 150°

100° 120°

14. t − (−5) = 9

15. MODELING WITH MATHEMATICS A discounted

18.

77°

48°

100°

19. 76°

92°

x°

20. 115°

122°

85°

x° 60°

In Exercises 21–30, solve the equation. Justify each step. Check your solution. (See Example 2.)

16. MODELING WITH MATHEMATICS You and a friend

are playing a board game. Your final score x is 12 points less than your friend’s final score. Write and solve an equation to find your final score. ROUND

9

ROUND

10

FINAL SCORE

Your Friend You

8

Chapter 1

Solving Linear Equations

21. 5g = 20

22. 4q = 52

23. p ÷ 5 = 3

24. y ÷ 7 = 1

25. −8r = 64

26. x ÷ (−2) = 8

x 6

w −3

27. — = 8

28. — = 6

29. −54 = 9s

30. −7 = —

t 7

In Exercises 31– 38, solve the equation. Check your solution. 1

3

45. REASONING Identify the property of equality that

makes Equation 1 and Equation 2 equivalent.

5

31. —32 + t = —2

32. b − — =— 16 16

33. —37 m = 6

34. −—5 y = 4

Equation 1

1 x x−—=—+3 2 4

35. 5.2 = a − 0.4

36. f + 3π = 7π

Equation 2

4x − 2 = x + 12

37. − 108π = 6π j

38. x ÷ (−2) = 1.4

2

ERROR ANALYSIS In Exercises 39 and 40, describe and

correct the error in solving the equation. 39.



46. PROBLEM SOLVING Tatami mats are used as a floor

covering in Japan. One possible layout uses four identical rectangular mats and one square mat, as shown. The area of the square mat is half the area of one of the rectangular mats.

−0.8 + r = 12.6 r = 12.6 + (−0.8) r = 11.8

40.

 3

Total area = 81 ft2

m −— = −4 3 m −— = 3 (−4) 3 m = −12

⋅( )

⋅

41. ANALYZING RELATIONSHIPS A baker orders 162 eggs.

Each carton contains 18 eggs. Which equation can you use to find the number x of cartons? Explain your reasoning and solve the equation.

A 162x = 18 ○

x B — = 162 ○ 18

C 18x = 162 ○

D x + 18 = 162 ○

MODELING WITH MATHEMATICS In Exercises 42– 44, write and solve an equation to answer the question. (See Examples 3 and 4.) 42. The temperature at 5 p.m. is 20°F. The temperature

at 10 p.m. is −5°F. How many degrees did the temperature fall?

47. PROBLEM SOLVING You spend $30.40 on 4 CDs.

Each CD costs the same amount and is on sale for 80% of the original price. a. Write and solve an equation to find how much you spend on each CD. b. The next day, the CDs are no longer on sale. You have $25. Will you be able to buy 3 more CDs? Explain your reasoning. 48. ANALYZING RELATIONSHIPS As c increases, does

the value of x increase, decrease, or stay the same for each equation? Assume c is positive.

43. The length of an

American flag is 1.9 times its width. What is the width of the flag?

a. Write and solve an equation to find the area of one rectangular mat. b. The length of a rectangular mat is twice the width. Use Guess, Check, and Revise to find the dimensions of one rectangular mat.

9.5 ft

Equation

Value of x

x−c=0

44. The balance of an investment account is $308 more

cx = 1

than the balance 4 years ago. The current balance of the account is $4708. What was the balance 4 years ago?

cx = c x c

—=1

Section 1.1

Solving Simple Equations

9

MATHEMATICAL CONNECTIONS In Exercises 53–56, find

49. USING STRUCTURE Use the values −2, 5, 9, and 10

the height h or the area of the base B of the solid.

to complete each statement about the equation ax = b − 5.

53.

54.

a. When a = ___ and b = ___, x is a positive integer.

h

b. When a = ___ and b = ___, x is a negative integer.

7 in. B = 147 cm2

B

50. HOW DO YOU SEE IT? The circle graph shows the

percents of different animals sold at a local pet store in 1 year.

Volume = 84π in.3 55.

Hamster: 5%

Volume = 1323 cm3 56.

5m

h

Rabbit: 9% Bird: 7%

B = 30 ft 2

B

Dog: 48%

Volume = 15π m3

Cat: x%

Volume = 35 ft3

57. MAKING AN ARGUMENT In baseball, a player’s

batting average is calculated by dividing the number of hits by the number of at-bats. The table shows Player A’s batting average and number of at-bats for three regular seasons.

a. What percent is represented by the entire circle? b. How does the equation 7 + 9 + 5 + 48 + x = 100 relate to the circle graph? How can you use this equation to find the percent of cats sold?

Season

Batting average

At-bats

2010

.312

596

2011

.296

446

2012

.295

599

51. REASONING One-sixth of the girls and two-sevenths

of the boys in a school marching band are in the percussion section. The percussion section has 6 girls and 10 boys. How many students are in the marching band? Explain.

a. How many hits did Player A have in the 2011 regular season? Round your answer to the nearest whole number. b. Player B had 33 fewer hits in the 2011 season than Player A but had a greater batting average. Your friend concludes that Player B had more at-bats in the 2011 season than Player A. Is your friend correct? Explain.

52. THOUGHT PROVOKING Write a real-life problem

that can be modeled by an equation equivalent to the equation 5x = 30. Then solve the equation and write the answer in the context of your real-life problem.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Use the Distributive Property to simplify the expression. (Skills Review Handbook)

(

1

59. —56 x + —2 + 4

58. 8(y + 3)

)

60. 5(m + 3 + n)

61. 4(2p + 4q + 6)

Copy and complete the statement. Round to the nearest hundredth, if necessary. (Skills Review Handbook) 5L min

L

h

63. — ≈ —

7 gal min

qt sec

65. — ≈ —

62. — = — 64. — ≈ —

10

Chapter 1

Solving Linear Equations

68 mi h

mi sec

8 km min

h

mi

1.2

Solving Multi-Step Equations Essential Question

How can you use multi-step equations to solve

real-life problems?

Solving for the Angle Measures of a Polygon Work with a partner. The sum S of the angle measures of a polygon with n sides can be found using the formula S = 180(n − 2). Write and solve an equation to find each value of x. Justify the steps in your solution. Then find the angle measures of each polygon. How can you check the reasonableness of your answers? a.

b.

c. 50°

(30 + x)°

(2x + 30)°

(x + 10)°

JUSTIFYING CONCLUSIONS To be proficient in math, you need to be sure your answers make sense in the context of the problem. For instance, if you find the angle measures of a triangle, and they have a sum that is not equal to 180°, then you should check your work for mistakes.

9x ° 30°

(2x + 20)°

(x + 20)°

50°

x°

d.

e.

(x − 17)°

(x + 35)°

(5x + 2)° (3x + 5)°

f.

(2x + 8)° (3x + 16)°

(8x + 8)° (5x + 10)°

(x + 42)° x°

(4x − 18)° (3x − 7)°

(4x + 15)° (2x + 25)°

Writing a Multi-Step Equation Work with a partner.

CONNECTIONS TO GEOMETRY You will learn more about angle measures of polygons in a future course.

a. Draw an irregular polygon. b. Measure the angles of the polygon. Record the measurements on a separate sheet of paper. c. Choose a value for x. Then, using this value, work backward to assign a variable expression to each angle measure, as in Exploration 1. d. Trade polygons with your partner. e. Solve an equation to find the angle measures of the polygon your partner drew. Do your answers seem reasonable? Explain.

Communicate Your Answer 3. How can you use multi-step equations to solve real-life problems? 4. In Exploration 1, you were given the formula for the sum S of the angle measures

of a polygon with n sides. Explain why this formula works. 5. The sum of the angle measures of a polygon is 1080º. How many sides does the

polygon have? Explain how you found your answer. Section 1.2

Solving Multi-Step Equations

11

1.2 Lesson

What You Will Learn Solve multi-step linear equations using inverse operations. Use multi-step linear equations to solve real-life problems.

Core Vocabul Vocabulary larry

Use unit analysis to model real-life problems.

Previous inverse operations mean

Solving Multi-Step Linear Equations

Core Concept Solving Multi-Step Equations To solve a multi-step equation, simplify each side of the equation, if necessary. Then use inverse operations to isolate the variable.

Solving a Two-Step Equation Solve 2.5x − 13 = 2. Check your solution.

SOLUTION 2.5x − 13 = + 13

Undo the subtraction.

2

Write the equation.

+ 13

2.5x =

Add 13 to each side.

15

15 2.5x —= — 2.5 2.5

Undo the multiplication.

Check

Simplify.

2.5x − 13 = 2 ? 2.5(6) − 13 = 2

Divide each side by 2.5.

x=6

2=2

Simplify.



The solution is x = 6.

Combining Like Terms to Solve an Equation Solve −12 = 9x − 6x + 15. Check your solution.

SOLUTION

Undo the addition.

Undo the multiplication.

−12 = 9x − 6x + 15

Write the equation.

−12 = 3x + 15

Combine like terms.

− 15

Subtract 15 from each side.

− 15

−27 = 3x

Simplify.

−27 3x —=— 3 3

Divide each side by 3.

−9 = x

Check

Simplify.

− 12 = − 12

The solution is x = −9.

Monitoring Progress

− 12 = 9x − 6x + 15 ? − 12 = 9(− 9) − 6(− 9) + 15



Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solution. 1. −2n + 3 = 9

12

Chapter 1

Solving Linear Equations

2. −21 = —12 c − 11

3. −2x − 10x + 12 = 18

Using Structure to Solve a Multi-Step Equation Solve 2(1 − x) + 3 = − 8. Check your solution.

SOLUTION Method 1 One way to solve the equation is by using the Distributive Property. 2(1 − x) + 3 = −8 2(1) − 2(x) + 3 = −8

REMEMBER The Distributive Property states the following for real numbers a, b, and c.

2 − 2x + 3 = −8 −2x + 5 = −8 −5

−5

−2x = −13

Sum a(b + c) = ab + ac

−2x −2

−13 −2

—=—

Difference a(b − c) = ab − ac

x = 6.5 The solution is x = 6.5.

Write the equation. Distributive Property Multiply. Combine like terms. Subtract 5 from each side. Simplify. Divide each side by −2. Simplify.

Check 2(1 − x) + 3 = − 8 ? 2(1− 6.5) + 3 = − 8 −8 = −8



Method 2 Another way to solve the equation is by interpreting the expression 1 − x as a single quantity. 2(1 − x) + 3 = −8

LOOKING FOR STRUCTURE First solve for the expression 1 − x, and then solve for x.

−3

Write the equation.

−3

Subtract 3 from each side.

2(1 − x) = −11

Simplify.

2(1 − x) 2

Divide each side by 2.

−11 2

—=—

1 − x = −5.5 −1

−1

Subtract 1 from each side.

−x = −6.5 −x −1

Simplify.

−6.5 −1

—=—

x = 6.5

Simplify. Divide each side by − 1. Simplify.

The solution is x = 6.5, which is the same solution obtained in Method 1.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solution. 4. 3(x + 1) + 6 = −9

5. 15 = 5 + 4(2d − 3)

6. 13 = −2(y − 4) + 3y

7. 2x(5 − 3) − 3x = 5

8. −4(2m + 5) − 3m = 35

9. 5(3 − x) + 2(3 − x) = 14

Section 1.2

Solving Multi-Step Equations

13

Solving Real-Life Problems Modeling with Mathematics U the table to find the number of miles x Use yyou need to bike on Friday so that the mean nnumber of miles biked per day is 5.

SOLUTION S 11. Understand the Problem You know how many miles you biked Monday through Thursday. You are asked to find the number of miles you need to bike on Friday so that the mean number of miles biked per day is 5.

Day

Miles

Monday

3.5

Tuesday

5.5

Wednesday

0

Thursday

5

Friday

x

2. 2 Make a Plan Use the definition of mean to write an equation that represents the problem. Then solve the equation. 3. 3 Solve the Problem The mean of a data set is the sum of the data divided by the number of data values. 3.5 + 5.5 + 0 + 5 + x 5

Write the equation.

14 + x 5

Combine like terms.

—— = 5 —=5

14 + x 5 —=5 5 5

⋅

14 + x = − 14

⋅

Multiply each side by 5.

25

Simplify.

− 14

Subtract 14 from each side.

x = 11

Simplify.

You need to bike 11 miles on Friday. 4. Look Back Notice that on the days that you did bike, the values are close to the mean. Because you did not bike on Wednesday, you need to bike about twice the mean on Friday. Eleven miles is about twice the mean. So, your solution is reasonable.

Monitoring Progress 10. The formula d =

Help in English and Spanish at BigIdeasMath.com

1 —2 n

+ 26 relates the nozzle pressure n (in pounds per square inch) of a fire hose and the maximum horizontal distance the water reaches d (in feet). How much pressure is needed to reach a fire 50 feet away?

d

14

Chapter 1

Solving Linear Equations

REMEMBER When you add miles to miles, you get miles. But, when you divide miles by days, you get miles per day.

Using Unit Analysis to Model Real-Life Problems When you write an equation to model a real-life problem, you should check that the units on each side of the equation balance. For instance, in Example 4, notice how the units balance. miles

miles per day

3.5 + 5.5 + 0 + 5 + x 5

mi day

—— = 5

per

mi day

—=—



days

Solving a Real-Life Problem Your school’s drama club charges $4 per person for admission to a play. The club borrowed $400 to pay for costumes and props. After paying back the loan, the club has a profit of $100. How many people attended the play?

SOLUTION 1. Understand the Problem You know how much the club charges for admission. You also know how much the club borrowed and its profit. You are asked to find how many people attended the play. 2. Make a Plan Use a verbal model to write an equation that represents the problem. Then solve the equation. 3. Solve the Problem

REMEMBER When you multiply dollars per person by people, you get dollars.

Words

Ticket price

Variable

Let x be the number of people who attended.

Equation

—

$4 person

Number of people

⋅ who attended

−

Amount = Profit of loan

⋅ x people − $400 = $100

4x − 400 = 100

$=$



Write the equation.

4x − 400 + 400 = 100 + 400

Add 400 to each side.

4x = 500

Simplify.

4x 4

Divide each side by 4.

500 4

—=—

x = 125

Simplify.

So, 125 people attended the play. 4. Look Back To check that your solution is reasonable, multiply $4 per person by 125 people. The result is $500. After paying back the $400 loan, the club has $100, which is the profit.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

11. You have 96 feet of fencing to enclose a rectangular pen for your dog. To provide

sufficient running space for your dog to exercise, the pen should be three times as long as it is wide. Find the dimensions of the pen.

Section 1.2

Solving Multi-Step Equations

15

1.2

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE To solve the equation 2x + 3x = 20, first combine 2x and 3x because

they are _________. 2. WRITING Describe two ways to solve the equation 2(4x − 11) = 10.

Monitoring Progress and Modeling with Mathematics In Exercises 3−14, solve the equation. Check your solution. (See Examples 1 and 2.)

23. −3(3 + x) + 4(x − 6) = − 4

3. 3w + 7 = 19

4. 2g − 13 = 3

24. 5(r + 9) − 2(1 − r) = 1

5. 11 = 12 − q

6. 10 = 7 − m

USING TOOLS In Exercises 25−28, find the value of the

z −4

a 3

7. 5 = — − 3

8. — + 4 = 6

h+6 5

d−8 −2

9. — = 2

10. — = 12

11. 8y + 3y = 44

variable. Then find the angle measures of the polygon. Use a protractor to check the reasonableness of your answer. 25.

26.

12. 36 = 13n − 4n

45°

2a° 2a°

k°

Sum of angle measures: 180°

13. 12v + 10v + 14 = 80

a° Sum of angle measures: 360°

14. 6c − 8 − 2c = −16 15. MODELING WITH MATHEMATICS The altitude a

(in feet) of a plane t minutes after liftoff is given by a = 3400t + 600. How many minutes after liftoff is the plane at an altitude of 21,000 feet?

16. MODELING WITH MATHEMATICS A repair bill for

your car is $553. The parts cost $265. The labor cost is $48 per hour. Write and solve an equation to find the number of hours of labor spent repairing the car. In Exercises 17−24, solve the equation. Check your solution. (See Example 3.) 17. 4(z + 5) = 32

18. − 2(4g − 3) = 30

19. 6 + 5(m + 1) = 26

20. 5h + 2(11 − h) = − 5

21. 27 = 3c − 3(6 − 2c) 22. −3 = 12y − 5(2y − 7)

16

Chapter 1

a°

2k°

Solving Linear Equations

27.

b° 3 b° 2

(b + 45)°

(2b − 90)° 90° Sum of angle measures: 540°

28.

120° x°

100°

120° 120°

(x + 10)° Sum of angle measures: 720°

In Exercises 29−34, write and solve an equation to find the number. 29. The sum of twice a number and 13 is 75. 30. The difference of three times a number and 4 is −19. 31. Eight plus the quotient of a number and 3 is −2. 32. The sum of twice a number and half the number is 10. 33. Six times the sum of a number and 15 is − 42. 34. Four times the difference of a number and 7 is 12.

USING EQUATIONS In Exercises 35−37, write and solve

ERROR ANALYSIS In Exercises 40 and 41, describe and

an equation to answer the question. Check that the units on each side of the equation balance. (See Examples 4 and 5.)

correct the error in solving the equation. 40.

35. During the summer, you work 30 hours per week at



a gas station and earn $8.75 per hour. You also work as a landscaper for $11 per hour and can work as many hours as you want. You want to earn a total of $400 per week. How many hours must you work as a landscaper? 36. The area of the surface of the swimming pool is

210 square feet. What is the length d of the deep end (in feet)?

deep end

shallow end

d

9 ft

−10 − 2y = −4 −2y = 6 y = −3 41.



1 4

—(x − 2) + 4 = 12

1 4

—(x − 2) = 8

x=4

10 ft

$2.50. You pay 8% sales tax and leave a $3 tip. You pay a total of $13.80. How much does one taco cost?

MATHEMATICAL CONNECTIONS In Exercises 42−44,

write and solve an equation to answer the question. 42. The perimeter of the tennis court is 228 feet. What are

the dimensions of the court?

JUSTIFYING STEPS In Exercises 38 and 39, justify each step of the solution.

1 2

−14 − 2y + 4 = −4

x−2=2

37. You order two tacos and a salad. The salad costs

38. −—(5x − 8) − 1 = 6

−2(7 − y) + 4 = −4

w

Write the equation.

1 −—(5x − 8) = 7 2

2w + 6 43. The perimeter of the Norwegian flag is 190 inches.

What are the dimensions of the flag?

5x − 8 = −14 5x = −6

y

6 x = −— 5 39.

2(x + 3) + x = −9 2(x) + 2(3) + x = −9

11 y 8

Write the equation.

44. The perimeter of the school crossing sign is

102 inches. What is the length of each side?

2x + 6 + x = −9 s+6

3x + 6 = −9

s+6

3x = −15 s

x = −5

s 2s

Section 1.2

Solving Multi-Step Equations

17

45. COMPARING METHODS Solve the equation

2(4 − 8x) + 6 = −1 using (a) Method 1 from Example 3 and (b) Method 2 from Example 3. Which method do you prefer? Explain.

46. PROBLEM SOLVING An online ticket agency charges

the amounts shown for basketball tickets. The total cost for an order is $220.70. How many tickets are purchased?

49. REASONING An even integer can be represented by

the expression 2n, where n is any integer. Find three consecutive even integers that have a sum of 54. Explain your reasoning. 50. HOW DO YOU SEE IT? The scatter plot shows the

attendance for each meeting of a gaming club.

Charge

Amount

Ticket price

$32.50 per ticket

Convenience charge

$3.30 per ticket

Processing charge

$5.90 per order

Students

Gaming Club Attendance y 25 20 15 10 5 0

18

1

48. THOUGHT PROVOKING You teach a math class and

assign a weight to each component of the class. You determine final grades by totaling the products of the weights and the component scores. Choose values for the remaining weights and find the necessary score on the final exam for a student to earn an A (90%) in the class, if possible. Explain your reasoning. Component

Student’s Weight score

Score × Weight

92% × 0.20 = 18.4%

3

4

x

a. The mean attendance for the first four meetings is 20. Is the number of students who attended the fourth meeting greater than or less than 20? Explain. b. Estimate the number of students who attended the fourth meeting. c. Describe a way you can check your estimate in part (b).

REASONING In Exercises 51−56, the letters a, b, and c

represent nonzero constants. Solve the equation for x.

Class Participation

92%

Homework

95%

52. x + a = —4

Midterm Exam

88%

53. ax − b = 12.5

0.20

2

17

Meeting

47. MAKING AN ARGUMENT You have quarters and

dimes that total $2.80. Your friend says it is possible that the number of quarters is 8 more than the number of dimes. Is your friend correct? Explain.

21

51. bx = −7 3

Final Exam Total

54. ax + b = c

1

55. 2bx − bx = −8 56. cx − 4b = 5b

Maintaining Mathematical Proficiency Simplify the expression. 57. 4m + 5 − 3m

Reviewing what you learned in previous grades and lessons

(Skills Review Handbook) 58. 9 − 8b + 6b

59. 6t + 3(1 − 2t) − 5

Determine whether (a) x = −1 or (b) x = 2 is a solution of the equation.

18

(Skills Review Handbook)

60. x − 8 = − 9

61. x + 1.5 = 3.5

62. 2x − 1 = 3

63. 3x + 4 = 1

64. x + 4 = 3x

65. − 2(x − 1) = 1 − 3x

Chapter 1

Solving Linear Equations

1.3

Solving Equations with Variables on Both Sides Essential Question

How can you solve an equation that has

variables on both sides?

Perimeter Work with a partner. The two polygons have the same perimeter. Use this information to write and solve an equation involving x. Explain the process you used to find the solution. Then find the perimeter of each polygon. 3 x 2

x

5

5

3 5

2

CONNECTIONS TO GEOMETRY You will learn about finding perimeters and areas of polygons in the coordinate plane in Chapter 8.

Work with a partner.

•

Each figure has the unusual property that the value of its perimeter (in feet) is equal to the value of its area (in square feet). Use this information to write an equation for each figure.

• •

Solve each equation for x. Explain the process you used to find the solution. Find the perimeter and area of each figure.

5

3

b.

5

4

To be proficient in math, you need to visualize complex things, such as composite figures, as being made up of simpler, more manageable parts.

4

Perimeter and Area

a.

LOOKING FOR STRUCTURE

2

x

c.

1

2

6

2

x

x x

Communicate Your Answer 3. How can you solve an equation that has variables on both sides? 4. Write three equations that have the variable x on both sides. The equations should

be different from those you wrote in Explorations 1 and 2. Have your partner solve the equations. Section 1.3

Solving Equations with Variables on Both Sides

19

1.3 Lesson

What You Will Learn Solve linear equations that have variables on both sides.

Core Vocabul Vocabulary larry

Identify special solutions of linear equations. Use linear equations to solve real-life problems.

identity, p. 21 Previous inverse operations

Solving Equations with Variables on Both Sides

Core Concept Solving Equations with Variables on Both Sides To solve an equation with variables on both sides, simplify one or both sides of the equation, if necessary. Then use inverse operations to collect the variable terms on one side, collect the constant terms on the other side, and isolate the variable.

Solving an Equation with Variables on Both Sides Solve 10 − 4x = −9x. Check your solution.

SOLUTION 10 − 4x = −9x + 4x

Write the equation.

+ 4x

Add 4x to each side.

10 = − 5x

Simplify.

−5x 10 —=— −5 −5

Divide each side by −5.

−2 = x

Simplify.

Check 10 − 4x = −9x ? 10 − 4(−2) = −9(−2) 18 = 18



The solution is x = −2.

Solving an Equation with Grouping Symbols 1 Solve 3(3x − 4) = —(32x + 56). 4

SOLUTION 3(3x − 4) = 9x − 12 =

— (32x + 56)

1 4

Write the equation.

8x + 14

Distributive Property

+ 12

Add 12 to each side.

+ 12 9x = − 8x

8x + 26

Simplify.

− 8x

Subtract 8x from each side.

x = 26

Simplify.

The solution is x = 26.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solution. 1. −2x = 3x + 10

20

Chapter 1

Solving Linear Equations

2. —12 (6h − 4) = −5h + 1

3

3. −—4 (8n + 12) = 3(n − 3)

Identifying Special Solutions of Linear Equations

Core Concept Special Solutions of Linear Equations Equations do not always have one solution. An equation that is true for all values of the variable is an identity and has infinitely many solutions. An equation that is not true for any value of the variable has no solution.

Identifying the Number of Solutions

REASONING The equation 15x + 6 = 15x is not true because the number 15x cannot be equal to 6 more than itself.

Solve each equation. a. 3(5x + 2) = 15x

b. −2(4y + 1) = −8y − 2

SOLUTION a. 3(5x + 2) =

15x

Write the equation.

15x + 6 =

15x

Distributive Property

− 15x

− 15x

Subtract 15x from each side.



6=0

Simplify.

The statement 6 = 0 is never true. So, the equation has no solution.

b. −2(4y + 1) = −8y − 2

READING All real numbers are solutions of an identity.

Write the equation.

−8y − 2 = −8y − 2

Distributive Property

+ 8y

Add 8y to each side.

+ 8y −2 = −2

Simplify.

The statement −2 = −2 is always true. So, the equation is an identity and has infinitely many solutions.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. 5

4. 4(1 − p) = −4p + 4

5. 6m − m = —6 (6m − 10)

6. 10k + 7 = −3 − 10k

7. 3(2a − 2) = 2(3a − 3)

Concept Summary Steps for Solving Linear Equations Here are several steps you can use to solve a linear equation. Depending on the equation, you may not need to use some steps. Step 1 Use the Distributive Property to remove any grouping symbols.

STUDY TIP To check an identity, you can choose several different values of the variable.

Step 2 Simplify the expression on each side of the equation. Step 3 Collect the variable terms on one side of the equation and the constant

terms on the other side. Step 4 Isolate the variable. Step 5 Check your solution.

Section 1.3

Solving Equations with Variables on Both Sides

21

Solving Real-Life Problems Modeling with Mathematics A boat leaves New Orleans and travels upstream on the Mississippi River for 4 hours. The return trip takes only 2.8 hours because the boat travels 3 miles per hour faster T ddownstream due to the current. How far does the boat travel upstream?

SOLUTION 1. Understand the Problem You are given the amounts of time the boat travels and the difference in speeds for each direction. You are asked to find the distance the boat travels upstream. 22. Make a Plan Use the Distance Formula to write expressions that represent the problem. Because the distance the boat travels in both directions is the same, you can use the expressions to write an equation. 33. Solve the Problem Use the formula (distance) = (rate)(time). Words

Distance upstream = Distance downstream

Variable

Let x be the speed (in miles per hour) of the boat traveling upstream.

Equation

x mi 1h

—

(x + 3) mi

⋅4 h = — ⋅ 2.8 h 1h

(mi = mi)



4x = 2.8(x + 3)

Write the equation.

4x = 2.8x + 8.4

Distributive Property

− 2.8x − 2.8x

Subtract 2.8x from each side.

1.2x = 8.4

Simplify.

1.2x 1.2

Divide each side by 1.2.

8.4 1.2

—=—

x=7

Simplify.

So, the boat travels 7 miles per hour upstream. To determine how far the boat travels upstream, multiply 7 miles per hour by 4 hours to obtain 28 miles. 4. Look Back To check that your solution is reasonable, use the formula for distance. Because the speed upstream is 7 miles per hour, the speed downstream is 7 + 3 = 10 miles per hour. When you substitute each speed into the Distance Formula, you get the same distance for upstream and downstream. Upstream

⋅

7 mi Distance = — 4 h = 28 mi 1h Downstream

⋅

10 mi Distance = — 2.8 h = 28 mi 1h

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

8. A boat travels upstream on the Mississippi River for 3.5 hours. The return trip

only takes 2.5 hours because the boat travels 2 miles per hour faster downstream due to the current. How far does the boat travel upstream? 22

Chapter 1

Solving Linear Equations

1.3

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY Is the equation − 2(4 − x) = 2x + 8 an identity? Explain your reasoning. 2. WRITING Describe the steps in solving the linear equation 3(3x − 8) = 4x + 6.

Monitoring Progress and Modeling with Mathematics In Exercises 3–16, solve the equation. Check your solution. (See Examples 1 and 2.) 3. 15 − 2x = 3x

4. 26 − 4s = 9s

5. 5p − 9 = 2p + 12

6. 8g + 10 = 35 + 3g

7. 5t + 16 = 6 − 5t

In Exercises 19–24, solve the equation. Determine whether the equation has one solution, no solution, or infinitely many solutions. (See Example 3.) 19. 3t + 4 = 12 + 3t

20. 6d + 8 = 14 + 3d

21. 2(h + 1) = 5h − 7 22. 12y + 6 = 6(2y + 1)

8. −3r + 10 = 15r − 8

23. 3(4g + 6) = 2(6g + 9)

9. 7 + 3x − 12x = 3x + 1

1

24. 5(1 + 2m) = —2 (8 + 20m)

10. w − 2 + 2w = 6 + 5w 11. 10(g + 5) = 2(g + 9)

ERROR ANALYSIS In Exercises 25 and 26, describe and correct the error in solving the equation.

12. −9(t − 2) = 4(t − 15)

25.

13. —23 (3x + 9) = −2(2x + 6)



3

14. 2(2t + 4) = —4 (24 − 8t)

2c − 6 = 4 2c = 10 c=5

15. 10(2y + 2) − y = 2(8y − 8) 16. 2(4x + 2) = 4x − 12(x − 1)

26.

17. MODELING WITH MATHEMATICS You and your

friend drive toward each other. The equation 50h = 190 − 45h represents the number h of hours until you and your friend meet. When will you meet? 18. MODELING WITH MATHEMATICS The equation

1.5r + 15 = 2.25r represents the number r of movies you must rent to spend the same amount at each movie store. How many movies must you rent to spend the same amount at each movie store?

VIDEO CITY MEM M MBE ERSH HIP

Membership Fee: $15

5c − 6 = 4 − 3c

Membership Fee: Free

Section 1.3



6(2y + 6) = 4(9 + 3y) 12y + 36 = 36 + 12y 12y = 12y

0=0 The equation has no solution. 27. MODELING WITH MATHEMATICS Write and solve an

equation to find the month when you would pay the same total amount for each Internet service. Installation fee

Price per month

Company A

$60.00

$42.95

Company B

$25.00

$49.95

Solving Equations with Variables on Both Sides

23

28. PROBLEM SOLVING One serving of granola provides

37. REASONING Two times the greater of two

4% of the protein you need daily. You must get the remaining 48 grams of protein from other sources. How many grams of protein do you need daily?

consecutive integers is 9 less than three times the lesser integer. What are the integers? 38. HOW DO YOU SEE IT? The table and the graph show

information about students enrolled in Spanish and French classes at a high school.

USING STRUCTURE In Exercises 29 and 30, find the value

of r. 29. 8(x + 6) − 10 + r = 3(x + 12) + 5x

Students enrolled this year

30. 4(x − 3) − r + 2x = 5(3x − 7) − 9x MATHEMATICAL CONNECTIONS In Exercises 31 and 32,

the value of the surface area of the cylinder is equal to the value of the volume of the cylinder. Find the value of x. Then find the surface area and volume of the cylinder. 32.

2.5 cm

355

French

229

1 5

7 ft

x ft

x cm

y 400 350 300 250 200 150 0

33. MODELING WITH MATHEMATICS A cheetah that

French

Spanish 1 2 3 4 5 6 7 8 9 10 x

Years from now

is running 90 feet per second is 120 feet behind an antelope that is running 60 feet per second. How long will it take the cheetah to catch up to the antelope? (See Example 4.)

a. Use the graph to determine after how many years there will be equal enrollment in Spanish and French classes. b. How does the equation 355 − 9x = 229 + 12x relate to the table and the graph? How can you use this equation to determine whether your answer in part (a) is reasonable?

34. MAKING AN ARGUMENT A cheetah can run at top

speed for only about 20 seconds. If an antelope is too far away for a cheetah to catch it in 20 seconds, the antelope is probably safe. Your friend claims the antelope in Exercise 33 will not be safe if the cheetah starts running 650 feet behind it. Is your friend correct? Explain.

39. WRITING EQUATIONS Give an example of a linear

equation that has (a) no solution and (b) infinitely many solutions. Justify your answers.

REASONING In Exercises 35 and 36, for what value of a is the equation an identity? Explain your reasoning.

40. THOUGHT PROVOKING Draw

a different figure that has the same perimeter as the triangle shown. Explain why your figure has the same perimeter.

35. a(2x + 3) = 9x + 15 + x 36. 8x − 8 + 3ax = 5ax − 2a

Maintaining Mathematical Proficiency 41. 9, ∣ −4 ∣, −4, 5, ∣ 2 ∣

42.

43. −18, ∣ −24 ∣, −19, ∣ −18 ∣, ∣ 22 ∣

44. −∣ −3 ∣, ∣ 0 ∣, −1, ∣ 2 ∣, −2

Chapter 1

Solving Linear Equations

x+3

2x + 1

3x

Reviewing what you learned in previous grades and lessons

Order the values from least to greatest. (Skills Review Handbook)

24

9 fewer students each year 12 more students each year

Predicted Language Class Enrollment Students enrolled

31.

Spanish

Average rate of change

∣ −32 ∣, 22, −16, −∣ 21 ∣, ∣ −10 ∣

1.1–1.3

What Did You Learn?

Core Vocabulary conjecture, p. 3 rule, p. 3 theorem, p. 3 equation, p. 4 linear equation in one variable, p. 4

solution, p. 4 inverse operations, p. 4 equivalent equations, p. 4 identity, p. 21

Core Concepts Section 1.1 Addition Property of Equality, p. 4 Subtraction Property of Equality, p. 4 Substitution Property of Equality, p. 4 Multiplication Property of Equality, p. 5

Division Property of Equality, p. 5 Four-Step Approach to Problem Solving, p. 6 Common Problem-Solving Strategies, p. 7

Section 1.2 Solving Multi-Step Equations, p. 12

Unit Analysis, p. 15

Section 1.3 Solving Equations with Variables on Both Sides, p. 20

Special Solutions of Linear Equations, p. 21

Mathematical Practices 1.

How did you make sense of the relationships between the quantities in Exercise 46 on page 9?

2.

What is the limitation of the tool you used in Exercises 25–28 on page 16?

3.

What definition did you use in your reasoning in Exercises 35 and 36 on page 24?

Using the Features of Yourr Textbook d Tests ests to Prepare for Quizzes and • Read and understand the core vocabulary and the contents of the Core Concept boxes. • Review the Examples and the Monitoring Progress questions. Use the tutorials at BigIdeasMath.com for additional help. • Review previously completed homework assignments.

25 5

1.1–1.3

Quiz

Solve the equation. Justify each step. Check your solution. (Section 1.1) 1. x + 9 = 7

2. 8.6 = z − 3.8

3. 60 = −12r

4. —34 p = 18

Solve the equation. Check your solution. (Section 1.2) 5. 2m − 3 = 13

6. 5 = 10 − v

7. 5 = 7w + 8w + 2

8. −21a + 28a − 6 = −10.2 1

9. 2k − 3(2k − 3) = 45

10. 68 = —5 (20x + 50) + 2

Solve the equation. (Section 1.3) 11. 3c + 1 = c + 1

12. −8 − 5n = 64 + 3n

13. 2(8q − 5) = 4q

14. 9(y − 4) − 7y = 5(3y − 2)

15. 4(g + 8) = 7 + 4g

16. −4(−5h − 4) = 2(10h + 8)

17. To estimate how many miles you are from a thunderstorm, count the seconds between

when you see lightning and when you hear thunder. Then divide by 5. Write and solve an equation to determine how many seconds you would count for a thunderstorm that is 2 miles away. (Section 1.1) 18. You want to hang three equally-sized travel posters on a wall so that the posters on the ends

are each 3 feet from the end of the wall. You want the spacing between posters to be equal. Write and solve an equation to determine how much space you should leave between the posters. (Section 1.2)

3 ft

2 ft

2 ft

2 ft

3 ft

15 ft

19. You want to paint a piece of pottery at an art studio. The total cost is the cost of the piece

plus an hourly studio fee. There are two studios to choose from. (Section 1.3)

a. After how many hours of painting are the total costs the same at both studios? Justify your answer. b. Studio B increases the hourly studio fee by $2. How does this affect your answer in part (a)? Explain.

26

Chapter 1

Solving Linear Equations

1.4

Solving Absolute Value Equations Essential Question

How can you solve an absolute value equation?

Solving an Absolute Value Equation Algebraically Work with a partner. Consider the absolute value equation

∣ x + 2 ∣ = 3.

MAKING SENSE OF PROBLEMS To be proficient in math, you need to explain to yourself the meaning of a problem and look for entry points to its solution.

a. Describe the values of x + 2 that make the equation true. Use your description to write two linear equations that represent the solutions of the absolute value equation. b. Use the linear equations you wrote in part (a) to find the solutions of the absolute value equation. c. How can you use linear equations to solve an absolute value equation?

Solving an Absolute Value Equation Graphically Work with a partner. Consider the absolute value equation

∣ x + 2 ∣ = 3. a. On a real number line, locate the point for which x + 2 = 0. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1

0

1

2

3

4

5

6

7

8

9 10

b. Locate the points that are 3 units from the point you found in part (a). What do you notice about these points? c. How can you use a number line to solve an absolute value equation?

Solving an Absolute Value Equation Numerically Work with a partner. Consider the absolute value equation

∣ x + 2 ∣ = 3. a. Use a spreadsheet, as shown, to solve the absolute value equation. b. Compare the solutions you found using the spreadsheet with those you found in Explorations 1 and 2. What do you notice? c. How can you use a spreadsheet to solve an absolute value equation?

Communicate Your Answer

1 2 3 4 5 6 7 8 9 10 11

A x -6 -5 -4 -3 -2 -1 0 1 2

B |x + 2| 4

abs(A2 + 2)

4. How can you solve an absolute value equation? 5. What do you like or dislike about the algebraic, graphical, and numerical methods

for solving an absolute value equation? Give reasons for your answers. Section 1.4

Solving Absolute Value Equations

27

1.4 Lesson

What You Will Learn Solve absolute value equations. Solve equations involving two absolute values.

Core Vocabul Vocabulary larry

Identify special solutions of absolute value equations.

absolute value equation, p. 28 extraneous solution, p. 31 Previous absolute value opposite

Solving Absolute Value Equations An absolute value equation is an equation that contains an absolute value expression. You can solve these types of equations by solving two related linear equations.

Core Concept Properties of Absolute Value Let a and b be real numbers. Then the following properties are true. 1. ∣ a ∣ ≥ 0 2. ∣ −a ∣ = ∣ a ∣ 3.

∣ ∣

∣a∣ a 4. — = —, b ≠ 0 b ∣b∣

∣ ab ∣ = ∣ a ∣ ∣ b ∣

Solving Absolute Value Equations To solve ∣ ax + b ∣ = c when c ≥ 0, solve the related linear equations ax + b = c

or

ax + b = − c.

When c < 0, the absolute value equation ∣ ax + b ∣ = c has no solution because absolute value always indicates a number that is not negative.

Solving Absolute Value Equations Solve each equation. Graph the solutions, if possible. a. ∣ x − 4 ∣ = 6

b. ∣ 3x + 1 ∣ = −5

SOLUTION a. Write the two related linear equations for ∣ x − 4 ∣ = 6. Then solve. x−4=6 x = 10

or

x − 4 = −6 x = −2

Write related linear equations. Add 4 to each side.

The solutions are x = 10 and x = −2. −4

−2

0

2

4

6

6

8

10

12

6

Each solution is 6 units from 4. Property of Absolute Value

b. The absolute value of an expression must be greater than or equal to 0. The expression ∣ 3x + 1 ∣ cannot equal −5. So, the equation has no solution.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. Graph the solutions, if possible. 1. ∣ x ∣ = 10

28

Chapter 1

Solving Linear Equations

2. ∣ x − 1 ∣ = 4

3. ∣ 3 + x ∣ = −3

Solving an Absolute Value Equation Solve ∣ 3x + 9 ∣ − 10 = −4.

SOLUTION First isolate the absolute value expression on one side of the equation.

ANOTHER WAY

∣ 3x + 9 ∣ − 10 = −4

Using the product property of absolute value, |ab| = |a| |b|, you could rewrite the equation as

∣ 3x + 9 ∣ = 6

3|x + 3| − 10 = −4 and then solve.

Write the equation. Add 10 to each side.

Now write two related linear equations for ∣ 3x + 9 ∣ = 6. Then solve. 3x + 9 =

6

3x + 9 = −6

or

3x = −3

3x = −15

x = −1

x = −5

Write related linear equations. Subtract 9 from each side. Divide each side by 3.

The solutions are x = −1 and x = −5.

Writing an Absolute Value Equation

REASONING Mathematically proficient students have the ability to decontextualize problem situations.

In a cheerleading competition, the minimum length of a routine is 4 minutes. The maximum length of a routine is 5 minutes. Write an absolute value equation that represents the minimum and maximum lengths.

SOLUTION 1. Understand the Problem You know the minimum and maximum lengths. You are asked to write an absolute value equation that represents these lengths. 2. Make a Plan Consider the minimum and maximum lengths as solutions to an absolute value equation. Use a number line to find the halfway point between the solutions. Then use the halfway point and the distance to each solution to write an absolute value equation. 33. Solve the Problem 4.0

4.1

4.2

4.3

4.4

4.5

4.6

0.5

4.7

4.8

4.9

5.0

0.5

halfway point

distance from halfway point

∣ x − 4.5 ∣ = 0.5 The equation is ∣ x − 4.5 ∣ = 0.5. 4. 4 Look Back To check that your equation is reasonable, substitute the minimum and maximum lengths into the equation and simplify. Minimum

∣ 4 − 4.5 ∣ = 0.5

Maximum



∣ 5 − 4.5 ∣ = 0.5

Monitoring Progress M



Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solutions. S 4. ∣ x − 2 ∣ + 5 = 9

5. 4∣ 2x + 7 ∣ = 16

6. −2∣ 5x − 1 ∣ − 3 = −11

7. For a poetry contest, the minimum length of a poem is 16 lines. The maximum

length is 32 lines. Write an absolute value equation that represents the minimum and maximum lengths. Section 1.4

Solving Absolute Value Equations

29

Solving Equations with Two Absolute Values If the absolute values of two algebraic expressions are equal, then they must either be equal to each other or be opposites of each other.

Core Concept Solving Equations with Two Absolute Values

To solve ∣ ax + b ∣ = ∣ cx + d ∣, solve the related linear equations ax + b = cx + d

or

ax + b = −(cx + d).

Solving Equations with Two Absolute Values Solve (a) ∣ 3x − 4 ∣ = ∣ x ∣ and (b) ∣ 4x − 10 ∣ = 2∣ 3x + 1 ∣.

SOLUTION a. Write the two related linear equations for ∣ 3x − 4 ∣ = ∣ x ∣. Then solve.

Check

3x − 4 =

∣ 3x − 4 ∣ = ∣ x ∣

−x

? ∣ 3(2) − 4 ∣ = ∣2∣

+4



? ∣ 3(1) − 4 ∣ = ∣1∣ ? ∣ −1 ∣ = ∣1∣ 1=1



+x

+x

4x − 4 =

0 +4

+4

0 +4

2x = 4

4x = 4

—=—

4 2

—=—

x=2

x=1

2x 2

∣ 3x − 4 ∣ = ∣ x ∣

3x − 4 = −x

or

−x

2x − 4 =

? ∣2∣ = ∣2∣ 2=2

x

4x 4

4 4

The solutions are x = 2 and x = 1. b. Write the two related linear equations for ∣ 4x − 10 ∣ = 2∣ 3x + 1 ∣. Then solve. 4x − 10 =

2(3x + 1)

4x − 10 =

6x + 2

− 6x

4x − 10 = 2[−(3x + 1)]

or

4x − 10 = 2(−3x − 1)

− 6x

− 2x − 10 = + 10

2

+ 6x

+ 10

−2x = −2x −2

4x − 10 = −6x − 2

+ 6x

10x − 10 = −2 + 10

12 12 −2

—=—

+ 10

10x = 8 10x 10

x = −6

8 10

—=—

x = 0.8 The solutions are x = −6 and x = 0.8.

Monitoring Progress

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Solve the equation. Check your solutions. 8. ∣ x + 8 ∣ = ∣ 2x + 1 ∣

30

Chapter 1

Solving Linear Equations

9. 3∣ x − 4 ∣ = ∣ 2x + 5 ∣

Identifying Special Solutions When you solve an absolute value equation, it is possible for a solution to be extraneous. An extraneous solution is an apparent solution that must be rejected because it does not satisfy the original equation.

Identifying Extraneous Solutions Solve ∣ 2x + 12 ∣ = 4x. Check your solutions.

SOLUTION Write the two related linear equations for ∣ 2x + 12 ∣ = 4x. Then solve.

Check

∣ 2x + 12 ∣ = 4x

2x + 12 = 4x 12 = 2x

? ∣ 2(6) + 12 ∣ = 4(6)



∣ 2x + 12 ∣ = 4x ? ∣ 2(−2) + 12 ∣ = 4(−2)

−2 = x



Write related linear equations. Subtract 2x from each side. Solve for x.

Check the apparent solutions to see if either is extraneous. The solution is x = 6. Reject x = −2 because it is extraneous. When solving equations of the form ∣ ax + b ∣ = ∣ cx + d ∣, it is possible that one of the related linear equations will not have a solution.

? ∣8∣ = −8 8 ≠ −8

2x + 12 = −4x 12 = −6x

6=x

? ∣ 24 ∣ = 24 24 = 24

or

Solving an Equation with Two Absolute Values Solve ∣ x + 5 ∣ = ∣ x + 11 ∣.

SOLUTION By equating the expression x + 5 and the opposite of x + 11, you obtain x + 5 = −(x + 11)

Write related linear equation.

x + 5 = −x − 11

Distributive Property

2x + 5 = −11

Add x to each side.

2x = −16

Subtract 5 from each side.

x = −8.

Divide each side by 2.

However, by equating the expressions x + 5 and x + 11, you obtain

REMEMBER Always check your solutions in the original equation to make sure they are not extraneous.

x + 5 = x + 11 x=x+6 0=6

Write related linear equation. Subtract 5 from each side.

✗

Subtract x from each side.

which is a false statement. So, the original equation has only one solution. The solution is x = −8.

Monitoring Progress

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Solve the equation. Check your solutions. 10. ∣ x + 6 ∣ = 2x

11.

∣ 3x − 2 ∣ = x

12. ∣ 2 + x ∣ = ∣ x − 8 ∣

13.

∣ 5x − 2 ∣ = ∣ 5x + 4 ∣

Section 1.4

Solving Absolute Value Equations

31

Exercises

1.4

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY What is an extraneous solution? 2. WRITING Without calculating, how do you know that the equation ∣ 4x − 7 ∣ = −1 has no solution?

Monitoring Progress and Modeling with Mathematics In Exercises 3−10, simplify the expression. 3.

∣ −9 ∣

4. −∣ 15 ∣

5.

∣ 14 ∣ − ∣ −14 ∣

6.

∣ −3 ∣ + ∣ 3 ∣

⋅

8.

∣ −0.8 ⋅ 10 ∣

7. −∣ −5 (−7) ∣ 9.

∣ −327 ∣

10.

—

26. WRITING EQUATIONS The shoulder heights of the

shortest and tallest miniature poodles are shown.

∣ −−124 ∣ —

15 in.

10 in.

In Exercises 11−24, solve the equation. Graph the solution(s), if possible. (See Examples 1 and 2.) 11.

∣w∣ = 6

12.

∣ r ∣ = −2

13.

∣ y ∣ = −18

14.

∣ x ∣ = 13

15.

∣m + 3∣ = 7

16.

∣ q − 8 ∣ = 14

18.

∣ ∣

17. 19.

∣ −3d ∣ = 15 ∣ 4b − 5 ∣ = 19

20.

t — =6 2

∣x − 1∣ + 5 = 2

21. −4∣ 8 − 5n ∣ = 13

∣

2 3

∣

USING STRUCTURE In Exercises 27−30, match the

absolute value equation with its graph without solving the equation. 27.

∣x + 2∣ = 4

28.

∣x − 4∣ = 2

29.

∣x − 2∣ = 4

30.

∣x + 4∣ = 2

A.

22. −3 1 − — v = −9

∣ 14

a. Represent these two heights on a number line. b. Write an absolute value equation that represents these heights.

−10

∣

−4

−8

−6

−4

−2

4

25. WRITING EQUATIONS The minimum distance from

b. Write an absolute value equation that represents the minimum and maximum distances.

Solving Linear Equations

0

2

0

2

4

6

8

8

10

4

C. −4

Earth to the Sun is 91.4 million miles. The maximum distance is 94.5 million miles. (See Example 3.) a. Represent these two distances on a number line.

−2

2

B.

24. 9∣ 4p + 2 ∣ + 8 = 35

Chapter 1

−6

2

23. 3 = −2 — s − 5 + 3

32

−8

−2

0

2

4

4

4

D. −2

0

2

4

2

6

2

In Exercises 31−34, write an absolute value equation that has the given solutions. 31. x = 8 and x = 18

32. x = −6 and x = 10

33. x = 2 and x = 9

34. x = −10 and x = −5

48. MODELING WITH MATHEMATICS The recommended

weight of a soccer ball is 430 grams. The actual weight is allowed to vary by up to 20 grams. a. Write and solve an absolute value equation to find the minimum and maximum acceptable soccer ball weights. b. A soccer ball weighs 423 grams. Due to wear and tear, the weight of the ball decreases by 16 grams. Is the weight acceptable? Explain.

In Exercises 35−44, solve the equation. Check your solutions. (See Examples 4, 5, and 6.) 35.

∣ 4n − 15 ∣ = ∣ n ∣

36.

∣ 2c + 8 ∣ = ∣ 10c ∣

37.

∣ 2b − 9 ∣ = ∣ b − 6 ∣

38.

∣ 3k − 2 ∣ = 2∣ k + 2 ∣

39. 4∣ p − 3 ∣ = ∣ 2p + 8 ∣

40. 2∣ 4w − 1 ∣ = 3∣ 4w + 2 ∣

41.

∣ 3h + 1 ∣ = 7h

42.

∣ 6a − 5 ∣ = 4a

43.

∣f − 6∣ = ∣f + 8∣

44.

∣ 3x − 4 ∣ = ∣ 3x − 5 ∣

ERROR ANALYSIS In Exercises 49 and 50, describe and

correct the error in solving the equation.



49.

45. MODELING WITH MATHEMATICS Starting from

300 feet away, a car drives toward you. It then passes by you at a speed of 48 feet per second. The distance d (in feet) of the car from you after t seconds is given by the equation d = ∣ 300 − 48t ∣. At what times is the car 60 feet from you? absolute value equation ∣ 3x + 8 ∣ − 9 = −5 has no solution because the constant on the right side of the equation is negative. Is your friend correct? Explain.

2x − 1 = −(−9)

or

2x = 10

x = −4

x=5

The solutions are x = −4 and x = 5.



∣ 5x + 8 ∣ = x 5x + 8 = x

5x + 8 = −x

or

4x + 8 = 0

47. MODELING WITH MATHEMATICS You randomly

6x + 8 = 0

4x = −8

survey students about year-round school. The results are shown in the graph.

6x = −8

4 x = −— 3 4 The solutions are x = −2 and x = −—. 3 x = −2

Year-Round School 68%

Oppose

51. ANALYZING EQUATIONS Without solving completely,

32% Error: ±5%

Favor 0%

2x − 1 = −9 2x = −8

50.

46. MAKING AN ARGUMENT Your friend says that the

∣ 2x − 1 ∣ = −9

20%

40%

60%

place each equation into one of the three categories.

80%

The error given in the graph means that the actual percent could be 5% more or 5% less than the percent reported by the survey. a. Write and solve an absolute value equation to find the least and greatest percents of students who could be in favor of year-round school. b. A classmate claims that —13 of the student body is actually in favor of year-round school. Does this conflict with the survey data? Explain.

Section 1.4

No solution

One solution

Two solutions

∣x − 2∣ + 6 = 0

∣x + 3∣ − 1 = 0

∣x + 8∣ + 2 = 7

∣x − 1∣ + 4 = 4

∣ x − 6 ∣ − 5 = −9

∣ x + 5 ∣ − 8 = −8

Solving Absolute Value Equations

33

52. USING STRUCTURE Fill in the equation

∣x −

60. HOW DO YOU SEE IT? The circle graph shows the

∣=

results of a survey of registered voters the day of an election.

with a, b, c, or d so that the equation is graphed correctly. a

b

d

Which Party’s Candidate Will Get Your Vote?

c

d

Other: 4% Green: 2% Libertarian: 5%

ABSTRACT REASONING In Exercises 53−56, complete

the statement with always, sometimes, or never. Explain your reasoning. 53. If x 2 = a 2, then ∣ x ∣ is ________ equal to ∣ a ∣.

Democratic: 47% Republican: 42%

54. If a and b are real numbers, then ∣ a − b ∣ is

_________ equal to ∣ b − a ∣.

Error: ±2%

55. For any real number p, the equation ∣ x − 4 ∣ = p will

The error given in the graph means that the actual percent could be 2% more or 2% less than the percent reported by the survey.

________ have two solutions. 56. For any real number p, the equation ∣ x − p ∣ = 4 will

________ have two solutions.

a. What are the minimum and maximum percents of voters who could vote Republican? Green? b. How can you use absolute value equations to represent your answers in part (a)? c. One candidate receives 44% of the vote. Which party does the candidate belong to? Explain.

57. WRITING Explain why absolute value equations can

have no solution, one solution, or two solutions. Give an example of each case. 58. THOUGHT PROVOKING Describe a real-life situation

that can be modeled by an absolute value equation with the solutions x = 62 and x = 72.

61. ABSTRACT REASONING How many solutions does

the equation a∣ x + b ∣ + c = d have when a > 0 and c = d? when a < 0 and c > d? Explain your reasoning.

59. CRITICAL THINKING Solve the equation shown.

Explain how you found your solution(s). 8∣ x + 2 ∣ − 6 = 5∣ x + 2 ∣ + 3

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Identify the property of equality that makes Equation 1 and Equation 2 equivalent. 62.

Equation 1

3x + 8 = x − 1

Equation 2

3x + 9 = x

Use a geometric formula to solve the problem.

63.

Equation 1

4y = 28

Equation 2

y=7

(Section 1.1)

(Skills Review Handbook)

64. A square has an area of 81 square meters. Find the side length. 65. A circle has an area of 36π square inches. Find the radius. 66. A triangle has a height of 8 feet and an area of 48 square feet. Find the base. 67. A rectangle has a width of 4 centimeters and a perimeter of 26 centimeters. Find the length.

34

Chapter 1

Solving Linear Equations

1.5

Rewriting Equations and Formulas Essential Question

How can you use a formula for one measurement to write a formula for a different measurement? Using an Area Formula Work with a partner. a. Write a formula for the area A of a parallelogram.

REASONING QUANTITATIVELY To be proficient in math, you need to consider the given units. For instance, in Exploration 1, the area A is given in square inches and the height h is given in inches. A unit analysis shows that the units for the base b are also inches, which makes sense.

A = 30 in.2 h = 5 in.

b. Substitute the given values into the formula. Then solve the equation for b. Justify each step.

b

c. Solve the formula in part (a) for b without first substituting values into the formula. Justify each step. d. Compare how you solved the equations in parts (b) and (c). How are the processes similar? How are they different?

Using Area, Circumference, and Volume Formulas Work with a partner. Write the indicated formula for each figure. Then write a new formula by solving for the variable whose value is not given. Use the new formula to find the value of the variable. a. Area A of a trapezoid

b. Circumference C of a circle

b1 = 8 cm h

C = 24π ft r

A = 63 cm2

b2 = 10 cm

c. Volume V of a rectangular prism

d. Volume V of a cone

V = 75 yd3

V = 24π m3 h

B = 12π m2

h B = 15 yd2

Communicate Your Answer 3. How can you use a formula for one measurement to write a formula for a

different measurement? Give an example that is different from those given in Explorations 1 and 2. Section 1.5

Rewriting Equations and Formulas

35

1.5 Lesson

What You Will Learn Rewrite literal equations.

Core Vocabul Vocabulary larry

Rewrite and use formulas for area. Rewrite and use other common formulas.

literal equation, p. 36 formula, p. 37

Rewriting Literal Equations

Previous surface area

An equation that has two or more variables is called a literal equation. To rewrite a literal equation, solve for one variable in terms of the other variable(s).

Rewriting a Literal Equation Solve the literal equation 3y + 4x = 9 for y.

SOLUTION 3y + 4x = 9

Write the equation.

3y + 4x − 4x = 9 − 4x

Subtract 4x from each side.

3y = 9 − 4x 3y 3

Simplify.

9 − 4x 3

—=—

Divide each side by 3.

4 y = 3 − —x 3

Simplify.

4 The rewritten literal equation is y = 3 − — x. 3

Rewriting a Literal Equation Solve the literal equation y = 3x + 5xz for x.

SOLUTION y = 3x + 5xz

Write the equation.

y = x(3 + 5z)

Distributive Property

x(3 + 5z) y 3 + 5z 3 + 5z y —=x 3 + 5z

—=—

REMEMBER Division by 0 is undefined.

Divide each side by 3 + 5z. Simplify.

y The rewritten literal equation is x = —. 3 + 5z 3

In Example 2, you must assume that z ≠ −—5 in order to divide by 3 + 5z. In general, if you have to divide by a variable or variable expression when solving a literal equation, you should assume that the variable or variable expression does not equal 0.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the literal equation for y. 1. 3y − x = 9

2. 2x − 2y = 5

3. 20 = 8x + 4y

Solve the literal equation for x. 4. y = 5x − 4x

36

Chapter 1

Solving Linear Equations

5. 2x + kx = m

6. 3 + 5x − kx = y

Rewriting and Using Formulas for Area A formula shows how one variable is related to one or more other variables. A formula is a type of literal equation.

Rewriting a Formula for Surface Area The formula for the surface area S of a rectangular prism is S = 2ℓw + 2ℓh + 2wh. Solve the formula for the lengthℓ.

SOLUTION h w

S = 2ℓw + 2ℓh + 2wh

Write the equation.

S − 2wh = 2ℓw + 2ℓh + 2wh − 2wh

Subtract 2wh from each side.

S − 2wh = 2ℓw + 2ℓh

Simplify.

S − 2wh =ℓ(2w + 2h)

Distributive Property

ℓ(2w + 2h) 2w + 2h

S − 2wh 2w + 2h

— = ——

Divide each side by 2w + 2h.

S − 2wh 2w + 2h

Simplify.

— =ℓ

S − 2wh When you solve the formula forℓ, you obtainℓ= —. 2w + 2h

Using a Formula for Area You own a rectangular lot that is 500 feet deep. It has an area of 100,000 square feet. To pay for a new water system, you are assessed $5.50 per foot of lot frontage. a. Find the frontage of your lot. b. How much are you assessed for the new water system?

SOLUTION a. In the formula for the area of a rectangle, let the width w represent the lot frontage.

w

frontage

500 ft

A =ℓw

Write the formula for area of a rectangle.

A ℓ

Divide each side byℓ to solve for w.

—=w

100,000 500

Substitute 100,000 for A and 500 forℓ.

—=w

200 = w

Simplify.

The frontage of your lot is 200 feet.

⋅

$5.50 b. Each foot of frontage costs $5.50, and — 200 ft = $1100. 1 ft So, your total assessment is $1100.

Monitoring Progress

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Solve the formula for the indicated variable. 1

7. Area of a triangle: A = —2 bh; Solve for h. 8. Surface area of a cone: S = πr 2 + π rℓ; Solve for ℓ.

Section 1.5

Rewriting Equations and Formulas

37

Rewriting and Using Other Common Formulas

Core Concept Common Formulas F = degrees Fahrenheit, C = degrees Celsius

Temperature

5 C = — (F − 32) 9 I = interest, P = principal, r = annual interest rate (decimal form), t = time (years)

Simple Interest

I = Prt d = distance traveled, r = rate, t = time d = rt

Distance

Rewriting the Formula for Temperature Solve the temperature formula for F.

SOLUTION 5 C = —(F − 32) 9 9 5

— C = F − 32

Write the temperature formula. 9 Multiply each side by —. 5

— C + 32 = F − 32 + 32

9 5

Add 32 to each side.

9 5

Simplify.

— C + 32 = F

The rewritten formula is F = —95 C + 32.

Using the Formula for Temperature Which has the greater surface temperature: Mercury or Venus?

SOLUTION Convert the Celsius temperature of Mercury to degrees Fahrenheit. 9 F = — C + 32 5

Mercury 427°C

Venus 864°F

Write the rewritten formula from Example 5.

9 = —(427) + 32 5

Substitute 427 for C.

= 800.6

Simplify.

Because 864°F is greater than 800.6°F, Venus has the greater surface temperature.

Monitoring Progress

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9. A fever is generally considered to be a body temperature greater than 100°F. Your

friend has a temperature of 37°C. Does your friend have a fever? 38

Chapter 1

Solving Linear Equations

Using the Formula for Simple Interest You deposit $5000 in an account that earns simple interest. After 6 months, the account earns $162.50 in interest. What is the annual interest rate?

COMMON ERROR The unit of t is years. Be sure to convert months to years.

⋅

1 yr — 6 mo = 0.5 yr 12 mo

SOLUTION To find the annual interest rate, solve the simple interest formula for r. I = Prt I Pt

—=r

Write the simple interest formula. Divide each side by Pt to solve for r.

162.50 (5000)(0.5)

—=r

Substitute 162.50 for I, 5000 for P, and 0.5 for t.

0.065 = r

Simplify.

The annual interest rate is 0.065, or 6.5%.

Solving a Real-Life Problem A truck driver averages 60 miles per hour while delivering freight to a customer. On the return trip, the driver averages 50 miles per hour due to construction. The total driving time is 6.6 hours. How long does each trip take?

SOLUTION Step 1

Rewrite the Distance Formula to write expressions that represent the two trip d d times. Solving the formula d = rt for t, you obtain t = —. So, — represents r 60 d the delivery time, and — represents the return trip time. 50

Step 2

Use these expressions and the total driving time to write and solve an equation to find the distance one way. d 60

d 50

— + — = 6.6

The sum of the two trip times is 6.6 hours.

— = 6.6

11d 300

Add the left side using the LCD.

11d = 1980

Multiply each side by 300 and simplify.

d = 180

Divide each side by 11 and simplify.

The distance one way is 180 miles. Step 3

Use the expressions from Step 1 to find the two trip times.

60 mi So, the delivery takes 180 mi ÷ — = 3 hours, and the return trip takes 1h 50 mi 180 mi ÷ — = 3.6 hours. 1h

Monitoring Progress

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10. How much money must you deposit in a simple interest account to earn $500 in

interest in 5 years at 4% annual interest? 11. A truck driver averages 60 miles per hour while delivering freight and 45 miles

per hour on the return trip. The total driving time is 7 hours. How long does each trip take? Section 1.5

Rewriting Equations and Formulas

39

1.5

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check π 5

1. VOCABULARY Is 9r + 16 = — a literal equation? Explain. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

Solve 3x + 6y = 24 for x.

Solve 24 − 3x = 6y for x.

Solve 6y = 24 − 3x for y in terms of x.

Solve 24 − 6y = 3x for x in terms of y.

Monitoring Progress and Modeling with Mathematics In Exercises 3–12, solve the literal equation for y. (See Example 1.) 3. y − 3x = 13

4. 2x + y = 7

5. 2y − 18x = −26

6. 20x + 5y = 15

7. 9x − y = 45

8. 6x − 3y = −6

9. 4x − 5 = 7 + 4y 1

11. 2 + —6 y = 3x + 4

24. MODELING WITH MATHEMATICS The penny

size of a nail indicates the length of the nail. The penny size d is given by the literal equation d = 4n − 2, where n is the length (in inches) of the nail. a. Solve the equation for n. b. Use the equation from part (a) to find the lengths of nails with the following penny sizes: 3, 6, and 10.

10. 16x + 9 = 9y − 2x 1

12. 11 − —2 y = 3 + 6x

In Exercises 13–22, solve the literal equation for x. (See Example 2.) 13. y = 4x + 8x

14. m = 10x − x

15. a = 2x + 6xz

16. y = 3bx − 7x

17. y = 4x + rx + 6

18. z = 8 + 6x − px

19. sx + tx = r

20. a = bx + cx + d

21. 12 − 5x − 4kx = y

22. x − 9 + 2wx = y

ERROR ANALYSIS In Exercises 25 and 26, describe and

correct the error in solving the equation for x. 25.

✗

C (in dollars) to participate in a ski club is given by the literal equation C = 85x + 60, where x is the number of ski trips you take.

12 − 2x = −2(y − x) −2x = −2(y − x) − 12 x = (y − x) + 6

26.

✗

10 = ax − 3b 10 = x(a − 3b) 10 a − 3b

—=x

23. MODELING WITH MATHEMATICS The total cost

In Exercises 27–30, solve the formula for the indicated variable. (See Examples 3 and 5.)

a. Solve the equation for x.

27. Profit: P = R − C; Solve for C.

b. How many ski trips do you take if you spend a total of $315? $485?

28. Surface area of a cylinder: S = 2π r 2 + 2π rh;

Solve for h. 1

29. Area of a trapezoid: A = —2 h(b1 + b2); Solve for b2.

v −v t

1 0; 30. Average acceleration of an object: a = —

40

Chapter 1

n

Solving Linear Equations

Solve for v1.

31. REWRITING A FORMULA A common statistic used in

35. PROBLEM SOLVING You deposit $2000 in an account

professional football is the quarterback rating. This rating is made up of four major factors. One factor is the completion rating given by the formula

that earns simple interest at an annual rate of 4%. How long must you leave the money in the account to earn $500 in interest? (See Example 7.)

(

C R = 5 — − 0.3 A

)

36. PROBLEM SOLVING A flight averages 460 miles per

hour. The return flight averages 500 miles per hour due to a tailwind. The total flying time is 4.8 hours. How long is each flight? Explain. (See Example 8.)

where C is the number of completed passes and A is the number of attempted passes. Solve the formula for C. 32. REWRITING A FORMULA Newton’s law of gravitation

is given by the formula

( )

m1m2 F=G — d2

where F is the force between two objects of masses m1 and m2, G is the gravitational constant, and d is the distance between the two objects. Solve the formula for m1.

37. USING STRUCTURE An athletic facility is building an

indoor track. The track is composed of a rectangle and two semicircles, as shown. x

33. MODELING WITH MATHEMATICS The sale price

S (in dollars) of an item is given by the formula S = L − rL, where L is the list price (in dollars) and r is the discount rate (in decimal form). (See Examples 4 and 6.)

r

r

a. Solve the formula for r. a. Write a formula for the perimeter of the indoor track.

b. The list price of the shirt is $30. What is the discount rate?

b. Solve the formula for x.

Sale price:

$18

34. MODELING WITH MATHEMATICS The density d of a

m substance is given by the formula d = —, where m is V its mass and V is its volume.

c. The perimeter of the track is 660 feet, and r is 50 feet. Find x. Round your answer to the nearest foot. 38. MODELING WITH MATHEMATICS The distance

d (in miles) you travel in a car is given by the two equations shown, where t is the time (in hours) and g is the number of gallons of gasoline the car uses.

Pyrite Density: 5.01g/cm3

Volume: 1.2 cm3 d = 55t d = 20g

a. Write an equation that relates g and t. b. Solve the equation for g. c. You travel for 6 hours. How many gallons of gasoline does the car use? How far do you travel? Explain.

a. Solve the formula for m. b. Find the mass of the pyrite sample.

Section 1.5

Rewriting Equations and Formulas

41

39. MODELING WITH MATHEMATICS One type of stone

41. MAKING AN ARGUMENT Your friend claims that

formation found in Carlsbad Caverns in New Mexico is called a column. This cylindrical stone formation connects to the ceiling and the floor of a cave.

Thermometer A displays a greater temperature than Thermometer B. Is your friend correct? Explain yyour reasoning. g 100 90 80 70 60 50 40 30 20 10 0 −10

column

°F

Thermometer A Thermometer B

stalagmite

42. THOUGHT PROVOKING Give a possible value for h.

a. Rewrite the formula for the circumference of a circle, so that you can easily calculate the radius of a column given its circumference.

Justify your answer. Draw and label the figure using your chosen value of h.

b. What is the radius (to the nearest tenth of a foot) of a column that has a circumference of 7 feet? 8 feet? 9 feet?

A = 40 cm2

h

c. Explain how you can find the area of a cross section of a column when you know its circumference.

8 cm

40. HOW DO YOU SEE IT? The rectangular prism shown

MATHEMATICAL CONNECTIONS In Exercises 43 and 44,

has bases with equal side lengths.

write a formula for the area of the regular polygon. Solve the formula for the height h. 43.

44. center

center b

h

b

h

b

b

a. Use the figure to write a formula for the surface area S of the rectangular prism.

REASONING In Exercises 45 and 46, solve the literal

equation for a.

b. Your teacher asks you to rewrite the formula by solving for one of the side lengths, b orℓ. Which side length would you choose? Explain your reasoning.

a+b+c ab

45. x = —

( a ab− b )

46. y = x —

Maintaining Mathematical Proficiency Evaluate the expression.

(Skills Review Handbook)

⋅

47. 15 − 5 + 52

48. 18 2 − 42 ÷ 8

⋅

49. 33 + 12 ÷ 3 5

Solve the equation. Graph the solutions, if possible. 51.

42

∣x − 3∣ + 4 = 9 Chapter 1

Reviewing what you learned in previous grades and lessons

52.

∣ 3y − 12 ∣ − 7 = 2

Solving Linear Equations

50. 25(5 − 6) + 9 ÷ 3

(Section 1.4)

53. 2∣ 2r + 4 ∣ = −16

54. −4∣ s + 9 ∣ = −24

1.4–1.5

What Did You Learn?

Core Vocabulary absolute value equation, p. 28 extraneous solution, p. 31

literal equation, p. 36 formula, p. 37

Core Concepts Section 1.4 Properties of Absolute Value, p. 28 Solving Absolute Value Equations, p. 28 Solving Equations with Two Absolute Values, p. 30 Special Solutions of Absolute Value Equations, p. 31

Section 1.5 Rewriting Literal Equations, p. 36 Common Formulas, p. 38

Mathematical Practices 1.

How did you decide whether your friend’s argument in Exercise 46 on page 33 made sense?

2.

How did you use the structure of the equation in Exercise 59 on page 34 to rewrite the equation?

3.

What entry points did you use to answer Exercises 43 and 44 on page 42?

Performance Task:

Dead Reckoning Have you ever wondered how sailors navigated the oceans before the Global Positioning System (GPS)? One method sailors used is called dead reckoning. How does dead reckoning use mathematics to track locations? Could you use this method today? To explore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.

43

1

Chapter Review 1.1

Dynamic Solutions available at BigIdeasMath.com

Solving Simple Equations (pp. 3–10)

a. Solve x − 5 = −9. Justify each step. x − 5 = −9 Addition Property of Equality

+5

Write the equation.

+5

Add 5 to each side.

x = −4

Simplify.

The solution is x = −4. b. Solve 4x = 12. Justify each step. 4x = 12

Write the equation.

4x 4

Divide each side by 4.

12 4

—=—

Division Property of Equality

x=3

Simplify.

The solution is x = 3. Solve the equation. Justify each step. Check your solution. 1. z + 3 = − 6

3.2 3 .2 2 1.2

n 5

3. −— = −2

2. 2.6 = −0.2t

Solving Multi-Step Equations (pp. 11–18)

Solve −6x + 23 + 2x = 15. −6x + 23 + 2x = 15

Write the equation.

−4x + 23 = 15

Combine like terms.

−4x = −8

Subtract 23 from each side.

x=2

Divide each side by −4.

The solution is x = 2. Solve the equation. Check your solution. 4. 3y + 11 = −16

5.

6=1−b

6. n + 5n + 7 = 43

7. −4(2z + 6) − 12 = 4

8. —32 (x − 2) − 5 = 19

1

7

9. 6 = —5 w + —5 w − 4

Find the value of x. Then find the angle measures of the polygon. 10.

110° 5x° 2x° Sum of angle measures: 180°

44

Chapter 1

Solving Linear Equations

11.

(x − 30)° (x − 30)°

x°

x°

(x − 30)°

Sum of angle measures: 540°

1.3

Solving Equations with Variables on Both Sides

(pp. 19–24)

Solve 2( y − 4) = −4( y + 8). 2( y − 4) = −4( y + 8)

Write the equation.

2y − 8 = −4y − 32

Distributive Property

6y − 8 = −32

Add 4y to each side.

6y = −24

Add 8 to each side.

y = −4

Divide each side by 6.

The solution is y = −4. Solve the equation. 12. 3n − 3 = 4n + 1

1.4

1

13. 5(1 + x) = 5x + 5

14. 3(n + 4) = —2 (6n + 4)

Solving Absolute Value Equations (pp. 27–34)

a. Solve ∣ x − 5 ∣ = 3. x−5= +5

3

x − 5 = −3

or

+5

+5

x=8

+5

x=2

Write related linear equations. Add 5 to each side. Simplify.

The solutions are x = 8 and x = 2. Check

b. Solve ∣ 2x + 6 ∣ = 4x. Check your solutions. 2x + 6 = 4x −2x

−2x

or

2x + 6 = −4x −2x

−2x

Write related linear equations. Subtract 2x from each side.

6 = 2x

6 = −6x

6 2x —=— 2 2

6 −6x —=— −6 −6

Solve for x.

3=x

−1 = x

Simplify.

Simplify.

∣ 2x + 6 ∣ = 4x

? ∣ 2(3) + 6 ∣ = 4(3) ? ∣ 12 ∣ = 12 12 = 12

✓

∣ 2x + 6 ∣ = 4x

? ∣ 2(−1) + 6 ∣ = 4(−1) ? ∣ 4 ∣ = −4

Check the apparent solutions to see if either is extraneous. The solution is x = 3. Reject x = −1 because it is extraneous.

∕ −4 4=

✗

Solve the equation. Check your solutions. 15.

∣ y + 3 ∣ = 17

16. −2∣ 5w − 7 ∣ + 9 = − 7

17. ∣ x − 2 ∣ = ∣ 4 + x ∣

18. The minimum sustained wind speed of a Category 1 hurricane is 74 miles per hour. The maximum

sustained wind speed is 95 miles per hour. Write an absolute value equation that represents the minimum and maximum speeds.

Chapter 1

Chapter Review

45

1.5

Rewriting Equations and Formulas (pp. 35–42)

a. The slope-intercept form of a linear equation is y = mx + b. Solve the equation for m. y = mx + b

Write the equation.

y − b = mx + b − b

Subtract b from each side.

y − b = mx

Simplify.

y − b mx x x y−b —=m x

—=—

Divide each side by x. Simplify.

y−b When you solve the equation for m, you obtain m = —. x

b. The formula for the surface area S of a cylinder is S = 2𝛑 r 2 + 2𝛑 rh. Solve the formula for the height h. 2π r 2 + 2πrh

S= − 2πr 2

− 2πr 2

Write the equation. Subtract 2πr 2 from each side.

S − 2πr 2 = 2πrh

Simplify.

—=—

S − 2πr 2 2πr

Divide each side by 2πr.

S − 2πr 2 2πr

Simplify.

2πrh 2πr

—=h

S − 2π r 2 When you solve the formula for h, you obtain h = —. 2πr Solve the literal equation for y. 19. 2x − 4y = 20

20. 8x − 3 = 5 + 4y

22. The volume V of a pyramid is given by the formula V =

base and h is the height. a. Solve the formula for h. b. Find the height h of the pyramid.

21. a = 9y + 3yx 1 —3 Bh,

where B is the area of the

V = 216 cm3

B = 36 cm2 9

23. The formula F = —5 (K − 273.15) + 32 converts a temperature from kelvin K to degrees

Fahrenheit F. a. Solve the formula for K. b. Convert 180°F to kelvin K. Round your answer to the nearest hundredth.

46

Chapter 1

Solving Linear Equations

1

Chapter Test

Solve the equation. Justify each step. Check your solution. 2. —23 x + 5 = 3

1. x − 7 = 15

3. 11x + 1 = −1 + x

Solve the equation. 4. 2∣ x − 3 ∣ − 5 = 7

5.

∣ 2x − 19 ∣ = 4x + 1

6. −2 + 5x − 7 = 3x − 9 + 2x

7. 3(x + 4) − 1 = −7

8.

∣ 20 + 2x ∣ = ∣ 4x + 4 ∣

9. —13 (6x + 12) − 2(x − 7) = 19

Describe the values of c for which the equation has no solution. Explain your reasoning. 10. 3x − 5 = 3x − c

11.

∣x − 7∣ = c

12. A safety regulation states that the minimum height of a handrail is 30 inches. The

maximum height is 38 inches. Write an absolute value equation that represents the minimum and maximum heights. 13. The perimeter P (in yards) of a soccer field is represented by the formula P = 2ℓ+ 2w,

whereℓ is the length (in yards) and w is the width (in yards). a. Solve the formula for w. b. Find the width of the field. c. About what percent of the field is inside the circle?

P = 330 yd

10 yd = 100 yd

14. Your car needs new brakes. You call a dealership and a local

mechanic for prices. Cost of parts

Labor cost per hour

Dealership

$24

$99

Local Mechanic

$45

$89

a. After how many hours are the total costs the same at both places? Justify your answer. b. When do the repairs cost less at the dealership? at the local mechanic? Explain. 15. Consider the equation ∣ 4x + 20 ∣ = 6x. Without calculating, how do you know that x = −2 is an

extraneous solution? 16. Your friend was solving the equation shown and was confused by the result

“−8 = −8.” Explain what this result means. 4(y − 2) − 2y = 6y − 8 − 4y 4y − 8 − 2y = 6y − 8 − 4y 2y − 8 = 2y − 8 −8 = −8 Chapter 1

Chapter Test

47

1

Cumulative Assessment

1. A mountain biking park has 48 trails, 37.5% of which are beginner trails. The rest are

divided evenly between intermediate and expert trails. How many of each kind of trail are there? A 12 beginner, 18 intermediate, 18 expert ○ B 18 beginner, 15 intermediate, 15 expert ○ C 18 beginner, 12 intermediate, 18 expert ○ D 30 beginner, 9 intermediate, 9 expert ○ 2. Which of the equations are equivalent to cx − a = b?

cx − a + b = 2b

0 = cx − a + b

b 2cx − 2a = — 2

b x−a=— c

a+b x=— c

b + a = cx

3. Let N represent the number of solutions of the equation 3(x − a) = 3x − 6. Complete

each statement with the symbol , or =. a. When a = 3, N ____ 1. b. When a = −3, N ____ 1. c. When a = 2, N ____ 1. d. When a = −2, N ____ 1. e. When a = x, N ____ 1. f. When a = −x, N ____ 1.

4. You are painting your dining room white and your living room blue. You spend

$132 on 5 cans of paint. The white paint costs $24 per can, and the blue paint costs $28 per can. a. Use the numbers and symbols to write an equation that represents how many cans of each color you bought. x

132

5

24

28

=

(

)

+

−

×

÷

b. How much would you have saved by switching the colors of the dining room and living room? Explain.

48

Chapter 1

Solving Linear Equations

5. Which of the equations are equivalent?

6x + 6 = −14

8x + 6 = −2x − 14

5x + 3 = −7

7x + 3 = 2x − 13

6. The perimeter of the triangle is 13 inches. What is the length of the shortest

side? A 2 in. ○

(x − 5) in.

B 3 in. ○

x 2

C 4 in. ○

in.

6 in.

D 8 in. ○

7. You pay $45 per month for cable TV. Your friend buys a satellite TV receiver for $99 and

pays $36 per month for satellite TV. Your friend claims that the expenses for a year of satellite TV are less than the expenses for a year of cable. a. Write and solve an equation to determine when you and your friend will have paid the same amount for TV services. b. Is your friend correct? Explain. 8. Place each equation into one of the four categories. No solution

One solution

Two solutions

Infinitely many solutions

∣ 8x + 3 ∣ = 0

−6 = 5x − 9

3x − 12 = 3(x − 4) + 1

−2x + 4 = 2x + 4

0 = ∣ x + 13 ∣ + 2 9 = 3∣ 2x − 11 ∣

−4(x + 4) = −4x − 16

12x − 2x = 10x − 8

7 − 2x = 3 − 2(x − 2)

9. A car travels 1000 feet in 12.5 seconds. Which of the expressions do not represent the

average speed of the car? second 80 — feet

feet 80 — second

80 feet second

—

—

second 80 feet

Chapter 1

Cumulative Assessment

49

2 Solving Linear Inequalities 2.1 2.2 2.3 2.4 2.5 2.6

Writing and Graphing Inequalities Solving Inequalities Using Addition or Subtraction Solving Inequalities Using Multiplication or Division Solving Multi-Step Inequalities Solving Compound Inequalities Solving Absolute Value Inequalities

Camel Physiology (p. 91)

Mountain Plant Life (p. 85)

SEE the Big Idea

Digital Camera (p. 70)

Microwave Electricity (p. 64)

Natural Arch (p. 59)

Maintaining Mathematical Proficiency Graphing Numbers on a Number Line

(6.NS.C.6c)

Example 1 Graph each number.

Example 2

a. a.

3 3

b.

−5

b.

−5

−5

−3

−2

−1

0

1

2

3

4

5

−2

−1

0

1

2

3

4

5

Graph each number. −5

−4

∣4∣

a. Example 2

−4

−3

Graph each ∣ number. b. ∣ −2 a. ∣ 4 ∣

The absolute value of a positive number is positive.

Graph the number. 1. 6

2.

4. 2 + ∣ −2 ∣

−5

−4

−3

−2

∣2∣

−1 1 −0∣ −4 1 ∣ 5.

∣ −2 ∣

b.

2

3

4

5

∣ −1 ∣

6. −5 + ∣ 3 ∣

The absolute value of a negative number is positive.

Comparing Real Numbers Example 3

3.

(6.NS.C.7a)

Complete the statement −1 −5 −4 −3 −2 −1

0

−5 with , or 4=. 1 2 3

5

Graph the number.

−1is to the right of −5. So, −1 > −5. 2. ∣ 2 ∣

1. 6

3.

4. 2 + ∣ −2 ∣ 5. 1 − ∣ −4 ∣ Example 4 Evaluate 15 ÷ (−3).

∣ −1 ∣

6. −5 + ∣ 3 ∣

Comparing Real Numbers Example 3

Complete the statement −1 −5 with , or =. 15 ÷ (−3) = −5 Graph –5.

Multiply or divide. 10. −3 (8)

−7

13. −24 ÷ (−6)

−6

−5

−4

Graph –1.

−3

−2

−1

0

⋅ −1 > −5. −1 is to the right of −5. So, 11. −7 (−9)

14. −16 ÷ 2

Complete the statement with 16. 6 8 17. 36, 6 or =.

⋅

1

2

3

12. 4

⋅ (−7)

15. 12 ÷ (−3) 18. −3(−4)

7. 2 9 8. −6 5 9. −12 −4 19. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers,

∣ −8 ∣ integers. ∣ 8 ∣ Give an example12.of each. 10. (c) −7multiplying −13 integers, and (d) 11.dividing −10

∣ −18 ∣

13. ABSTRACT REASONING A number a is to the left of a number b on the number line.

How do the numbers −a and −b compare?

Dynamic Solutions available at BigIdeasMath.com

51

Mathematical Practices

Mathematically proficient students use technology tools to explore concepts.

Using a Graphing Calculator

Core Concept Solving an Inequality in One Variable You can use a graphing calculator to solve an inequality. 1.

Enter the inequality into a graphing calculator.

2.

Graph the inequality.

3.

Use the graph to write the solution.

Using a Graphing Calculator Use a graphing calculator to solve (a) 2x − 1 < x + 2 and (b) 2x − 1 ≤ x + 2.

SOLUTION a.

Enter the inequality 2x − 1 < x + 2 into a graphing calculator. Press graph. Y1=2X-1 ●

●

is greater than is more than

●

is less than or equal to

≥ ●

is greater than or equal to

●

is at most

●

is at least

●

is no more than

●

is no less than

Writing Inequalities Write each sentence as an inequality. a. A number w minus 3.5 is less than or equal to −2. b. Three is less than a number n plus 5. c. Zero is greater than or equal to twice a number x plus 1.

SOLUTION a. A number w minus 3.5 is less than or equal to −2. ≤

w − 3.5

−2

An inequality is w − 3.5 ≤ −2.

READING

b. Three is less than a number n plus 5.

The inequality 3 < n + 5 is the same as n + 5 > 3.

3

n+5


−21

SOLUTION a.

x + 8 < −3

Write the inequality.

?

−4 + 8 < −3 4 < −3

✗

Substitute −4 for x. Simplify.

4 is not less than −3. So, −4 is not a solution of the inequality.

b.

−4.5x > −21

Write the inequality.

?

−4.5(−4) > −21 18 > −21

✓

Substitute −4 for x. Simplify.

18 is greater than −21. So, −4 is a solution of the inequality.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Tell whether −6 is a solution of the inequality. 3. c + 4 < −1

4. 10 ≤ 3 − m

5. 21 ÷ x ≥ −3.5

6. 4x − 25 > −2

Section 2.1

Writing and Graphing Inequalities

55

The graph of an inequality shows the solution set of the inequality on a number line. An open circle, ○, is used when a number is not a solution. A closed circle, ●, is used when a number is a solution. An arrow to the left or right shows that the graph continues in that direction.

Graphing Inequalities Graph each inequality. a. y ≤ −3

b. 2 < x

c. x > 0

SOLUTION a. Test a number to the left of −3.

y = −4 is a solution.

Test a number to the right of −3.

y = 0 is not a solution.

Use a closed circle because – 3 is a solution.

ANOTHER WAY

−6

Another way to represent the solutions of an inequality is to use set-builder notation. In Example 3b, the solutions can be written as {x | x > 2}, which is read as “the set of all numbers x such that x is greater than 2.”

−5

−4

−3

−2

−1

0

1

2

3

4

Shade the number line on the side where you found a solution.

b. Test a number to the left of 2.

x = 0 is not a solution.

Test a number to the right of 2.

x = 4 is a solution. Use an open circle because 2 is not a solution.

−2

−1

0

1

2

3

4

5

6

7

8

Shade the number line on the side where you found a solution.

c. Just by looking at the inequality, you can see that it represents the set of all positive numbers. Use an open circle because 0 is not a solution. −5

−4

−3

−2

−1

0

1

2

3

4

5

Shade the number line on the positive side of 0.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Graph the inequality. 7. b > −8 1

9. r < —2

56

Chapter 2

Solving Linear Inequalities

8. 1.4 ≥ g —

10. v ≥ √ 36

Writing Linear Inequalities from Graphs

YOU MUST BE THIS TALL TO RIDE

Writing Inequalities from Graphs The graphs show the height restrictions h (in inches) for two rides at an amusement park. Write an inequality that represents the height restriction of each ride. Ride A

44

46

48

50

52

54

44

46

48

50

52

54

Ride B

SOLUTION Ride A

The closed circle means that 48 is a solution.

44

46

48

50

52

54

Because the arrow points to the right, all numbers greater than 48 are solutions. Ride B

The open circle means that 52 is not a solution.

44

46

48

50

52

54

Because the arrow points to the left, all numbers less than 52 are solutions.

So, h ≥ 48 represents the height restriction for Ride A, and h < 52 represents the height restriction for Ride B.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

11. Write an inequality that represents the graph. –8

–7

–6

–5

–4

–3

–2

–1

0

1

2

Concept Summary Representing Linear Inequalities Words

x is less than 2

Algebra

Graph

x2 x≤2 x≥2

Writing and Graphing Inequalities

57

2.1

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE A mathematical sentence using the symbols , ≤, or ≥ is

called a(n)_______. 2. VOCABULARY Is 5 in the solution set of x + 3 > 8? Explain. 3. ATTENDING TO PRECISION Describe how to graph an inequality. 4. DIFFERENT WORDS, SAME QUESTION Which is different? Write “both” inequalities.

w is greater than or equal to −7.

w is no less than −7.

w is no more than −7.

w is at least −7.

Monitoring Progress and Modeling with Mathematics In Exercises 5–12, write the sentence as an inequality. (See Example 1.) 5. A number x is greater than 3.

14. MODELING WITH MATHEMATICS There are

430 people in a wave pool. Write an inequality that represents how many more people can enter the pool.

HOURS

6. A number n plus 7 is less than or equal to 9. 7. Fifteen is no more than a number t divided by 5.

Monday－Friday: 10 A.M.－6 P.M. Saturday－Sunday: 10 A.M.－7 P.M. Maximum Capacity: 600

8. Three times a number w is less than 18. 9. One-half of a number y is more than 22. 10. Three is less than the sum of a number s and 4. 11. Thirteen is at least the difference of a number v and 1. 12. Four is no less than the quotient of a number x

and 2.1.

In Exercises 15–24, tell whether the value is a solution of the inequality. (See Example 2.) 15. r + 4 > 8; r = 2

16. 5 − x < 8; x = −3

17. 3s ≤ 19; s = −6

18. 17 ≥ 2y; y = 7

x 2

4 z

20. — ≥ 3; z = 2

19. −1 > −—; x = 3

13. MODELING WITH MATHEMATICS

On a fishing trip, you catch two fish. The weight of the first fish is shown. The second fish weighs at least 0.5 pound more than the first fish. Write an inequality that represents the possible weights of the second fish.

1.2 LB MODE

1.2 LB

21. 14 ≥ −2n + 4; n = −5 22. −5 ÷ (2s) < −1; s = 10

10 2z

23. 20 ≤ — + 20; z = 5

3m 6

24. — − 2 > 3; m = 8

25. MODELING WITH MATHEMATICS The tallest person

who ever lived was approximately 8 feet 11 inches tall. a. Write an inequality that represents the heights of every other person who has ever lived. b. Is 9 feet a solution of the inequality? Explain. 58

Chapter 2

Solving Linear Inequalities

26. DRAWING CONCLUSIONS The winner of a

43.

weight-lifting competition bench-pressed 400 pounds. The other competitors all bench-pressed at least 23 pounds less.

b. Was one of the other competitors able to bench-press 379 pounds? Explain.

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

A ○

−y + 7 < −4

?

B ○

−8 + 7 < −4 −1 < −4

C ○

8 is in the solution set.

✗

2

of a swimming pool must be no less than 76°F. The temperature is currently 74°F. Which graph correctly shows how much the temperature needs to increase? Explain your reasoning.

correct the error in determining whether 8 is in the solution set of the inequality.

28.

1

45. ANALYZING RELATIONSHIPS The water temperature

ERROR ANALYSIS In Exercises 27 and 28, describe and

✗

0

44.

a. Write an inequality that represents the weights that the other competitors bench-pressed.

27.

−5 −4 −3 −2 −1

1 — x+2 ≤ 6 2

D ○

? ≤6 ? 4+2 ≤ 6 6≤6 8 is not in the solution set. 1 (8) + 2 — 2

30. z ≤ 5

31. −1 > t

32. −2 < w

33. v ≤ −4

34. s < 1

35. —14 < p

36. r ≥ −∣ 5 ∣

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

46. MODELING WITH MATHEMATICS According to a

state law for vehicles traveling on state roads, the maximum total weight of a vehicle and its contents depends on the number of axles on the vehicle. For each type of vehicle, write and graph an inequality that represents the possible total weights w (in pounds) of the vehicle and its contents.

In Exercises 29 –36, graph the inequality. (See Example 3.) 29. x ≥ 2

−5 −4 −3 −2 −1

Maximum Total Weights

2 axles, 40,000 lb

In Exercises 37– 40, write and graph an inequality for the given solution set. 37. {x ∣ x < 7}

38. {n ∣ n ≥ −2}

39. {z ∣ 1.3 ≤ z}

40. {w ∣ 5.2 > w}

3 axles, 60,000 lb

4 axles, 80,000 lb

47. PROBLEM SOLVING The Xianren Bridge is located

in Guangxi Province, China. This arch is the world’s longest natural arch, with a length of 400 feet. Write and graph an inequality that represents the lengthsℓ (in inches) of all other natural arches.

In Exercises 41– 44, write an inequality that represents the graph. (See Example 4.) 41. −5 −4 −3 −2 −1

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

42.

Section 2.1

Writing and Graphing Inequalities

59

48. THOUGHT PROVOKING A student works no more

53. CRITICAL THINKING Describe a real-life situation that

than 25 hours each week at a part-time job. Write an inequality that represents how many hours the student can work each day.

can be modeled by more than one inequality. 54. MODELING WITH MATHEMATICS In 1997,

Superman’s cape from the 1978 movie Superman was sold at an auction. The winning bid was $17,000. Write and graph an inequality that represents the amounts all the losing bids.

49. WRITING Describe a real-life situation modeled by

the inequality 23 + x ≤ 31.

50. HOW DO YOU SEE IT? The graph represents the

MATHEMATICAL CONNECTIONS In Exercises 55–58,

known melting points of all metallic elements (in degrees Celsius).

write an inequality that represents the missing dimension x. 55. The area is less than

−38.93 −38.91 −38.89 −38.87 −38.85 −38.83

56. The area is greater than

or equal to 8 square feet.

42 square meters.

a. Write an inequality represented by the graph. b. Is it possible for a metallic element to have a melting point of −38.87°C? Explain.

x 6m

10 ft

51. DRAWING CONCLUSIONS A one-way ride on a

subway costs $0.90. A monthly pass costs $24. Write an inequality that represents how many one-way rides you can buy before it is cheaper to buy the monthly pass. Is it cheaper to pay the one-way fare for 25 rides? Explain.

x

57. The area is less than

58. The area is greater than

18 square centimeters.

12 square inches. 2 in.

4 cm

Subway Prices

x

Onee-way ride ...................... $ŘšŘ Monthly pass ................. $ŚŜŘŘ

8 cm

x

59. WRITING A runner finishes a 200-meter dash in 52. MAKING AN ARGUMENT The inequality x ≤ 1324

represents the weights (in pounds) of all mako sharks ever caught using a rod and reel. Your friend says this means no one using a rod and reel has ever caught a mako shark that weighs 1324 pounds. Your cousin says this means someone using a rod and reel has caught a mako shark that weighs 1324 pounds. Who is correct? Explain your reasoning.

Maintaining Mathematical Proficiency

35 seconds. Let r represent any speed (in meters per second) faster than the runner’s speed. a. Write an inequality that represents r. Then graph the inequality. b. Every point on the graph represents a speed faster than the runner’s speed. Do you think every point could represent the speed of a runner? Explain. Reviewing what you learned in previous grades and lessons

Solve the equation. Check your solution. (Section 1.1) 60. x + 2 = 3

61. y − 9 = 5

62. 6 = 4 + y

63. −12 = y − 11

Solve the literal equation for x.

(Section 1.5)

⋅⋅

60

64. v = x y z

65. s = 2r + 3x

66. w = 5 + 3(x − 1)

67. n = —

Chapter 2

Solving Linear Inequalities

2x + 1 2

2.2

Solving Inequalities Using Addition or Subtraction Essential Question

How can you use addition or subtraction to

solve an inequality?

Quarterback Passing Efficiency Work with a partner. The National Collegiate Athletic Association (NCAA) uses the following formula to rank the passing efficiencies P of quarterbacks.

8.4Y + 100C + 330T − 200N P = ——— A

MODELING WITH MATHEMATICS To be proficient in math, you need to identify and analyze important relationships and then draw conclusions, using tools such as diagrams, flowcharts, and formulas.

Y = total length of all completed passes (in Yards)

C = Completed passes

T = passes resulting in a Touchdown

N = iNtercepted passes

A = Attempted passes

M = incoMplete passes

Completed Intercepted Incomplete

Attempts

Touchdown Not Touchdown

Determine whether each inequality must be true. Explain your reasoning. b. C + N ≤ A

a. T < C

c. N < A

d. A − C ≥ M

Finding Solutions of Inequalities Work with a partner. Use the passing efficiency formula to create a passing record that makes each inequality true. Record your results in the table. Then describe the values of P that make each inequality true. Attempts

Completions

Yards

Touchdowns

Interceptions

a. P < 0 b. P + 100 ≥ 250 c. P − 250 > −80

Communicate Your Answer 3. How can you use addition or subtraction to solve an inequality? 4. Solve each inequality.

a. x + 3 < 4

b. x − 3 ≥ 5

c. 4 > x − 2

d. −2 ≤ x + 1

Section 2.2

Solving Inequalities Using Addition or Subtraction

61

2.2 Lesson

What You Will Learn Solve inequalities using addition. Solve inequalities using subtraction.

Core Vocabul Vocabulary larry

Use inequalities to solve real-life problems.

equivalent inequalities, p. 62 Previous inequality

Solving Inequalities Using Addition Just as you used the properties of equality to produce equivalent equations, you can use the properties of inequality to produce equivalent inequalities. Equivalent inequalities are inequalities that have the same solutions.

Core Concept Addition Property of Inequality Words

Adding the same number to each side of an inequality produces an equivalent inequality. −3
b, then a + c > b + c.

If a ≥ b, then a + c ≥ b + c.

If a < b, then a + c < b + c.

If a ≤ b, then a + c ≤ b + c.

The diagram shows one way to visualize the Addition Property of Inequality when c > 0.


y − —4

Solving Inequalities Using Subtraction

Core Concept Subtraction Property of Inequality Words

Subtracting the same number from each side of an inequality produces an equivalent inequality. −3 ≤

Numbers

−5

7 > −20

1 −5

−7

−8 ≤ −4 Algebra

−7

0 > −27

If a > b, then a − c > b − c.

If a ≥ b, then a − c ≥ b − c.

If a < b, then a − c < b − c.

If a ≤ b, then a − c ≤ b − c.

The diagram shows one way to visualize the Subtraction Property of Inequality when c > 0.

a−c


– 9.4 −9.9 −9.8 −9.7 −9.6 −9.5 −9.4 −9.3 −9.2 −9.1 −9.0 −8.9

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the inequality. Graph the solution. 4. k + 5 ≤ −3

Section 2.2

1

5. —56 ≤ z + —6

6. p + 0.7 > −2.3

Solving Inequalities Using Addition or Subtraction

63

Solving Real-Life Problems Modeling with Mathematics A circuit overloads at 1800 watts of electricity. You plug a microwave oven that uses 11100 watts of electricity into the circuit. aa. Write and solve an inequality that represents how many watts you can add to the

circuit without overloading the circuit. b. In addition to the microwave oven, which of the following appliances can you plug b into the circuit at the same time without overloading the circuit? Appliance

Watts

Clock radio

50

Blender

300

Hot plate

1200

Toaster

800

SOLUTION S 1. Understand the Problem You know that the microwave oven uses 1100 watts out of a possible 1800 watts. You are asked to write and solve an inequality that represents how many watts you can add without overloading the circuit. You also know the numbers of watts used by four other appliances. You are asked to identify the appliances you can plug in at the same time without overloading the circuit. 2. Make a Plan Use a verbal model to write an inequality. Then solve the inequality and identify other appliances that you can plug into the circuit at the same time without overloading the circuit. 3. Solve the Problem Words

Additional Watts used by Overload + < watts microwave oven wattage

Variable

Let w be the additional watts you can add to the circuit. 1100

+

1100 + w
−4

11. r + 4 < 5

12. −8 ≤ 8 + y

13. 9 + w > 7

14. 15 ≥ q + 3

15. h − (−2) ≥ 10

16. −6 > t − (−13)

17. j + 9 − 3 < 8

18. 1 − 12 + y ≥ −5

correct the error in solving the inequality or graphing the solution.

19. 10 ≥ 3p − 2p − 7

20. 18 − 5z + 6z > 3 + 6

27.

Price: $19.76

ERROR ANALYSIS In Exercises 27 and 28, describe and

In Exercises 21−24, write the sentence as an inequality. Then solve the inequality.

✗

−17 < x − 14 −17 + 14 < x − 14 + 14 −3 < x

21. A number plus 8 is greater than 11. −6 −5 −4 −3 −2 −1

0

22. A number minus 3 is at least −5. 28.

23. The difference of a number and 9 is fewer than 4. 24. Six is less than or equal to the sum of a number

and 15.

−12 −11 −10 −9 −8 −7 −6

25. MODELING WITH MATHEMATICS You are riding a

train. Your carry-on bag can weigh no more than 50 pounds. Your bag weighs 38 pounds. (See Example 3.)

29. PROBLEM SOLVING An NHL hockey player has

a. Write and solve an inequality that represents how much weight you can add to your bag. Section 2.2

✗

−10 + x ≥ − 9 −10 + 10 + x ≥ − 9 x ≥ −9

59 goals so far in a season. What are the possible numbers of additional goals the player can score to match or break the NHL record of 92 goals in a season?

Solving Inequalities Using Addition or Subtraction

65

30. MAKING AN ARGUMENT In an aerial ski competition,

34. THOUGHT PROVOKING Write an inequality that has

you perform two acrobatic ski jumps. The scores on the two jumps are then added together. Ski jump

Competitor’s score

Your score

1

117.1

119.5

2

119.8

the solution shown in the graph. Describe a real-life situation that can be modeled by the inequality. 10

11

12

13

14

15

16

17

18

19

20

35. WRITING Is it possible to check all the numbers in

the solution set of an inequality? When you solve the inequality x − 11 ≥ −3, which numbers can you check to verify your solution? Explain your reasoning.

a. Describe the score that you must earn on your second jump to beat your competitor. b. Your coach says that you will beat your competitor if you score 118.4 points. A teammate says that you only need 117.5 points. Who is correct? Explain.

36. HOW DO YOU SEE IT? The diagram represents the

numbers of students in a school with brown eyes, brown hair, or both.

31. REASONING Which of the following inequalities are

equivalent to the inequality x − b < 3, where b is a constant? Justify your answer.

A x−b−3 < 0 ○

B 0 > b−x+3 ○

C x < 3−b ○

D −3 < b − x ○

Brown hair, H

MATHEMATICAL CONNECTIONS In Exercises 32 and 33,

Both, X

Brown eyes, E

Determine whether each inequality must be true. Explain your reasoning.

write and solve an inequality to nd the possible values of x. 32. Perimeter < 51.3 inches

a. H ≥ E

b. H + 10 ≥ E

c. H ≥ X

d. H + 10 ≥ X

e. H > X

f. H + 10 > X

x in.

14.2 in.

37. REASONING Write and graph an inequality that

represents the numbers that are not solutions of each inequality.

15.5 in.

33. Perimeter ≤ 18.7 feet

a. x + 8 < 14 b. x − 12 ≥ 5.7

4.1 ft

38. PROBLEM SOLVING Use the inequalities c − 3 ≥ d,

4.9 ft

b + 4 < a + 1, and a − 2 ≤ d − 7 to order a, b, c, and d from least to greatest.

x ft 6.4 ft

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Find the product or quotient. (Skills Review Handbook)

⋅

39. 7 (−9)

⋅

40. −11 (−12)

41. −27 ÷ (−3)

42. 20 ÷ (−5)

Solve the equation. Check your solution. (Section 1.1) 43. 6x = 24

66

Chapter 2

44. −3y = −18

Solving Linear Inequalities

s −8

45. — = 13

n 4

46. — = −7.3

2.3

Solving Inequalities Using Multiplication or Division Essential Question

How can you use division to solve an

inequality?

Writing a Rule Work with a partner. a. Copy and complete the table. Decide which graph represents the solution of the inequality 6 < 3x. Write the solution of the inequality.

LOOKING FOR A PATTERN To be proficient in math, you need to investigate relationships, observe patterns, and use your observations to write general rules.

x

−1

3x

−3

?

6 < 3x

−1

0

0

1

2

3

4

5

No

1

2

3

4

−1

5

0

1

2

3

4

5

b. Use a table to solve each inequality. Then write a rule that describes how to use division to solve the inequalities. ii. 3 ≥ 3x

i. 2x < 4

iv. 6 ≥ 3x

iii. 2x < 8

Writing a Rule Work with a partner. a. Copy and complete the table. Decide which graph represents the solution of the inequality 6 < −3x. Write the solution of the inequality. −5

x

−4

−3

−2

−1

0

1

−3x

?

6 < −3x

−5 −4 −3 −2 −1

0

−5 −4 −3 −2 −1

1

0

1

b. Use a table to solve each inequality. Then write a rule that describes how to use division to solve the inequalities. ii. 3 ≥ −3x

i. −2x < 4

iii. −2x < 8

iv. 6 ≥ −3x

Communicate Your Answer 3. How can you use division to solve an inequality? 4. Use the rules you wrote in Explorations 1(b) and 2(b) to solve each inequality.

a. 7x < −21 Section 2.3

b. 12 ≤ 4x

c. 10 < −5x

d. −3x ≤ 0

Solving Inequalities Using Multiplication or Division

67

2.3 Lesson

What You Will Learn Solve inequalities by multiplying or dividing by positive numbers. Solve inequalities by multiplying or dividing by negative numbers. Use inequalities to solve real-life problems.

Multiplying or Dividing by Positive Numbers

Core Concept Multiplication and Division Properties of Inequality (c > 0) Words

Multiplying or dividing each side of an inequality by the same positive number produces an equivalent inequality. −6 < 8

Numbers

⋅

6 > −8

⋅

6 2

−8 2

— > —

2 (−6) < 2 8 −12 < 16

3 > −4

a b If a > b and c > 0, then ac > bc. If a > b and c > 0, then — > —. c c

Algebra

a b If a < b and c > 0, then ac < bc. If a < b and c > 0, then — < —. c c These properties are also true for ≤ and ≥.

Multiplying or Dividing by Positive Numbers x Solve (a) — > −5 and (b) −24 ≥ 3x. Graph each solution. 8

SOLUTION x 8

a. Multiplication Property of Inequality

— > −5

⋅

⋅

x 8 — > 8 (−5) 8 x > −40

Write the inequality. Multiply each side by 8. Simplify.

x > –40

The solution is x > −40.

b. −24 ≥ 3x Division Property of Inequality

−24 3

3x 3

— ≥ —

−8 ≥ x

−42

−41

−40

−39

−38

−8

−7

−6

Write the inequality. Divide each side by 3. Simplify. x ≤ –8

The solution is x ≤ −8. −10

Monitoring Progress

−9

Help in English and Spanish at BigIdeasMath.com

Solve the inequality. Graph the solution. n 7

1. — ≥ −1

68

Chapter 2

Solving Linear Inequalities

1 5

2. −6.4 ≥ —w

3. 4b ≥ 36

4. −18 > 1.5q

Multiplying or Dividing by Negative Numbers

Core Concept Multiplication and Division Properties of Inequality (c < 0) Words

When multiplying or dividing each side of an inequality by the same negative number, the direction of the inequality symbol must be reversed to produce an equivalent inequality. −6 < 8

Numbers

⋅

6 > −8

⋅

— < —

12 > −16

COMMON ERROR

Algebra

A negative sign in an inequality does not necessarily mean you must reverse the inequality symbol, as shown in Example 1. Only reverse the inequality symbol when you multiply or divide each side by a negative number.

−8 −2

6 −2

−2 (−6) > −2 8

−3 < 4

a b If a > b and c < 0, then ac < bc. If a > b and c < 0, then — < —. c c a b If a < b and c < 0, then ac > bc. If a < b and c < 0, then — > —. c c

These properties are also true for ≤ and ≥.

Multiplying or Dividing by Negative Numbers Solve each inequality. Graph each solution. y a. 2 < — −3

b. −7y ≤ −35

SOLUTION a. Multiplication Property of Inequality

⋅

y 2 −3

Write the inequality.

y ⋅— −3

Multiply each side by −3. Reverse the inequality symbol.

−6 > y

Simplify. y < –6

The solution is y < −6. −8

b. −7y ≤ −35 Division Property of Inequality

−7y ≥ −7 y≥

—

−7

−6

−35 −7 5

Divide each side by −7. Reverse the inequality symbol. Simplify. y≥5

The solution is y ≥ 5. 3

Monitoring Progress

4

5

6

7

Help in English and Spanish at BigIdeasMath.com

Solve the inequality. Graph the solution. p x 5. — < 7 6. — ≤ −5 −4 −5

Section 2.3

−4

Write the inequality.

—

8. −9m > 63

−5

9. −2r ≥ −22

1 10

7. 1 ≥ − — z 10. −0.4y ≥ −12

Solving Inequalities Using Multiplication or Division

69

Solving Real-Life Problems Modeling with Mathematics You earn $9.50 per hour at your summer job. Write and solve an inequality that represents the numbers of hours you need to work to buy a digital camera that costs $247.

SOLUTION 1. Understand the Problem You know your hourly wage and the cost of the digital camera. You are asked to write and solve an inequality that represents the numbers of hours you need to work to buy the digital camera. 2. Make a Plan Use a verbal model to write an inequality. Then solve the inequality. 3. Solve the Problem

⋅ worked Hours

Words

Hourly wage

Variable

Let n be the number of hours worked.

Inequality

9.5

⋅

n

≥

≥

Cost of camera

247

9.5n ≥ 247

Write the inequality.

9.5n 9.5

Divide each side by 9.5.

247 9.5

— ≥ —

Division Property of Inequality

n ≥ 26

Simplify.

You need to work at least 26 hours for your gross pay to be at least $247. If you have payroll deductions, such as Social Security taxes, you need to work more than 26 hours.

REMEMBER Compatible numbers are numbers that are easy to compute mentally.

4. Look Back You can use estimation to check that your answer is reasonable. $247

÷

$9.50/h

$250

÷

$10/h = 25 h

Use compatible numbers.

Your hourly wage is about $10 per hour. So, to earn about $250, you need to work about 25 hours. Unit Analysis Each time you set up an equation or inequality to represent a real-life problem, be sure to check that the units balance. $9.50 h

— × 26 h = $247

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

11. You have at most $3.65 to make copies. Each copy costs $0.25. Write and solve

an inequality that represents the numbers of copies you can make. 12. The maximum speed limit for a school bus is 55 miles per hour. Write and solve

an inequality that represents the numbers of hours it takes to travel 165 miles in a school bus.

70

Chapter 2

Solving Linear Inequalities

Exercises

2.3

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Explain how solving 2x < −8 is different from solving −2x < 8. 2. OPEN-ENDED Write an inequality that is solved using the Division Property of Inequality where the

inequality symbol needs to be reversed.

Monitoring Progress and Modeling with Mathematics 2 9

In Exercises 3–10, solve the inequality. Graph the solution. (See Example 1.) 3. 4x < 8

4. 3y ≤ −9

5. −20 ≤ 10n

6. 35 < 7t

x 2

7. — > −2

4 5

9. 20 ≥ — w

24. 4 > —

3 4

25. 2x > —

26. 1.1y < 4.4

ERROR ANALYSIS In Exercises 27 and 28, describe and

correct the error in solving the inequality.

a 4

8. — < 10.2

27.

8 3

10. −16 ≤ — t

✗

In Exercises 11–18, solve the inequality. Graph the solution. (See Example 2.) 11. −6t < 12

12. −9y > 9

13. −10 ≥ −2z

14. −15 ≤ −3c

n −3

n −4

23. 2 ≤ − — x

16. — ≤ 16

1 17. −8 < −— m 4

2 18. −6 > −— y 3

⋅

⋅

−9 < x The solution is x > −9.

w −5

15. — ≥ 1

2 −6 > — x 3 3 2 3 — (−6) < — — x 2 2 3 18 −— < x 2

28.

19. MODELING WITH MATHEMATICS You have $12 to

buy five goldfish for your new fish tank. Write and solve an inequality that represents the prices you can pay per fish. (See Example 3.) 20. MODELING WITH MATHEMATICS A weather

✗

−4y ≤ −32 −4y −4

−32 −4

—≤—

y≤8 The solution is y ≤ 8.

29. ATTENDING TO PRECISION You have $700 to buy

forecaster predicts that the temperature in Antarctica will decrease 8°F each hour for the next 6 hours. Write and solve an inequality to determine how many hours it will take for the temperature to drop at least 36°F.

new carpet for your bedroom. Write and solve an inequality that represents the costs per square foot that you can pay for the new carpet. Specify the units of measure in each step.

USING TOOLS In Exercises 21–26, solve the inequality.

Use a graphing calculator to verify your answer. 21. 36 < 3y

14 ft

22. 17v ≥ 51

14 ft

Section 2.3

Solving Inequalities Using Multiplication or Division

71

30. HOW DO YOU SEE IT? Let m > 0. Match each

35. ANALYZING RELATIONSHIPS Consider the number

line shown.

inequality with its graph. Explain your reasoning. x a. — < −1 m

x b. — > 1 m

x c. — < 1 m

x d. −— < 1 m

−A

c. Use the results from parts (a) and (b) to explain why the direction of the inequality symbol must be reversed when multiplying or dividing each side of an inequality by the same negative number.

−m

C. −m

36. REASONING Why might solving the inequality

4 x error? (Hint: Consider x > 0 and x < 0.)

D.

— ≥ 2 by multiplying each side by x lead to an

31. MAKING AN ARGUMENT You run for 2 hours at a

37. MATHEMATICAL CONNECTIONS The radius of a

speed no faster than 6.3 miles per hour.

C circle is represented by the formula r = —. Write and 2π solve an inequality that represents the possible circumferences C of the circle.

a. Write and solve an inequality that represents the possible numbers of miles you run. b. A marathon is approximately 26.2 miles. Your friend says that if you continue to run at this speed, you will not be able to complete a marathon in less than 4 hours. Is your friend correct? Explain.

r>5

x 4 solution of x = p. Write a second inequality that also has a solution of x = p.

32. THOUGHT PROVOKING The inequality — ≤ 5 has a

38. CRITICAL THINKING A water-skiing instructor

recommends that a boat pulling a beginning skier has a speed less than 18 miles per hour. Write and solve an inequality that represents the possible distances d (in miles) that a beginner can travel in 45 minutes of practice time.

33. PROBLEM SOLVING The U.S. Mint pays $0.02

to produce every penny. How many pennies are produced when the U.S. Mint pays more than $6 million in production costs?

39. CRITICAL THINKING A local zoo employs 36 people

to take care of the animals each day. At most, 24 of the employees work full time. Write and solve an inequality that represents the fraction of employees who work part time. Graph the solution.

2

34. REASONING Are x ≤ —3 and −3x ≤ −2 equivalent?

Explain your reasoning.

Maintaining Mathematical Proficiency Solve the equation. Check your solution.

Reviewing what you learned in previous grades and lessons

(Section 1.2 and Section 1.3) 1

40. 5x + 3 = 13

41.

—2 y − 8 = −10

42. −3n + 2 = 2n − 3

43.

—2 z + 4 = —2 z − 8

Tell which number is greater. 44. 0.8, 85%

72

A

b. Write an inequality relating −A and −B.

m

B.

B

a. Write an inequality relating A and B.

m

A.

−B

Chapter 2

1

5

(Skills Review Handbook)

16 45. — , 50% 30

Solving Linear Inequalities

46. 120%, 0.12

2

47. 60%, —3

2.4

Solving Multi-Step Inequalities Essential Question

How can you solve a multi-step inequality?

Solving a Multi-Step Inequality Work with a partner. ●

●

Use what you already know about solving equations and inequalities to solve each multi-step inequality. Justify each step. Match each inequality with its graph. Use a graphing calculator to check your answer.

a. 2x + 3 ≤ x + 5

b. −2x + 3 > x + 9

JUSTIFYING STEPS

c. 27 ≥ 5x + 4x

d. −8x + 2x − 16 < −5x + 7x

To be proficient in math, you need to justify each step in a solution and communicate your justification to others.

e. 3(x − 3) − 5x > −3x − 6

f. −5x − 6x ≤ 8 − 8x − x

A.

B.

4

−6

6

4

6

6

−4

C.

4

D.

4

−6

4

−6

6

6

−4

E.

−4 4

F.

4

−6

−6

6

6

−4

−4

Communicate Your Answer 2. How can you solve a multi-step inequality? 3. Write two different multi-step inequalities whose solutions are represented

by the graph. −6

−5

−4

−3

Section 2.4

−2

−1

0

1

2

Solving Multi-Step Inequalities

73

2.4 Lesson

What You Will Learn Solve multi-step inequalities. Use multi-step inequalities to solve real-life problems.

Solving Multi-Step Inequalities To solve a multi-step inequality, simplify each side of the inequality, if necessary. Then use inverse operations to isolate the variable. Be sure to reverse the inequality symbol when multiplying or dividing by a negative number.

Solving Multi-Step Inequalities Solve each inequality. Graph each solution. y a. — + 7 < 9 −6

b. 2v − 4 ≥ 8

SOLUTION y a. — + 7 < −6 −7

9

Write the inequality.

−7

Subtract 7 from each side.

y −6

— < 2

Simplify.

⋅

⋅

y −6 — > −6 2 −6

Multiply each side by −6. Reverse the inequality symbol.

y > −12

Simplify.

The solution is y > −12. y > – 12 −20 −18 −16 −14 −12 −10

b. 2v − 4 ≥ +4

8

−8

−6

−4

−2

0

16

18

Write the inequality.

+4

Add 4 to each side.

2v ≥ 12

Simplify.

2v 2

Divide each side by 2.

12 2

— ≥ —

v≥6

Simplify.

The solution is v ≥ 6. v≥6 −2

0

2

4

Monitoring Progress

6

8

10

12

14

Help in English and Spanish at BigIdeasMath.com

Solve the inequality. Graph the solution. 1. 4b − 1 < 7

n −2

3. — + 11 > 12

74

Chapter 2

Solving Linear Inequalities

2. 8 − 9c ≥ −28

v 3

4. 6 ≥ 5 − —

Solving an Inequality with Variables on Both Sides Solve 6x − 5 < 2x + 11.

SOLUTION 6x − 5


4(2b + 3)

Write the inequality.

8b − 3 >

8b + 12

Distributive Property

− 8b

− 8b −3 > 12

Subtract 8b from each side.

✗

Simplify.

The inequality −3 > 12 is false. So, there is no solution.

b. 2(5w − 1) ≤ 7 + 10w

Write the inequality.

10w − 2 ≤ 7 + 10w

Distributive Property

− 10w

− 10w −2 ≤ 7

Subtract 10w from each side. Simplify.

The inequality −2 ≤ 7 is true. So, all real numbers are solutions.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the inequality. 5. 5x − 12 ≤ 3x − 4

6. 2(k − 5) < 2k + 5

7. −4(3n − 1) > −12n + 5.2

8. 3(2a − 1) ≥ 10a − 11

Section 2.4

Solving Multi-Step Inequalities

75

Solving Real-Life Problems Modeling with Mathematics You need a mean score of at least 90 points to advance to the next round of the touch-screen trivia game. What scores in the fifth game will allow you to advance?

Game 1: Game 2: Game 3: Game 4:

SOLUTION 1. Understand the Problem You know the scores of your first four games. You are asked to find the scores in the fifth game that will allow you to advance.

REMEMBER The mean in Example 4 is equal to the sum of the game scores divided by the number of games.

2. Make a Plan Use the definition of the mean of a set of numbers to write an inequality. Then solve the inequality and answer the question. 3. Solve the Problem Let x be your score in the fifth game. 95 + 91 + 77 + 89 + x 5

—— ≥ 90

Write an inequality.

352 + x 5

— ≥ 90

Simplify.

352 + x 5 — ≥ 5 90 5

⋅

352 + x ≥ − 352

⋅

Multiply each side by 5.

450

Simplify.

− 352

Subtract 352 from each side.

x ≥ 98

Simplify.

A score of at least 98 points will allow you to advance.

4. Look Back You can draw a diagram to check that your answer is reasonable. The horizontal bar graph shows the differences between the game scores and the desired mean of 90. +5

Game 1 +1

Game 2 Game 3

−13 −1

Game 4

+8

Game 5 75

78

81

84

87

90

93

96

99

To have a mean of 90, the sum of the differences must be zero. 5 + 1 − 13 − 1 + 8 = 0

Monitoring Progress

✓ Help in English and Spanish at BigIdeasMath.com

9. WHAT IF? You need a mean score of at least 85 points to advance to the next

round. What scores in the fifth game will allow you to advance? 76

Chapter 2

Solving Linear Inequalities

2.4

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Compare solving multi-step inequalities and solving multi-step equations. 2. WRITING Without solving, how can you tell that the inequality 4x + 8 ≤ 4x − 3 has no solution?

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, match the inequality with its graph. 3. 7b − 4 ≤ 10

4. 4p + 4 ≥ 12

5. −6g + 2 ≥ 20

6. 3(2 − f ) ≤ 15

23. 6(ℓ+ 3) < 3(2ℓ + 6) 24. 2(5c − 7) ≥ 10(c − 3)

(

0

1

2

3

4

5

6

1

(

)

7

1

)

26. 15 —3 b + 3 ≤ 6(b + 9)

25. 4 —2 t − 2 > 2(t − 3)

A. −3 −2 −1

22. 3w − 5 > 2w + w − 7

27. 9j − 6 + 6j ≥ 3(5j − 2) 28. 6h − 6 + 2h < 2(4h − 3)

B.

−3 −2 −1

0

1

2

3

4

5

6

7

ERROR ANALYSIS In Exercises 29 and 30, describe and

correct the error in solving the inequality. C.

−8 −7 −6 −5 −4 −3 −2 −1

0

1

2

−8 −7 −6 −5 −4 −3 −2 −1

0

1

2

29.

✗

D.

In Exercises 7–16, solve the inequality. Graph the solution. (See Example 1.)

9. −9 ≤ 7 − 8v

30.

✗

m 3

12. 1 + — ≤ 6

13. — + 9 > 13

p −8

14. 3 + — ≤ 6

15. 6 ≥ −6(a + 2)

16. 18 ≤ 3(b − 4)

r −4

−2(1 − x) ≤ 2x − 7 −2 + 2x ≤ 2x − 7 −2 ≤ −7

10. 2 > −3t − 10

w 2

11. — + 4 > 5

x≥6

8. 5y + 9 ≤ 4

7. 2x − 3 > 7

x 4 x + 6 ≥ 12

—+6 ≥ 3

All real numbers are solutions.

31. MODELING WITH MATHEMATICS Write and solve an

inequality that represents how many $20 bills you can withdraw from the account without going below the minimum balance. (See Example 4.)

In Exercises 17–28, solve the inequality. (See Examples 2 and 3.) 17. 4 − 2m > 7 − 3m

18. 8n + 2 ≤ 8n − 9

19. −2d − 2 < 3d + 8

20. 8 + 10f > 14 − 2f

21. 8g − 5g − 4 ≤ −3 + 3g

Section 2.4

Solving Multi-Step Inequalities

77

36. HOW DO YOU SEE IT? The graph shows your budget

32. MODELING WITH MATHEMATICS

and the total cost of x gallons of gasoline and a car wash. You want to determine the possible amounts (in gallons) of gasoline you can buy within your budget.

A woodworker wants to earn at least $25 an hour making and selling cabinets. He pays $125 for materials. Write and solve an inequality that represents how many hours the woodworker can spend building the cabinet.

500

Gas Station Costs

Dollars

$

y 50 40 30 20 10 0

y = 40

y = 3.55x + 8 0 1 2 3 4 5 6 7 8 9 10 11 x

Amount of gasoline (gallons)

33. MATHEMATICAL CONNECTIONS The area of the

rectangle is greater than 60 square feet. Write and solve an inequality to nd the possible values of x.

a. What is your budget? b. How much does a gallon of gasoline cost? How much does a car wash cost? c. Write an inequality that represents the possible amounts of gasoline you can buy.

(2x − 3) ft

d. Use the graph to estimate the solution of your inequality in part (c).

12 ft

34. MAKING AN ARGUMENT Forest Park Campgrounds

charges a $100 membership fee plus $35 per night. Woodland Campgrounds charges a $20 membership fee plus $55 per night. Your friend says that if you plan to camp for four or more nights, then you should choose Woodland Campgrounds. Is your friend correct? Explain.

37. PROBLEM SOLVING For what

values of r will the area of the shaded region be greater than or equal to 9(π − 2)?

r

35. PROBLEM SOLVING The height of one story of a

building is about 10 feet. The bottom of the ladder on the fire truck must be at least 24 feet away from the building. How many stories can the ladder reach? Justify your answer.

74 ft

38. THOUGHT PROVOKING A runner’s times (in minutes)

in the four races he has completed are 25.5, 24.3, 24.8, and 23.5. The runner plans to run at least one more race and wants to have an average time less than 24 minutes. Write and solve an inequality to show how the runner can achieve his goal. REASONING In Exercises 39 and 40, find the value of

a for which the solution of the inequality is all real numbers. 39. a(x + 3) < 5x + 15 − x 8 ft

40. 3x + 8 + 2ax ≥ 3ax − 4a

Maintaining Mathematical Proficiency Write the sentence as an inequality. 41. Six times a number y is

less than or equal to 10. 78

Chapter 2

Reviewing what you learned in previous grades and lessons

(Section 2.1)

42. A number p plus 7 is

greater than 24.

Solving Linear Inequalities

43. The quotient of a number r

and 7 is no more than 18.

2.1–2.4

What Did You Learn?

Core Vocabulary inequality, p. 54 solution of an inequality, p. 55 solution set, p. 55

graph of an inequality, p. 56 equivalent inequalities, p. 62

Core Concepts Section 2.1 Representing Linear Inequalities, p. 57

Section 2.2 Addition Property of Inequality, p. 62

Subtraction Property of Inequality, p. 63

Section 2.3 Multiplication and Division Properties of Inequality (c > 0), p. 68 Multiplication and Division Properties of Inequality (c < 0), p. 69

Section 2.4 Solving Multi-Step Inequalities, p. 74 Special Solutions of Linear Inequalities, p. 75

Mathematical Practices 1.

Explain the meaning of the inequality symbol in your answer to Exercise 47 on page 59. How did you know which symbol to use?

2.

In Exercise 30 on page 66, why is it important to check the reasonableness of your answer in part (a) before answering part (b)?

3.

Explain how considering the units involved in Exercise 29 on ppage g 71 helped p yyou answer the question.

Analyzing Your Errors Application Errors What Happens: You can do numerical problems, but you struggle with problems that have context. How to Avoid This Error: Do not just mimic the steps of solving an application problem. Explain out loud what the question is asking and why you are doing each step. After solving the problem, ask yourself, “Does my solution make sense?” 79 9

2.1–2.4

Quiz

Write the sentence as an inequality. (Section 2.1) 1. A number z minus 6 is greater than or equal to 11. 2. Twelve is no more than the sum of −1.5 times a number w and 4.

Write an inequality that represents the graph. (Section 2.1) 3. −5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

4.

Solve the inequality. Graph the solution. (Section 2.2 and Section 2.3) 5. 9 + q ≤ 15

6. z − (−7) < 5

7. −3 < y − 4

8. 3p ≥ 18

w −2

10. −20x > 5

9. 6 > —

Solve the inequality. (Section 2.4) 11. 3y − 7 ≥ 17

12. 8(3g − 2) ≤ 12(2g + 1)

14. Three requirements for a lifeguard training course are shown.

(Section 2.1) a. Write and graph three inequalities that represent the requirements. b. You can swim 250 feet, tread water for 6 minutes, and swim 35 feet underwater without taking a breath. Do you satisfy the requirements of the course? Explain. 15. The maximum volume of an American white pelican’s bill is about

700 cubic inches. A pelican scoops up 100 cubic inches of water. Write and solve an inequality that represents the additional volumes the pelican’s bill can contain. (Section 2.2) 16. You save $15 per week to purchase one of the

bikes shown. (Section 2.3 and Section 2.4) a. Write and solve an inequality to find the numbers of weeks you need to save to purchase a bike. b. Your parents give you $65 to help you buy the new bike. How does this affect you answer in part (a)? Use an inequality to justify your answer.

80

Chapter 2

Solving Linear Inequalities

13. 6(2x − 1) ≥ 3(4x + 1)

LIFEGUARDS NEEDED Take Our Training Course NOW!!!

Lifeguard Training Requirements Swim at least 100 yards. Tread water for at least 5 minutes. Swim 10 yards or more underwater without taking a breath.

2.5

Solving Compound Inequalities Essential Question

How can you use inequalities to describe intervals on the real number line? Describing Intervals on the Real Number Line Work with a partner. In parts (a)–(d), use two inequalities to describe the interval. a.

REASONING ABSTRACTLY To be proficient in math, you need to create a clear representation of the problem at hand.

Half-Open Interval –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

b.

1

2

3

4

5

6

7

8

9

10

3

4

5

6

7

8

9

10

3

4

5

6

7

8

9

10

4

5

6

7

8

9

10

Half-Open Interval –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

c.

0

0

1

2

Closed Interval –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

d.

0

1

2

Open Interval –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

0

1

2

3

e. Do you use “and” or “or” to connect the two inequalities in parts (a)–(d)? Explain.

Describing Two Infinite Intervals Work with a partner. In parts (a)–(d), use two inequalities to describe the interval. a. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

7

8

9

10

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

7

8

9

10

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

7

8

9

10

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

7

8

9

10

b.

c.

d.

e. Do you use “and” or “or” to connect the two inequalities in parts (a)–(d)? Explain.

Communicate Your Answer 3. How can you use inequalities to describe intervals on the real number line?

Section 2.5

Solving Compound Inequalities

81

2.5 Lesson

What You Will Learn Write and graph compound inequalities. Solve compound inequalities.

Core Vocabul Vocabulary larry

Use compound inequalities to solve real-life problems.

compound inequality, p. 82

Writing and Graphing Compound Inequalities A compound inequality is an inequality formed by joining two inequalities with the word “and” or the word “or.” The graph of a compound inequality with “and” is the intersection of the graphs of the inequalities. The graph shows numbers that are solutions of both inequalities.

The graph of a compound inequality with “or” is the union of the graphs of the inequalities. The graph shows numbers that are solutions of either inequality.

x≥2

y ≤ −2

x1

2 ≤ x and x < 5 2 ≤ x 1

5

−3 −2 −1

0

1

2

Writing and Graphing Compound Inequalities Write each sentence as an inequality. Graph each inequality. a. A number x is greater than −8 and less than or equal to 4.

REMEMBER

b. A number y is at most 0 or at least 2.

A compound inequality with “and” can be written as a single inequality. For example, you can write x > −8 and x ≤ 4 as −8 < x ≤ 4.

SOLUTION a. A number x is greater than −8 and less than or equal to 4. x > −8

x≤4

and

An inequality is −8 < x ≤ 4. −10 −8 −6 −4 −2

0

2

4

6

Graph the intersection of the graphs of x > −8 and x ≤ 4.

6

Graph the union of the graphs of y ≤ 0 and y ≥ 2.

b. A number y is at most 0 or at least 2. y≤0

y≥2

or

An inequality is y ≤ 0 or y ≥ 2. −2 −1

0

1

2

Monitoring Progress

3

4

5

Help in English and Spanish at BigIdeasMath.com

Write the sentence as an inequality. Graph the inequality. 1. A number d is more than 0 and less than 10. 2. A number a is fewer than −6 or no less than −3.

82

Chapter 2

Solving Linear Inequalities

Solving Compound Inequalities

LOOKING FOR STRUCTURE To be proficient in math, you need to see complicated things as single objects or as being composed of several objects.

You can solve a compound inequality by solving two inequalities separately. When a compound inequality with “and” is written as a single inequality, you can solve the inequality by performing the same operation on each expression.

Solving Compound Inequalities with “And” Solve each inequality. Graph each solution. b. −3 < −2x + 1 ≤ 9

a. −4 < x − 2 < 3

SOLUTION a. Separate the compound inequality into two inequalities, then solve. −4 < x − 2 +2

x−2


x

1

2

3

4

5

1

2

3

Write the inequality.

−1

−4 < −2x

0

Subtract 1 from each expression. Simplify.

8 −2

Divide each expression by −2. Reverse each inequality symbol.

≥ −4

Simplify.

The solution is −4 ≤ x < 2. −5 −4 −3 −2 −1

0

Solving a Compound Inequality with “Or” Solve 3y − 5 < −8 or 2y − 1 > 5. Graph the solution.

SOLUTION 3y − 5 < −8 +5

or

2y − 1 >

+5

+1

3y < −3 3y 3

−3 3

2y 2

Write the inequality.

+1

2y >

— < —

y < −1

5

Addition Property of Inequality

6

Simplify.

6 2

— > —

Division Property of Inequality

y>3

or

Simplify.

The solution is y < −1 or y > 3. −2 −1

Monitoring Progress

0

1

2

3

4

5

6

Help in English and Spanish at BigIdeasMath.com

Solve the inequality. Graph the solution. 3. 5 ≤ m + 4 < 10

4. −3 < 2k − 5 < 7

5. 4c + 3 ≤ −5 or c − 8 > −1

6. 2p + 1 < −7 or 3 − 2p ≤ −1

Section 2.5

Solving Compound Inequalities

83

Solving Real-Life Problems Modeling with Mathematics Electrical devices should operate effectively within a specified temperature range. Outside the operating temperature range, the device may fail. a. Write and solve a compound inequality that represents the possible operating temperatures (in degrees Fahrenheit) of the smartphone. b. Describe one situation in which the surrounding temperature could be below the operating range and one in which it could be above.

SOLUTION Operating temperature: 0ºC to 35ºC

1. Understand the Problem You know the operating temperature range in degrees Celsius. You are asked to write and solve a compound inequality that represents the possible operating temperatures (in degrees Fahrenheit) of the smartphone. Then you are asked to describe situations outside this range. 2. Make a Plan Write a compound inequality in degrees Celsius. Use the formula C = —59 (F − 32) to rewrite the inequality in degrees Fahrenheit. Then solve the inequality and describe the situations.

STUDY TIP You can also solve the inequality by first multiplying each expression by —95.

3. Solve the Problem Let C be the temperature in degrees Celsius, and let F be the temperature in degrees Fahrenheit.

⋅

0≤

C

≤ 35

Write the inequality using C.

0≤

5 —9 (F − 32) 5 —9 (F − 32)

≤ 35

Substitute —59 (F − 32) for C.

≤ 9 35

Multiply each expression by 9.

9 0≤9 0≤ 0≤ + 160

⋅

⋅

5(F − 32) ≤ 315 5F − 160 + 160

≤

Simplify.

315 + 160

Distributive Property Add 160 to each expression.

160 ≤

5F

≤ 475

Simplify.

160 5

—

5F 5

475 ≤— 5

Divide each expression by 5.

F

≤ 95

Simplify.

—≤

32 ≤

The solution is 32 ≤ F ≤ 95. So, the operating temperature range of the smartphone is 32°F to 95°F. One situation when the surrounding temperature could be below this range is winter in Alaska. One situation when the surrounding temperature could be above this range is daytime in the Mojave Desert of the American Southwest.

4. Look Back You can use the formula C = —59(F − 32) to check that your answer is correct. Substitute 32 and 95 for F in the formula to verify that 0°C and 35°C are the minimum and maximum operating temperatures in degrees Celsius.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

7. Write and solve a compound inequality that represents the temperature rating (in

degrees Fahrenheit) of the winter boots. −40ºC to 15ºC

84

Chapter 2

Solving Linear Inequalities

Exercises

2.5

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Compare the graph of −6 ≤ x ≤ −4 with the graph of x ≤ −6 or x ≥ −4. 2. WHICH ONE DOESN’T BELONG? Which compound inequality does not belong with the other three?

Explain your reasoning. a < −2 or a > 8

a > 4 or a < −3

a < 6 or a > −9

a > 7 or a < −5

Monitoring Progress and Modeling with Mathematics In Exercises 3– 6, write a compound inequality that is represented by the graph. 3. −3 −2 −1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

0

−2 −1

8

4. 5

6

7

12. MODELING WITH MATHEMATICS The life zones on

Mount Rainier, a mountain in Washington, can be approximately classified by elevation, as follows. Low-elevation forest: above 1700 feet to 2500 feet Mid-elevation forest: above 2500 feet to 4000 feet Subalpine: above 4000 feet to 6500 feet Alpine: above 6500 feet to the summit

5.

6. 0

1

2

3

4

5

6

7

In Exercises 7–10, write the sentence as an inequality. Graph the inequality. (See Example 1.)

Elevation of Mount Rainier: 14,410 ft

7. A number p is less than 6 and greater than 2.

Write a compound inequality that represents the elevation range for each type of plant life.

8. A number n is less than or equal to −7 or greater

a. trees in the low-elevation forest zone b. flowers in the subalpine and alpine zones

than 12. 2

9. A number m is more than −7 —3 or at most −10. 10. A number r is no less than −1.5 and fewer than 9.5.

13. 6 < x + 5 ≤ 11

11. MODELING WITH MATHEMATICS

Slitsnails are large mollusks that live in deep waters. They have been found in the range of elevations shown. Write and graph a compound inequality that represents this range.

In Exercises 13–20, solve the inequality. Graph the solution. (See Examples 2 and 3.)

−100 ft

14. 24 > −3r ≥ − 9

15. v + 8 < 3 or −8v < −40 16. −14 > w + 3 or 3w ≥ −27 17. 2r + 3 < 7 or −r + 9 ≤ 2 18. −6 < 3n + 9 < 21

−2500 ft

1

19. −12 < —2 (4x + 16) < 18 1

20. 35 < 7(2 − b) or —3 (15b − 12) ≥ 21

Section 2.5

Solving Compound Inequalities

85

ERROR ANALYSIS In Exercises 21 and 22, describe and correct the error in solving the inequality or graphing the solution. 21.

✗

30. 3x − 18 < 4x − 23 and x − 16 < −22 31. REASONING Fill in the compound inequality

4(x − 6) 2(x − 10) and 5(x + 2) ≥ 2(x + 8) with , or ≥ so that the solution is only one value.

4 < −2x + 3 < 9 4 < −2x < 6 −2 > x > −3

32. THOUGHT PROVOKING Write a real-life story that

can be modeled by the graph. −4

−3

−2

−1

0 2

22.

✗ −15

x−2>3

or

x>5

or

−10

−5

0

x + 8 < −2

5

6

7

8

9

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Use the triangle shown to write and solve three inequalities. Your friend claims the value of x can be 1. Is your friend correct? Explain.

10

23. MODELING WITH MATHEMATICS

Write and solve a compound inequality that represents the possible temperatures (in degrees Fahrenheit) of the interior of the iceberg. (See Example 4.)

4

10

11

12

33. MAKING AN ARGUMENT

x < −10 5

3

−20ºC −2 20 0ºC to to −15ºC −15 15ºC

x

7

5

34. HOW DO YOU SEE IT? The graph shows the annual

profits of a company from 2006 to 2013. Annual Profit Profit (millions of dollars)

24. PROBLEM SOLVING A ski shop sells skis with lengths

ranging from 150 centimeters to 220 centimeters. The shop says the length of the skis should be about 1.16 times a skier’s height (in centimeters). Write and solve a compound inequality that represents the heights of skiers the shop does not provide skis for.

100 90 80 70 60 50 0

2006 2007 2008 2009 2010 2011 2012 2013

Year

In Exercises 25–30, solve the inequality. Graph the solution, if possible. 25. 22 < −3c + 4 < 14

a. Write a compound inequality that represents the annual profits from 2006 to 2013.

26. 2m − 1 ≥ 5 or 5m > −25

b. You can use the formula P = R − C to find the profit P, where R is the revenue and C is the cost. From 2006 to 2013, the company’s annual cost was about $125 million. Is it possible the company had an annual revenue of $160 million from 2006 to 2013? Explain.

27. −y + 3 ≤ 8 and y + 2 > 9 28. x − 8 ≤ 4 or 2x + 3 > 9 29. 2n + 19 ≤ 10 + n or −3n + 3 < −2n + 33

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Solve the equation. Graph the solutions, if possible. (Section 1.4) 35.

∣ d9 ∣ = 6 —

36. 7∣ 5p − 7 ∣ = −21

37.

∣ r + 2 ∣ = ∣ 3r − 4 ∣

38.

∣ 12 w − 6 ∣ = ∣ w + 7 ∣ —

Find and interpret the mean absolute deviation of the data. (Skills Review Handbook) 39. 1, 1, 2, 5, 6, 8, 10, 12, 12, 13

86

Chapter 2

Solving Linear Inequalities

40. 24, 26, 28, 28, 30, 30, 32, 32, 34, 36

2.6

Solving Absolute Value Inequalities Essential Question

How can you solve an absolute value inequality?

Solving an Absolute Value Inequality Algebraically Work with a partner. Consider the absolute value inequality

∣ x + 2 ∣ ≤ 3. a. Describe the values of x + 2 that make the inequality true. Use your description to write two linear inequalities that represent the solutions of the absolute value inequality.

MAKING SENSE OF PROBLEMS To be proficient in math, you need to explain to yourself the meaning of a problem and look for entry points to its solution.

b. Use the linear inequalities you wrote in part (a) to find the solutions of the absolute value inequality. c. How can you use linear inequalities to solve an absolute value inequality?

Solving an Absolute Value Inequality Graphically Work with a partner. Consider the absolute value inequality

∣ x + 2 ∣ ≤ 3. a. On a real number line, locate the point for which x + 2 = 0. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

7

8

9

10

b. Locate the points that are within 3 units from the point you found in part (a). What do you notice about these points? c. How can you use a number line to solve an absolute value inequality?

Solving an Absolute Value Inequality Numerically Work with a partner. Consider the absolute value inequality

∣ x + 2 ∣ ≤ 3. a. Use a spreadsheet, as shown, to solve the absolute value inequality. b. Compare the solutions you found using the spreadsheet with those you found in Explorations 1 and 2. What do you notice? c. How can you use a spreadsheet to solve an absolute value inequality?

1 2 3 4 5 6 7 8 9 10 11

A x -6 -5 -4 -3 -2 -1 0 1 2

B |x + 2| 4

abs(A2 + 2)

Communicate Your Answer 4. How can you solve an absolute value inequality? 5. What do you like or dislike about the algebraic, graphical, and numerical methods

for solving an absolute value inequality? Give reasons for your answers. Section 2.6

Solving Absolute Value Inequalities

87

2.6 Lesson

What You Will Learn Solve absolute value inequalities. Use absolute value inequalities to solve real-life problems.

Core Vocabul Vocabulary larry absolute value inequality, p. 88 absolute deviation, p. 90 Previous compound inequality mean

Solving Absolute Value Inequalities An absolute value inequality is an inequality that contains an absolute value expression. For example, ∣ x ∣ < 2 and ∣ x ∣ > 2 are absolute value inequalities. Recall that ∣ x ∣ = 2 means the distance between x and 0 is 2. The inequality ∣ x ∣ < 2 means the distance between x and 0 is less than 2. −4 −3 −2 −1

0

1

2

3

The inequality ∣ x ∣ > 2 means the distance between x and 0 is greater than 2. 4

−4 −3 −2 −1

The graph of ∣ x ∣ < 2 is the graph of x > −2 and x < 2.

0

1

2

3

4

The graph of ∣ x ∣ > 2 is the graph of x < −2 or x > 2.

You can solve these types of inequalities by solving a compound inequality.

Core Concept Solving Absolute Value Inequalities

To solve ∣ ax + b ∣ < c for c > 0, solve the compound inequality ax + b > − c

and

ax + b < c.

To solve ∣ ax + b ∣ > c for c > 0, solve the compound inequality ax + b < − c

or

ax + b > c.

In the inequalities above, you can replace < with ≤ and > with ≥.

Solving Absolute Value Inequalities Solve each inequality. Graph each solution, if possible. a. ∣ x + 7 ∣ ≤ 2

b. ∣ 8x − 11 ∣ < 0

SOLUTION a. Use ∣ x + 7 ∣ ≤ 2 to write a compound inequality. Then solve. x + 7 ≥ −2

REMEMBER A compound inequality with “and” can be written as a single inequality.

−7

and

−7

x ≥ −9

x+7 ≤ −7

and

2 −7

x ≤ −5

Write a compound inequality. Subtract 7 from each side. Simplify.

The solution is −9 ≤ x ≤ −5. −10 −9 −8 −7 −6 −5 −4 −3 −2

b. By definition, the absolute value of an expression must be greater than or equal to 0. The expression ∣ 8x − 11 ∣ cannot be less than 0. So, the inequality has no solution. 88

Chapter 2

Solving Linear Inequalities

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the inequality. Graph the solution, if possible. 1. ∣ x ∣ ≤ 3.5

3. ∣ 2w − 1 ∣ < 11

2. ∣ k − 3 ∣ < −1

Solving Absolute Value Inequalities Solve each inequality. Graph each solution. a. ∣ c − 1 ∣ ≥ 5

b. ∣ 10 − m ∣ ≥ − 2

c. 4∣ 2x − 5 ∣ + 1 > 21

SOLUTION a. Use ∣ c − 1 ∣ ≥ 5 to write a compound inequality. Then solve. c − 1 ≤ −5 +1

c−1 ≥

or

+1

c ≤ −4

+1

5

Write a compound inequality.

+1

Add 1 to each side.

c≥6

or

Simplify.

The solution is c ≤ −4 or c ≥ 6. −6 −4 −2

0

2

4

6

8

10

b. By definition, the absolute value of an expression must be greater than or equal to 0. The expression ∣ 10 − m ∣ will always be greater than −2. So, all real numbers are solutions. −2

−1

0

1

2

c. First isolate the absolute value expression on one side of the inequality. 4∣ 2x − 5 ∣ + 1 > 21 −1

Write the inequality.

−1

Subtract 1 from each side.

4∣ 2x − 5 ∣ > 20 4∣ 2x − 5 ∣

Simplify.

20 4

— > —

4

Divide each side by 4.

∣ 2x − 5 ∣ > 5 Simplify. Then use ∣ 2x − 5 ∣ > 5 to write a compound inequality. Then solve. 2x − 5 < −5 +5

+5

2x < 2x 2

2x − 5 >

or

+5

0 0 2

— < —

x 10

Simplify.

2x 2

Divide each side by 2.

10 2

— > —

x>5

or

Simplify.

The solution is x < 0 or x > 5. −2 −1

Monitoring Progress

0

1

2

3

4

5

6

Help in English and Spanish at BigIdeasMath.com

Solve the inequality. Graph the solution. 4. ∣ x + 3 ∣ > 8

5. ∣ n + 2 ∣ − 3 ≥ −6

Section 2.6

6. 3∣ d + 1 ∣ − 7 ≥ −1

Solving Absolute Value Inequalities

89

Solving Real-Life Problems The absolute deviation of a number x from a given value is the absolute value of the difference of x and the given value. absolute deviation = ∣ x − given value ∣

Modeling with Mathematics Computer prices

$890

$750

$650

$370

$660

$670

$450

$650

$725

$825

You are buying a new computer. The table shows the prices of computers in a store advertisement. You are willing to pay the mean price with an absolute deviation of at most $100. How many of the computer prices meet your condition?

SOLUTION 1. Understand the Problem You know the prices of 10 computers. You are asked to find how many computers are at most $100 from the mean price. 2. Make a Plan Calculate the mean price by dividing the sum of the prices by the number of prices, 10. Use the absolute deviation and the mean price to write an absolute value inequality. Then solve the inequality and use it to answer the question. 3. Solve the Problem 6640 The mean price is — = $664. Let x represent a price you are willing to pay. 10

∣ x − 664 ∣ ≤ 100

STUDY TIP

−100 ≤ x − 664 ≤ 100

The absolute deviation of at most $100 from the mean, $664, is given by the inequality ∣ x – 664 ∣ ≤ 100.

564 ≤ x ≤ 764

Write the absolute value inequality. Write a compound inequality. Add 664 to each expression and simplify.

The prices you will consider must be at least $564 and at most $764. Six prices meet your condition: $750, $650, $660, $670, $650, and $725.

4. Look Back You can check that your answer is correct by graphing the computer prices and the mean on a number line. Any point within 100 of 664 represents a price that you will consider.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

7. WHAT IF? You are willing to pay the mean price with an absolute deviation of at

most $75. How many of the computer prices meet your condition?

Concept Summary Solving Inequalities One-Step and Multi-Step Inequalities ●

Follow the steps for solving an equation. Reverse the inequality symbol when multiplying or dividing by a negative number.

Compound Inequalities ●

If necessary, write the inequality as two separate inequalities. Then solve each inequality separately. Include and or or in the solution.

Absolute Value Inequalities ●

90

Chapter 2

If necessary, isolate the absolute value expression on one side of the inequality. Write the absolute value inequality as a compound inequality. Then solve the compound inequality.

Solving Linear Inequalities

2.6

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept p Check 1. REASONING Can you determine the solution of ∣ 4x − 2 ∣ ≥ −6 without solving? Explain. 2. WRITING Describe how solving ∣ w − 9 ∣ ≤ 2 is different from solving ∣ w − 9 ∣ ≥ 2.

Monitoring Progress and Modeling with Mathematics ERROR ANALYSIS In Exercises 21 and 22, describe

In Exercises 3 –18, solve the inequality. Graph the solution, if possible. (See Examples 1 and 2.) 3.

∣x∣ < 3

4.

∣ y ∣ ≥ 4.5

5.

∣d + 9∣ > 3

6.

∣ h − 5 ∣ ≤ 10

7.

∣ 2s − 7 ∣ ≥ −1

8.

∣ 4c + 5 ∣ > 7

9.

∣ 5p + 2 ∣ < −4

10.

∣ 9 − 4n ∣ < 5

∣ 6t − 7 ∣ − 8 ≥ 3

12.

∣ 3j − 1 ∣ + 6 > 0

11.

13. 3∣ 14 − m ∣ > 18

and correct the error in solving the absolute value inequality. 21.

|x − 5| < 20 x − 5 < 20 x < 25

22.

14. −4∣ 6b − 8 ∣ ≤ 12

✗

|x + 4| > 13 x + 4 > −13 and x > −17

15. 2∣ 3w + 8 ∣ − 13 ≤ −5

and

x 7

✗

23. A number is less than 6 units from 0. 24. A number is more than 9 units from 3. 25. Half of a number is at most 5 units from 14. 26. Twice a number is no less than 10 units from −1. 27. PROBLEM SOLVING An auto parts manufacturer

throws out gaskets with weights that are not within 0.06 pound of the mean weight of the batch. The weights (in pounds) of the gaskets in a batch are 0.58, 0.63, 0.65, 0.53, and 0.61. Which gasket(s) should be thrown out? 28. PROBLEM SOLVING Six students measure the

acceleration (in meters per second per second) of an object in free fall. The measured values are shown. The students want to state that the absolute deviation of each measured value x from the mean is at most d. Find the value of d. 10.56, 9.52, 9.73, 9.80, 9.78, 10.91 Section 2.6

Solving Absolute Value Inequalities

91

MATHEMATICAL CONNECTIONS In Exercises 29 and 30,

35. MAKING AN ARGUMENT One of your classmates

claims that the solution of ∣ n ∣ > 0 is all real numbers. Is your classmate correct? Explain your reasoning.

write an absolute value inequality that represents the situation. Then solve the inequality. 29. The difference between the areas of the figures is less

than 2.

36. THOUGHT PROVOKING Draw and label a geometric

gure so that the perimeter P of the gure is a solution of the inequality ∣ P − 60 ∣ ≤ 12.

x+6

37. REASONING What is the solution of the inequality

∣ ax + b ∣ < c, where c < 0? What is the solution of the inequality ∣ ax + b ∣ > c, where c < 0? Explain.

6

4

2

38. HOW DO YOU SEE IT? Write an absolute value

inequality for each graph.

30. The difference between the perimeters of the figures is

less than or equal to 3.

x

3 x+1

x

REASONING In Exercises 31–34, tell whether the statement is true or false. If it is false, explain why.

−4 −3 −2 −1

0

1

2

3

4

5

6

−4 −3 −2 −1

0

1

2

3

4

5

6

−4 −3 −2 −1

0

1

2

3

4

5

6

−4 −3 −2 −1

0

1

2

3

4

5

6

How did you decide which inequality symbol to use for each inequality?

31. If a is a solution of ∣ x + 3 ∣ ≤ 8, then a is also a

solution of x + 3 ≥ −8.

32. If a is a solution of ∣ x + 3 ∣ > 8, then a is also a

solution of x + 3 > 8.

39. WRITING Explain why the solution set of the

33. If a is a solution of ∣ x + 3 ∣ ≥ 8, then a is also a

inequality ∣ x ∣ < 5 is the intersection of two sets, while the solution set of the inequality ∣ x ∣ > 5 is the union of two sets.

34. If a is a solution of x + 3 ≤ −8, then a is also a

40. PROBLEM SOLVING Solve the compound inequality

solution of x + 3 ≥ −8. solution of ∣ x + 3 ∣ ≥ 8.

below. Describe your steps.

∣ x − 3 ∣ < 4 and ∣ x + 2 ∣ > 8

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Plot the ordered pair in a coordinate plane. Describe the location of the point. (Skills Review Handbook) 42. B(0, −3)

41. A(1, 3)

43. C(−4, −2)

44. D(−1, 2)

Copy and complete the table. (Skills Review Handbook) 45.

x

0

1

2

3

4

Chapter 2

x −2x − 3

5x + 1

92

46.

Solving Linear Inequalities

−2

−1

0

1

2

2.5–2.6

What Did You Learn?

Core Vocabulary compound inequality, p. 82 absolute value inequality, p. 88 absolute deviation, p. 90

Core Concepts Section 2.5 Writing and Graphing Compound Inequalities, p. 82 Solving Compound Inequalities, p. 83

Section 2.6 Solving Absolute Value Inequalities, p. 88

Mathematical Practices 1.

How can you use a diagram to help you solve Exercise 12 on page 85?

2.

In Exercises 13 and 14 on page 85, how can you use structure to break down the compound inequality into two inequalities?

3.

Describe the given information and the overall goal of Exercise 27 on page 91.

4.

For false statements in Exercises 31–34 on page 92, use examples to show the statements are false.

Performance Task:

Designing for Electricity How many watts do the electrical appliances in your home use? What would happen if all of your appliances were on at the same time? Designers use inequalities to calculate the electrical needs of a home. What information about watts would you need to calculate the electrical load when planning electrical circuits for your dream home? To explore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com. 93

2

Chapter Review 2.1

Dynamic Solutions available at BigIdeasMath.com

Writing and Graphing Inequalities (pp. 53–60)

a. A number x plus 36 is no more than 40. Write this sentence as an inequality. A number x plus 36 is no more than 40. ≤

x + 36

40

An inequality is x + 36 ≤ 40. b. Graph w > −3. Test a number to the left of −3.

w = −4 is not a solution.

Test a number to the right of −3.

w = 0 is a solution. Use an open circle because −3 is not a solution.

−6

−5

−4

−3

−2

−1

0

1

2

3

4

Shade the number line on the side where you found a solution.

Write the sentence as an inequality. 1. A number d minus 2 is less than −1. 2. Ten is at least the product of a number h and 5.

Graph the inequality. 4. y ≤ 2

3. x > 4

2.2

5. −1 ≥ z

Solving Inequalities Using Addition or Subtraction (pp. 61–66)

Solve x + 2.5 ≤ −6. Graph the solution. x + 2.5 ≤ −6 Subtraction Property of Inequality

− 2.5

Write the inequality.

− 2.5

Subtract 2.5 from each side.

x ≤ −8.5

Simplify.

The solution is x ≤ −8.5. x ≤ – 8.5 −12 −11 −10

−9

−8

−7

−6

−5

−4

−3

−2

Solve the inequality. Graph the solution. 6. p + 4 < 10

94

Chapter 2

Solving Linear Inequalities

7. r − 4 < −6

8. 2.1 ≥ m − 6.7

2.3

Solving Inequalities Using Multiplication or Division (pp. 67–72)

n Solve — > 5. Graph the solution. −10 n — > 5 −10

⋅

Write the inequality.

⋅

n −10 — < −10 5 −10

Multiplication Property of Inequality

Multiply each side by −10. Reverse the inequality symbol.

n < −50

Simplify.

The solution is n < −50. n < –50 −80 −70

−60 −50 −40 −30

−20 −10

0

10

20

Solve the inequality. Graph the solution.

s −8

12. — ≥ 11

2.4

3 4

g 5

9. 3x > −21

10. −4 ≤ —

11. −—n ≤ 3

13. 36 < 2q

14. −1.2k > 6

Solving Multi-Step Inequalities (pp. 73–78)

Solve 22 + 3y ≥ 4. Graph the solution. 22 + 3y ≥ −22

4

Write the inequality.

−22

Subtract 22 from each side.

3y ≥ −18 3y 3

Simplify.

−18 3

— ≥ —

Divide each side by 3.

y ≥ −6

Simplify.

The solution is y ≥ −6. y ≥ –6 −9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

Solve the inequality. Graph the solution, if possible. b 2

15. 3x − 4 > 11

16. −4 < — + 9

17. 7 − 3n ≤ n + 3

18. 2(−4s + 2) ≥ −5s − 10

19. 6(2t + 9) ≤ 12t − 1

20. 3r − 8 > 3(r − 6)

Chapter 2

Chapter Review

95

2.5

Solving Compound Inequalities (pp. 81–86)

Solve −1 ≤ −2d + 7 ≤ 9. Graph the solution. −1 ≤ −2d + 7 ≤ −7

−7

9

Write the inequality.

−7

Subtract 7 from each expression.

−8 ≤ −2d

≤

−8 −2

−2d −2

2 ≥ — −2

Divide each expression by −2. Reverse each inequality symbol.

d

≥ −1

Simplify.

— ≥ —

4≥

2

Simplify.

The solution is −1 ≤ d ≤ 4. −3 −2 −1

0

1

2

3

4

5

21. A number x is more than −6 and at most 8. Write this sentence as an inequality.

Graph the inequality. Solve the inequality. Graph the solution. r 4

22. 19 ≥ 3z + 1 ≥ −5

2.6

23. — < −5 or −2r − 7 ≤ 3

Solving Absolute Value Inequalities (pp. 87–92)

Solve ∣ 2x + 11 ∣ + 3 > 8. Graph the solution.

∣ 2x + 11 ∣ + 3 > 8 −3

Write the inequality.

−3

Subtract 3 from each side.

∣ 2x + 11 ∣ > 5 2x + 11 < −5 − 11

or

− 11

Simplify.

2x + 11 > − 11

5 − 11

2x < −16

2x > −6

−16 2

— > —

2x 2

2x 2

— < —

x < −8

or

Write a compound inequality. Subtract 11 from each side. Simplify.

−6 2

Divide each side by 2.

x > −3

Simplify.

The solution is x < −8 or x > −3. −9 −8 −7 −6 −5 −4 −3 −2 −1

Solve the inequality. Graph the solution, if possible. 24. 27.

∣ m ∣ ≥ 10 5∣ b + 8 ∣ − 7 > 13

25. 28.

∣ k − 9 ∣ < −4 ∣ −3g − 2 ∣ + 1 < 6

26. 4∣ f − 6 ∣ ≤ 12 29. ∣ 9 − 2j ∣ + 10 ≥ 2

30. A safety regulation states that the height of a guardrail should be 106 centimeters

with an absolute deviation of no more than 7 centimeters. Write and solve an absolute value inequality that represents the acceptable heights of a guardrail.

96

Chapter 2

Solving Linear Inequalities

2

Chapter Test

Write the sentence as an inequality. 1. The sum of a number y and 9 is at least −1. 2. A number r is more than 0 or less than or equal to −8. 3. A number k is less than 3 units from 10.

Solve the inequality. Graph the solution, if possible. x 2

4. — − 5 ≥ −9

5. −4s < 6s + 1

6. 4p + 3 ≥ 2(2p + 1)

7. −7 < 2c − 1 < 10

8. −2 ≤ 4 − 3a ≤ 13

9. −5 < 2 − h or 6h + 5 > 71

10.

∣ 2q + 8 ∣ > 4

11. −2∣ y − 3 ∣ − 5 ≥ −4

12. 4∣ −3b + 5 ∣ − 9 < 7

13. You start a small baking business, and you want to earn a profit of at least

$250 in the first month. The expenses in the first month are $155. What are the possible revenues that you need to earn to meet the profit goal? 14. A manufacturer of bicycle parts requires that a bicycle chain have a width

of 0.3 inch with an absolute deviation of at most 0.0003 inch. Write and solve an absolute value inequality that represents the acceptable widths. 15. Let a, b, c, and d be constants. Describe the possible solution sets of the

inequality ax + b < cx + d.

Write and graph a compound inequality that represents the numbers that are not solutions of the inequality represented by the graph shown. Explain your reasoning. 16. −4 −3 −2 −1

0

1

2

3

4

0

1

2

17. −6 −5 −4 −3 −2 −1

18. A state imposes a sales tax on items of clothing that cost more than $175. The tax applies

only to the difference of the price of the item and $175.

a. Use the receipt shown to find the tax rate (as a percent). b. A shopper has $430 to spend on a winter coat. Write and solve an inequality to find the prices p of coats that the shopper can afford. Assume that p ≥ 175. c. Another state imposes a 5% sales tax on the entire price of an item of clothing. For which prices would paying the 5% tax be cheaper than paying the tax described above? Write and solve an inequality to find your answer and list three prices that are solutions.

The

TYLE store

PURCHASE DATE: 03/29/14 STORE#: 1006

ITEM: SUIT PRICE: TAX: TOTAL:

Chapter 2

$295.00 $ 7.50 $302.50

Chapter Test

97

2

Cumulative Assessment

1. The expected attendance at a school event is 65 people. The actual attendance can vary

by up to 30 people. Which equation can you use to find the minimum and maximum attendances?

A ∣ x − 65 ∣ = 30 ○ C ∣ x − 30 ∣ = 65 ○

B ∣ x + 65 ∣ = 30 ○ D ∣ x + 30 ∣ = 65 ○

2. Fill in values for a and b so that each statement is true for the inequality

ax + 4 ≤ 3x + b.

a. When a = 5 and b = _____, x ≤ −3. b. When a = _____ and b = _____, the solution of the inequality is all real numbers. c. When a = _____ and b = _____, the inequality has no solution. 3. Place each inequality into one of the two categories. At least one integer solution

No integer solutions

5x − 6 + x ≥ 2x − 8

x − 8 + 4x ≤ 3(x − 3) + 2x

2(3x + 8) > 3(2x + 6)

9x − 3 < 12 or 6x + 2 > −10

17 < 4x + 5 < 21

5(x − 1) ≤ 5x − 3

4. Admission to a play costs $25. A season pass costs $180.

a. Write an inequality that represents the numbers x of plays you must attend for the season pass to be a better deal. b. Select the numbers of plays for which the season pass is not a better deal.

98

Chapter 2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Solving Linear Inequalities

5. Select the values of a that make the solution of the equation 3(2x − 4) = 4(ax − 2)

positive. −2

−1

0

1

3

2

4

5

6. Fill in the compound inequality with so the solution is shown in

the graph. 4x − 18

−5

−4

−3

−x − 3 and −3x − 9

−2

−1

0

1

−3

2

3

4

5

7. You have a $250 gift card to use at a sporting goods store.

LE

SA

$ 12

LE

SA

$ 80

a. Write an inequality that represents the possible numbers x of pairs of socks you can buy when you buy 2 pairs of sneakers. Can you buy 8 pairs of socks? Explain. b. Describe what the inequality 60 + 80x ≤ 250 represents in this context. 8. Consider the equation shown, where a, b, c, and d are integers.

ax + b = cx + d Student A claims the equation will always have one solution. Student B claims the equation will always have no solution. Use the numbers shown to answer parts (a)–(c). −1

0

1

2

3

4

5

6

a. Select values for a, b, c, and d to create an equation that supports Student A’s claim. b. Select values for a, b, c, and d to create an equation that supports Student B’s claim. c. Select values for a, b, c, and d to create an equation that shows both Student A and Student B are incorrect. Chapter 2

Cumulative Assessment

99

3 Graphing Linear Functions 3.1 3.2 3.3 3.4 3.5 3.6

Functions Linear Functions Function Notation Graphing Linear Equations in Standard Form Graphing Linear Equations in Slope-Intercept Form Transformations of Graphs of Linear Functions

Submersible (p. 140)

SEE the Big Idea Basketball (p. 134)

Speed S d off Light Li ht (p. ( 125)

Coins (p. 116)

Taxi Ride (p. 109)

Maintaining Mathematical Proficiency Plotting Points Example 1

Plot the point A (−3, 4) in a coordinate plane. Describe the location of the point.

Start at the origin. Move 3 units left and 4 units up. Then plot the point. The point is in Quadrant II. A(−3, 4)

4

4

y

2

−3 −4

−2

2

4 x

−2 −4

Plot the point in a coordinate plane. Describe the location of the point. 1. A(3, 2)

2. B(−5, 1)

3. C(0, 3)

4. D(−1, −4)

5. E(−3, 0)

6. F(2, −1)

Evaluating Expressions Example 2

Evaluate 4x − 5 when x = 3. 4x − 5 = 4(3) − 5

Example 3

Substitute 3 for x.

= 12 − 5

Multiply.

=7

Subtract.

Evaluate −2x + 9 when x = −8. −2x + 9 = −2(−8) + 9

Substitute −8 for x.

= 16 + 9

Multiply.

= 25

Add.

Evaluate the expression for the given value of x. 7. 3x − 4; x = 7 10. −9x − 2; x = −4

8. −5x + 8; x = 3

9. 10x + 18; x = 5

11. 24 − 8x; x = −2

12. 15x + 9; x = −1

13. ABSTRACT REASONING Let a and b be positive real numbers. Describe how to plot

(a, b), (−a, b), (a, −b), and (−a, −b).

Dynamic Solutions available at BigIdeasMath.com

101

Mathematical Practices

Mathematically proficient students use technological tools to explore concepts.

Using a Graphing Calculator

Core Concept Standard and Square Viewing Windows A typical graphing calculator screen has a height to width ratio of 2 to 3. This means that when you use the standard viewing window of −10 to 10 (on each axis), the graph will not be in its true perspective. To see a graph in its true perspective, you need to use a square viewing window, in which the tick marks on the x-axis are spaced the same as the tick marks on the y-axis.

WINDOW Xmin=-10 Xmax=10 Xscl=1 Ymin=-10 Ymax=10 Yscl=1

This is the standard viewing window.

WINDOW Xmin=-9 Xmax=9 Xscl=1 Ymin=-6 Ymax=6 Yscl=1

This is a square viewing window.

Using a Graphing Calculator Use a graphing calculator to graph y = 2x + 5.

This is the graph in the standard viewing window.

10

SOLUTION

y = 2x + 5

Enter the equation y = 2x + 5 into your calculator. Then graph the equation. The standard viewing window does not show the graph in its true perspective. Notice that the tick marks on the y-axis are closer together than the tick marks on the x-axis. To see the graph in its true perspective, use a square viewing window.

−10 0

10

−10 7

This is the graph in a square viewing window.

y = 2x + 5

−7

5 −1

Monitoring Progress Determine whether the viewing window is square. Explain. 1. −8 ≤ x ≤ 7, −3 ≤ y ≤ 7

2. −6 ≤ x ≤ 6, −9 ≤ y ≤ 9 102

3. −18 ≤ x ≤ 18, −12 ≤ y ≤ 12

Use a graphing calculator to graph the equation. Use a square viewing window. 4. y = x + 3

5. y = −x − 2

7. y = −2x + 1

8. y =

1 −—3 x

−4

6. y = 2x − 1 1

9. y = —2 x + 2

10. How does the appearance of the slope of a line change between a standard viewing window and a

square viewing window?

102

Chapter 3

Graphing Linear Functions

3.1

Functions Essential Question

What is a function?

A relation pairs inputs with outputs. When a relation is given as ordered pairs, the x-coordinates are inputs and the y-coordinates are outputs. A relation that pairs each input with exactly one output is a function.

Describing a Function Work with a partner. Functions can be described in many ways. M M M M

ANALYZING RELATIONSHIPS To be proficient in math, you need to analyze relationships mathematically to draw conclusions.

M

by an equation by an input-output table using words by a graph as a set of ordered pairs

y 8 6 4

a. Explain why the graph shown represents a function. b. Describe the function in two other ways.

2 0

0

2

6

4

8

x

Identifying Functions Work with a partner. Determine whether each relation represents a function. Explain your reasoning. a.

b.

Input, x

0

1

2

3

4

Output, y

8

8

8

8

8

Input, x

8

8

8

8

8

Output, y

0

1

2

3

4

c. Input, x 1 2 3

Output, y

d.

8 9 10 11

y 8 6 4 2

e. (− 2, 5), (−1, 8), (0, 6), (1, 6), (2, 7)

0

0

2

4

6

8

x

f. (−2, 0), (−1, 0), (−1, 1), (0, 1), (1, 2), (2, 2) g. Each radio frequency x in a listening area has exactly one radio station y. h. The same television station x can be found on more than one channel y. i. x = 2 j. y = 2x + 3

Communicate Your Answer 3. What is a function? Give examples of relations, other than those in

Explorations 1 and 2, that (a) are functions and (b) are not functions. Section 3.1

Functions

103

3.1

Lesson

What You Will Learn Determine whether relations are functions. Find the domain and range of a function.

Core Vocabul Vocabulary larry

Identify the independent and dependent variables of functions.

relation, p. 104 function, p. 104 domain, p. 106 range, p. 106 independent variable, p. 107 dependent variable, p. 107 Previous ordered pair mapping diagram

Determining Whether Relations Are Functions A relation pairs inputs with outputs. When a relation is given as ordered pairs, the x-coordinates are inputs and the y-coordinates are outputs. A relation that pairs each input with exactly one output is a function.

Determining Whether Relations Are Functions Determine whether each relation is a function. Explain. a. (−2, 2), (−1, 2), (0, 2), (1, 0), (2, 0) b. (4, 0), (8, 7), (6, 4), (4, 3), (5, 2) c.

Output, y

REMEMBER A relation can be represented by a mapping diagram.

Input, x

d. Input, x

−2

−1

0

0

1

2

3

4

5

6

7

8

Output, y

−1 3 11

4 15

SOLUTION a. Every input has exactly one output. So, the relation is a function.

b. The input 4 has two outputs, 0 and 3. So, the relation is not a function.

c. The input 0 has two outputs, 5 and 6. So, the relation is not a function.

d. Every input has exactly one output. So, the relation is a function.

Monitoring Progress

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Determine whether the relation is a function. Explain.

104

Chapter 3

1. (−5, 0), (0, 0), (5, 0), (5, 10)

2. (−4, 8), (−1, 2), (2, −4), (5, −10)

3.

4. Input, x

Input, x

Output, y

2

2.6

4

5.2

6

7.8

Graphing Linear Functions

1 — 2

Output, y −2 0 4

Core Concept Vertical Line Test Words

A graph represents a function when no vertical line passes through more than one point on the graph.

Examples

Function

Not a function y

y

x

x

Using the Vertical Line Test Determine whether each graph represents a function. Explain. a.

b.

y

y

4

4

2

2

0

0

2

4

6

0

x

0

2

4

6

x

SOLUTION a. You can draw a vertical line through (2, 2) and (2, 5).

b. No vertical line can be drawn through more than one point on the graph.

So, the graph does not represent a function.

So, the graph represents a function.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Determine whether the graph represents a function. Explain. 5.

y 6

y 6

4

4

2

2

0

7.

6.

0

2

4

6

0

x

y 6

8.

4

2

2

0

2

4

6

x

2

4

6

x

0

2

4

6

x

y 6

4

0

0

0

Section 3.1

Functions

105

Finding the Domain and Range of a Function

Core Concept The Domain and Range of a Function The domain of a function is the set of all possible input values. The range of a function is the set of all possible output values.

input

−2 −6

output

Finding the Domain and Range from a Graph Find the domain and range of the function represented by the graph. a.

b.

y

4

y 2

2 −3

−2

2

x

1

3x

−2

−2

SOLUTION a. Write the ordered pairs. Identify the inputs and outputs.

b. Identify the x- and y-values represented by the graph. y

inputs 2

(−3, −2), (−1, 0), (1, 2), (3, 4)

STUDY TIP

range −3

outputs

A relation also has a domain and a range.

3x

−2

The domain is −3, −1, 1, and 3. The range is −2, 0, 2, and 4.

Monitoring Progress

1

domain

The domain is −2 ≤ x ≤ 3. The range is −1 ≤ y ≤ 2.

Help in English and Spanish at BigIdeasMath.com

Find the domain and range of the function represented by the graph. 9.

10.

y

y

4

6

2

4

−2

2

x

2 0

106

Chapter 3

Graphing Linear Functions

0

2

4

6

x

Identifying Independent and Dependent Variables The variable that represents the input values of a function is the independent variable because it can be any value in the domain. The variable that represents the output values of a function is the dependent variable because it depends on the value of the independent variable. When an equation represents a function, the dependent variable is defined in terms of the independent variable. The statement “y is a function of x” means that y varies depending on the value of x. y = −x + 10 independent variable, x

dependent variable, y

Identifying Independent and Dependent Variables The function y = −3x + 12 represents the amount y (in fluid ounces) of juice remaining in a bottle after you take x gulps. a. Identify the independent and dependent variables. b. The domain is 0, 1, 2, 3, and 4. What is the range?

SOLUTION a. The amount y of juice remaining depends on the number x of gulps. So, y is the dependent variable, and x is the independent variable. b. Make an input-output table to find the range. Input, x

−3x + 12

Output, y

0

−3(0) + 12

12

1

−3(1) + 12

9

2

−3(2) + 12

6

3

−3(3) + 12

3

4

−3(4) + 12

0

The range is 12, 9, 6, 3, and 0.

Monitoring Progress

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11. The function a = −4b + 14 represents the number a of avocados you have left

after making b batches of guacamole. a. Identify the independent and dependent variables. b. The domain is 0, 1, 2, and 3. What is the range? 12. The function t = 19m + 65 represents the temperature t (in degrees Fahrenheit)

of an oven after preheating for m minutes. a. Identify the independent and dependent variables. b. A recipe calls for an oven temperature of 350°F. Describe the domain and range of the function.

Section 3.1

Functions

107

Exercises

3.1

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING How are independent variables and dependent variables different? 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

Find the inputs of the function represented by the table.

Find the range of the function represented by the table.

Find the x-values of the function represented by (−1, 7), (0, 5), and (1, −1).

x

−1

0

1

y

7

5

−1

Find the domain of the function represented by (−1, 7), (0, 5), and (1, −1).

Monitoring Progress and Modeling with Mathematics In Exercises 3– 8, determine whether the relation is a function. Explain. (See Example 1.)

11.

7.

8.

Output, y

0 1 2 3

−3 0 3 2 1

6.

Input, x

108

1

2

Output, y

−10 −8 −6 −4 −2

0

2

0

2

6

4

13.

14.

y

1

0

1

16

Output, y

−2

−1

0

1

2

Input, x

−3

0

3

6

9

Output, y

11

5

−1

−7

−13

10.

y

6

4

4

2

2

0

0

2

4

Chapter 3

6

x

y

2

2

x

−2

15.

−2

6

16.

y

4x

2

y 6

4 4 2 −4

−2

2x

0

0

2

4

6

x

y

6

0

7x

4

−2

16

5

x

2

Input, x

3

−2

In Exercises 13–16, find the domain and range of the function represented by the graph. (See Example 3.)

1

In Exercises 9 –12, determine whether the graph represents a function. Explain. (See Example 2.) 9.

2

4

4. (7, 4), (5, −1), (3, −8), (1, −5), (3, 6)

Input, x

y

6

3. (1, −2), (2, 1), (3, 6), (4, 13), (5, 22)

5.

12.

y

17. MODELING WITH MATHEMATICS The function

y = 25x + 500 represents your monthly rent y (in dollars) when you pay x days late. (See Example 4.)

a. Identify the independent and dependent variables. 0

2

4

6

x

Graphing Linear Functions

b. The domain is 0, 1, 2, 3, 4, and 5. What is the range?

18. MODELING WITH MATHEMATICS The function

y = 3.5x + 2.8 represents the cost y (in dollars) of a taxi ride of x miles.

24. MULTIPLE REPRESENTATIONS The function

1.5x + 0.5y = 12 represents the number of hardcover books x and softcover books y you can buy at a used book sale. a. Solve the equation for y. b. Make an input-output table to find ordered pairs for the function. c. Plot the ordered pairs in a coordinate plane.

25. ATTENDING TO PRECISION The graph represents a

function. Find the input value corresponding to an output of 2. a. Identify the independent and dependent variables.

y

b. You have enough money to travel at most 20 miles in the taxi. Find the domain and range of the function.

2 −2

2

x

−2

ERROR ANALYSIS In Exercises 19 and 20, describe and

correct the error in the statement about the relation shown in the table. 26. OPEN-ENDED Fill in the table so that when t is the

19.

20.

Input, x

1

2

3

4

5

Output, y

6

7

8

6

9

 

The relation is not a function. One output is paired with two inputs.

independent variable, the relation is a function, and when t is the dependent variable, the relation is not a function. t v 27. ANALYZING RELATIONSHIPS You select items in a

vending machine by pressing one letter and then one number.

The relation is a function. The range is 1, 2, 3, 4, and 5.

ANALYZING RELATIONSHIPS In Exercises 21 and 22, identify the independent and dependent variables. 21. The number of quarters you put into a parking meter

affects the amount of time you have on the meter. 22. The battery power remaining on your MP3 player is

based on the amount of time you listen to it. 23. MULTIPLE REPRESENTATIONS The balance

y (in dollars) of your savings account is a function of the month x. Month, x Balance (dollars), y

0

1

2

3

4

100

125

150

175

200

a. Describe this situation in words.

a. Explain why the relation that pairs letter-number combinations with food or drink items is a function.

b. Write the function as a set of ordered pairs.

b. Identify the independent and dependent variables.

c. Plot the ordered pairs in a coordinate plane.

c. Find the domain and range of the function. Section 3.1

Functions

109

28. HOW DO YOU SEE IT? The graph represents the

34. A function pairs each chaperone on a school trip

height h of a projectile after t seconds.

with 10 students.

Height (feet)

Height of a Projectile h 25 20 15 10 5 0

REASONING In Exercises 35–38, tell whether the

statement is true or false. If it is false, explain why. 35. Every function is a relation. 36. Every relation is a function. 0

0.5

1

1.5

2

2.5

t

37. When you switch the inputs and outputs of any

Time (seconds)

function, the resulting relation is a function.

a. Explain why h is a function of t.

38. When the domain of a function has an infinite number

of values, the range always has an infinite number of values.

b. Approximate the height of the projectile after 0.5 second and after 1.25 seconds. c. Approximate the domain of the function.

39. MATHEMATICAL CONNECTIONS Consider the

triangle shown.

d. Is t a function of h? Explain.

29. MAKING AN ARGUMENT Your friend says that a line

13

h

always represents a function. Is your friend correct? Explain. 30. THOUGHT PROVOKING Write a function in which

10

the inputs and/or the outputs are not numbers. Identify the independent and dependent variables. Then find the domain and range of the function.

a. Write a function that represents the perimeter of the triangle. b. Identify the independent and dependent variables.

ATTENDING TO PRECISION In Exercises 31–34,

c. Describe the domain and range of the function. (Hint: The sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.)

determine whether the statement uses the word function in a way that is mathematically correct. Explain your reasoning. 31. The selling price of an item is a function of the cost of

making the item.

REASONING In Exercises 40–43, find the domain and

range of the function.

32. The sales tax on a purchased item in a given state is a

function of the selling price. 33. A function pairs each student in your school with a

40. y = ∣ x ∣

41. y = −∣ x ∣

42. y = ∣ x ∣ − 6

43. y = 4 − ∣ x ∣

homeroom teacher.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Write the sentence as an inequality. (Section 2.1) 44. A number y is less than 16.

45. Three is no less than a number x.

46. Seven is at most the quotient of a number d and −5. 47. The sum of a number w and 4 is more than −12.

Evaluate the expression. (Skills Review Handbook) 48.

110

112 Chapter 3

49.

(−3)4

Graphing Linear Functions

50.

−52

51.

25

3.2

Linear Functions Essential Question

How can you determine whether a function is

linear or nonlinear?

Finding Patterns for Similar Figures Work with a partner. Copy and complete each table for the sequence of similar figures. (In parts (a) and (b), use the rectangle shown.) Graph the data in each table. Decide whether each pattern is linear or nonlinear. Justify your conclusion. a. perimeters of similar rectangles x

1

x

2

3

b. areas of similar rectangles

4

5

1

x

P

A

P

A

40

40

30

30

20

20

10

10

2

3

4

6

8

5

2x

CONNECTIONS TO GEOMETRY You will learn more about perimeters and areas of similar figures in a future course.

0

0

2

4

6

8

0

x

c. circumferences of circles of radius r 1

r

USING TOOLS STRATEGICALLY To be proficient in math, you need to identify relationships using tools, such as tables and graphs.

2

3

4

5

C

A

40

80

30

60

20

40

10

20

2

4

6

8

1

r A

0

2

4

x

d. areas of circles of radius r

C

0

0

0

r

0

2

2

4

3

4

6

8

5

r

Communicate Your Answer 2. How do you know that the patterns you found in Exploration 1 represent

functions? 3. How can you determine whether a function is linear or nonlinear? 4. Describe two real-life patterns: one that is linear and one that is nonlinear.

Use patterns that are different from those described in Exploration 1. Section 3.2

Linear Functions

111

3.2 Lesson

What You Will Learn Identify linear functions using graphs, tables, and equations. Graph linear functions using discrete and continuous data.

Core Vocabul Vocabulary larry

Write real-life problems to fit data.

linear equation in two variables, p. 112 linear function, p. 112 nonlinear function, p. 112 solution of a linear equation in two variables, p. 114 discrete domain, p. 114 continuous domain, p. 114

Identifying Linear Functions A linear equation in two variables, x and y, is an equation that can be written in the form y = mx + b, where m and b are constants. The graph of a linear equation is a line. Likewise, a linear function is a function whose graph is a nonvertical line. A linear function has a constant rate of change and can be represented by a linear equation in two variables. A nonlinear function does not have a constant rate of change. So, its graph is not a line.

Previous whole number

Identifying Linear Functions Using Graphs Does the graph represent a linear or nonlinear function? Explain. a. 3

b.

y

3

y

1

1 −2

2

−2

x

2

x

−3

−3

SOLUTION a. The graph is not a line.

b. The graph is a line.

So, the function is nonlinear.

So, the function is linear.

Identifying Linear Functions Using Tables Does the table represent a linear or nonlinear function? Explain. a.

x

3

6

9

12

y

36

30

24

18

b.

x

1

3

5

7

y

2

9

20

35

SOLUTION +3

+3

a.

REMEMBER A constant rate of change describes a quantity that changes by equal amounts over equal intervals.

+3

x

3

6

9

12

y

36

30

24

18

−6

−6

+2

+2

−6

As x increases by 3, y decreases by 6. The rate of change is constant. So, the function is linear.

b.

+2

x

1

3

5

7

y

2

9

20

35

+7

+ 11

+ 15

As x increases by 2, y increases by different amounts. The rate of change is not constant. So, the function is nonlinear.

112

Chapter 3

Graphing Linear Functions

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Does the graph or table represent a linear or nonlinear function? Explain. 1.

2.

y

3

y

2 1 −2

2

−2

x

2

x

−2 −3

3.

x

0

1

2

3

y

3

5

7

9

4.

x

1

2

3

4

y

16

8

4

2

Identifying Linear Functions Using Equations Which of the following equations represent linear functions? Explain. 2 — y = 3.8, y = √ x , y = 3x, y = —, y = 6(x − 1), and x 2 − y = 0 x

SOLUTION 2 — You cannot rewrite the equations y = √x , y = 3x, y = —, and x 2 − y = 0 in the form x y = mx + b. So, these equations cannot represent linear functions. You can rewrite the equation y = 3.8 as y = 0x + 3.8 and the equation y = 6(x − 1) as y = 6x − 6. So, they represent linear functions.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Does the equation represent a linear or nonlinear function? Explain. 3x 5

5. y = x + 9

6. y = —

7. y = 5 − 2x 2

Concept Summary Representations of Functions Words

An output is 3 more than the input.

Equation

y=x+3

Input-Output Table Input, x

Output, y

−1

2

0

3

1

4

2

5

Mapping Diagram

Input, x

Output, y

−1 0 1 2

2 3 4 5

Graph 6 4 2

−2

Section 3.2

y

2

Linear Functions

4 x

113

Graphing Linear Functions A solution of a linear equation in two variables is an ordered pair (x, y) that makes the equation true. The graph of a linear equation in two variables is the set of points (x, y) in a coordinate plane that represents all solutions of the equation. Sometimes the points are distinct, and other times the points are connected.

Core Concept Discrete and Continuous Domains A discrete domain is a set of input values that consists of only certain numbers in an interval. Example: Integers from 1 to 5

−2 −1

0

1

2

3

4

5

6

A continuous domain is a set of input values that consists of all numbers in an interval. Example: All numbers from 1 to 5

−2 −1

0

1

2

3

4

5

6

Graphing Discrete Data The linear function y = 15.95x represents the cost y (in dollars) of x tickets for a museum. Each customer can buy a maximum of four tickets.

STUDY TIP

a. Find the domain of the function. Is the domain discrete or continuous? Explain.

The domain of a function depends on the real-life context of the function, not just the equation that represents the function.

b. Graph the function using its domain.

SOLUTION a. You cannot buy part of a ticket, only a certain number of tickets. Because x represents the number of tickets, it must be a whole number. The maximum number of tickets a customer can buy is four. So, the domain is 0, 1, 2, 3, and 4, and it is discrete. b. Step 1 Make an input-output table to find the ordered pairs.

Cost (dollars)

Museum Tickets y 70 60 50 40 30 20 10 0

(4, 63.8) (3, 47.85) (2, 31.9) (1, 15.95) (0, 0) 0 1 2 3 4 5 6 x

Number of tickets

Input, x

15.95x

Output, y

(x, y)

0

15.95(0)

0

(0, 0)

1

15.95(1)

15.95

(1, 15.95)

2

15.95(2)

31.9

(2, 31.9)

3

15.95(3)

47.85

(3, 47.85)

4

15.95(4)

63.8

(4, 63.8)

Step 2 Plot the ordered pairs. The domain is discrete. So, the graph consists of individual points.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

8. The linear function m = 50 − 9d represents the amount m (in dollars) of money

you have after buying d DVDs. (a) Find the domain of the function. Is the domain discrete or continuous? Explain. (b) Graph the function using its domain.

114

Chapter 3

Graphing Linear Functions

Graphing Continuous Data A cereal bar contains 130 calories. The number c of calories consumed is a function of the number b of bars eaten. a. Does this situation represent a linear function? Explain. b. Find the domain of the function. Is the domain discrete or continuous? Explain.

STUDY TIP When the domain of a linear function is not specified or cannot be obtained from a real-life context, it is understood to be all real numbers.

c. Graph the function using its domain.

SOLUTION a. As b increases by 1, c increases by 130. The rate of change is constant. So, this situation represents a linear function. b. You can eat part of a cereal bar. The number b of bars eaten can be any value greater than or equal to 0. So, the domain is b ≥ 0, and it is continuous. c. Step 1 Make an input-output table to find ordered pairs.

Calories consumed

Cereal Bar Calories c 700 600 500 400 300 200 100 0

(4, 520) (3, 390) (2, 260) (1, 130) (0, 0) 0 1 2 3 4 5 6b

Number of bars eaten

Input, b

Output, c

(b, c)

0

0

(0, 0)

1

130

(1, 130)

2

260

(2, 260)

3

390

(3, 390)

4

520

(4, 520)

Step 2 Plot the ordered pairs. Step 3 Draw a line through the points. The line should start at (0, 0) and continue to the right. Use an arrow to indicate that the line continues without end, as shown. The domain is continuous. So, the graph is a line with a domain of b ≥ 0.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

9. Is the domain discrete or continuous? Explain. Input Number of stories, x

1

2

3

Output Height of building (feet), y

12

24

36

10. A 20-gallon bathtub is draining at a rate of 2.5 gallons per minute. The number g

of gallons remaining is a function of the number m of minutes. a. Does this situation represent a linear function? Explain. b. Find the domain of the function. Is the domain discrete or continuous? Explain. c. Graph the function using its domain.

Section 3.2

Linear Functions

115

Writing Real-Life Problems Writing Real-Life Problems Write a real-life problem to fit the data shown in each graph. Is the domain of each function discrete or continuous? Explain. a.

b.

y

8

8

6

6

4

4

2

2

4

2

6

y

4

2

8 x

6

8 x

SOLUTION a. You want to think of a real-life situation in which there are two variables, x and y. Using the graph, notice that the sum of the variables is always 6, and the value of each variable must be a whole number from 0 to 6. x

0

1

2

3

4

5

6

y

6

5

4

3

2

1

0

Discrete domain

One possibility is two people bidding against each other on six coins at an auction. Each coin will be purchased by one of the two people. Because it is not possible to purchase part of a coin, the domain is discrete. b. You want to think of a real-life situation in which there are two variables, x and y. Using the graph, notice that the sum of the variables is always 6, and the value of each variable can be any real number from 0 to 6. x+y=6

or

y = −x + 6

Continuous domain

One possibility is two people bidding against each other on 6 ounces of gold dust at an auction. All the dust will be purchased by the two people. Because it is possible to purchase any portion of the dust, the domain is continuous.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Write a real-life problem to fit the data shown in the graph. Is the domain of the function discrete or continuous? Explain. 11.

8

12.

y

6

6

4

4

2

2

2

116

Chapter 3

8

Graphing Linear Functions

4

6

8 x

y

2

4

6

8 x

Exercises

3.2

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE A linear equation in two variables is an equation that can be written

in the form ________, where m and b are constants. 2. VOCABULARY Compare linear functions and nonlinear functions. 3. VOCABULARY Compare discrete domains and continuous domains. 4. WRITING How can you tell whether a graph shows a discrete domain or a continuous domain?

Monitoring Progress and Modeling with Mathematics In Exercises 5–10, determine whether the graph represents a linear or nonlinear function. Explain. (See Example 1.) 5.

6.

y

3

13.

y

14.

2 −2

2

−3

x

−1

−2

7.

3x

8.

3

2 2

y

15.

−2

x

−2

16

y

16

12

7

1

x

−1

0

1

2

y

35

20

5

−10

2

x



+2

+2

6

y

2

4

6

8

y

4

16

64

256

×4

×4

As x increases by 2, y increases by a constant factor of 4. So, the function is linear.

4

x 2

+2

x

×4

10.

y 1

2

−3

2

4

6x

In Exercises 11–14, determine whether the table represents a linear or nonlinear function. Explain. (See Example 2.)

12.

12

−3

−2

11.

8

correct the error in determining whether the table or graph represents a linear function.

1

−2

4

ERROR ANALYSIS In Exercises 15 and 16, describe and

−3

y

9.

1

x

x

1

2

3

4

y

5

10

15

20

x

5

7

9

11

y

−9

−3

−1

3

16.



y 2 −2

2

x

−2

The graph is a line. So, the graph represents a linear function.

Section 3.2

Linear Functions

117

In Exercises 17–24, determine whether the equation represents a linear or nonlinear function. Explain. (See Example 3.) 17. y = x 2 + 13 3—

21. 2 +

20. y = 4x(8 − x)

= 3x + 4

32. 2 —3 y

22. y − x = 2x −

23. 18x − 2y = 26

24. 2x + 3y = 9xy

25. CLASSIFYING FUNCTIONS Which of the following

equations do not represent linear functions? Explain.

A 12 = 2x2 + 4y 2 ○

B y−x+3=x ○

C x=8 ○

3 D x = 9 − —y ○ 4

5x E y=— ○ 11

F y = √x + 3 ○

Input Time (hours), x Output Distance (miles), y

18. y = 7 − 3x

19. y = √ 8 − x 1 —6 y

31.

y

−1

20

Output Athletes, y

0

4

8

correct the error in the statement about the domain. 33.



4

y

3 2 1

25

28. y

40 30

12

20

6

10 12

y

118

8

y

6

2

4

6

8x

The graph ends at x = 6, so the domain is discrete. 2

16 x

Input Bags, x

2

4

6

Output Marbles, y

20

40

60

4

6

8x

35. MODELING WITH MATHEMATICS The linear function

m = 55 − 8.5b represents the amount m (in dollars) of money that you have after buying b books. (See Example 4.)

a. Find the domain of the function. Is the domain discrete or continuous? Explain. b. Graph the function using its domain.

Input Years, x

1

2

3

Output Height of tree (feet), y

6

9

12

Chapter 3



2

In Exercises 29–32, determine whether the domain is discrete or continuous. Explain.

30.

8x

4

18

29.

6

11

27.

8

4

2.5 is in the domain.

In Exercises 27 and 28, find the domain of the function represented by the graph. Determine whether the domain is discrete or continuous. Explain.

4

450

ERROR ANALYSIS In Exercises 33 and 34, describe and

34.

24

300

2

2

15

150

1

linear function. 10

9

0

26. USING STRUCTURE Fill in the table so it represents a

5

6

Input Relay teams, x

—

x

3

Graphing Linear Functions

36. MODELING WITH MATHEMATICS The number y

of calories burned after x hours of rock climbing is represented by the linear function y = 650x.

a. Find the domain of the function. Is the domain discrete or continuous? Explain.

WRITING In Exercises 39–42, write a real-life problem

to fit the data shown in the graph. Determine whether the domain of the function is discrete or continuous. Explain. (See Example 6.) 39.

40. 8

y

y 4

b. Graph the function using its domain.

6 2 4

37. MODELING WITH MATHEMATICS You are researching

the speed of sound waves in dry air at 86°F. The table shows the distances d (in miles) sound waves travel in t seconds. (See Example 5.) Time (seconds), t

Distance (miles), d

2

0.434

4

0.868

6

1.302

8

1.736

10

2.170

a. Does this situation represent a linear function? Explain. b. Find the domain of the function. Is the domain discrete or continuous? Explain. c. Graph the function using its domain. 38. MODELING WITH MATHEMATICS The function

y = 30 + 5x represents the cost y (in dollars) of having your dog groomed and buying x extra services.

Pampered Pups Extra Grooming Services Paw Treatment Teeth Brushing Nail Polish

Deshedding Ear Treatment

a. Does this situation represent a linear function? Explain. b. Find the domain of the function. Is the domain discrete or continuous? Explain. c. Graph the function using its domain.

4

2

2

7x

−2 2

6

4

8x

41.

42. y

40 10

20

x

y

30

−100 20 −200 10 4

12

8

16 x

43. USING STRUCTURE The table shows your earnings

y (in dollars) for working x hours.

a. What is the missing y-value that makes the table represent a linear function?

Time (hours), x

Earnings (dollars), y

4

40.80

5

b. What is your hourly pay rate?

6

61.20

7

71.40

44. MAKING AN ARGUMENT The linear function

d = 50t represents the distance d (in miles) Car A is from a car rental store after t hours. The table shows the distances Car B is from the rental store. Time (hours), t

Distance (miles), d

1

60

3

180

5

310

a. Does the table represent a linear or nonlinear function? Explain. b. Your friend claims Car B is moving at a faster rate. Is your friend correct? Explain.

Section 3.2

Linear Functions

119

MATHEMATICAL CONNECTIONS In Exercises 45– 48, tell

51. CLASSIFYING A FUNCTION Is the function

whether the volume of the solid is a linear or nonlinear function of the missing dimension(s). Explain. 45.

represented by the ordered pairs linear or nonlinear? Explain your reasoning.

46.

(0, 2), (3, 14), (5, 22), (9, 38), (11, 46) 52. HOW DO YOU SEE IT? You and your friend go

s

9m

4 in.

s

b Running Distance

r

48.

2 cm

Distance (miles)

47.

running. The graph shows the distances you and your friend run.

3 in.

15 ft

h

y 6 5 4 3 2 1 0

You Friend

0

10

20

30

40

50

x

Minutes

49. REASONING A water company fills two different-

sized jugs. The first jug can hold x gallons of water. The second jug can hold y gallons of water. The company fills A jugs of the first size and B jugs of the second size. What does each expression represent? Does each expression represent a set of discrete or continuous values?

a. Describe your run and your friend’s run. Who runs at a constant rate? How do you know? Why might a person not run at a constant rate? b. Find the domain of each function. Describe the domains using the context of the problem.

a. x + y b. A + B

WRITING In Exercises 53 and 54, describe a real-life

c. Ax

situation for the constraints.

d. Ax + By

53. The function has at least one negative number in the

domain. The domain is continuous. 54. The function gives at least one negative number as an

50. THOUGHT PROVOKING You go to a farmer’s market

output. The domain is discrete.

to buy tomatoes. Graph a function that represents the cost of buying tomatoes. Explain your reasoning.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Tell whether x and y show direct variation. Explain your reasoning.

(Skills Review Handbook)

55.

57.

56.

y

3

2 −3

−2

3x

−2

Chapter 3

59.

y

1 2

x

−2

−3

Evaluate the expression when x = 2.

120

3

1 1

58. 6x + 8

y

2 −3

(Skills Review Handbook)

10 − 2x + 8

Graphing Linear Functions

60.

4(x + 2 − 5x)

61.

x 2

— + 5x − 7

x

3.3

Function Notation Essential Question

How can you use function notation to

represent a function?

The notation f(x), called function notation, is another name for y. This notation is read as “the value of f at x” or “f of x.” The parentheses do not imply multiplication. You can use letters other than f to name a function. The letters g, h, j, and k are often used to name functions.

Matching Functions with Their Graphs Work with a partner. Match each function with its graph.

ATTENDING TO PRECISION To be proficient in math, you need to use clear definitions and state the meanings of the symbols you use.

a. f (x) = 2x − 3

b. g(x) = −x + 2

c. h(x) = x 2 − 1

d. j(x) = 2x 2 − 3 4

A.

4

B.

−6

−6

6

6

−4

−4 4

C.

4

D.

−6

−6

6

6

−4

−4

Evaluating a Function Work with a partner. Consider the function f(x) = −x + 3. 5

Locate the points (x, f(x)) on the graph. Explain how you found each point.

f(x) = −x + 3

a. (−1, f(−1))

−6

6

b. (0, f (0)) c. (1, f (1))

−3

d. (2, f (2))

Communicate Your Answer 3. How can you use function notation to represent a function? How are standard

notation and function notation similar? How are they different? Standard Notation

Function Notation

y = 2x + 5

f(x) = 2x + 5 Section 3.3

Function Notation

121

3.3 Lesson

What You Will Learn Use function notation to evaluate and interpret functions. Use function notation to solve and graph functions.

Core Vocabul Vocabulary larry

Solve real-life problems using function notation.

function notation, p. 122

Using Function Notation to Evaluate and Interpret

Previous linear function quadrant

You know that a linear function can be written in the form y = mx + b. By naming a linear function f, you can also write the function using function notation. f(x) = mx + b

Function notation

The notation f(x) is another name for y. If f is a function, and x is in its domain, then f(x) represents the output of f corresponding to the input x. You can use letters other than f to name a function, such as g or h.

Evaluating a Function

READING The notation f (x) is read as “the value of f at x” or “f of x.” It does not mean “f times x.”

Evaluate f(x) = −4x + 7 when x = 2 and x = −2.

SOLUTION f(x) = −4x + 7

f (x) = −4x + 7

Write the function.

f(2) = −4(2) + 7 = −8 + 7 = −1

Substitute for x.

f (−2) = −4(−2) + 7

Multiply.

=8+7 = 15

Add.

When x = 2, f (x) = −1, and when x = −2, f (x) = 15.

Interpreting Function Notation Let f(t) be the outside temperature (°F) t hours after 6 a.m. Explain the meaning of each statement. a. f (0) = 58

b. f (6) = n

c. f (3) < f(9)

SOLUTION a. The initial value of the function is 58. So, the temperature at 6 a.m. is 58°F. b. The output of f when t = 6 is n. So, the temperature at noon (6 hours after 6 a.m.) is n°F. c. The output of f when t = 3 is less than the output of f when t = 9. So, the temperature at 9 a.m. (3 hours after 6 A.M.) is less than the temperature at 3 p.m. (9 hours after 6 a.m.).

Monitoring Progress

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Evaluate the function when x = −4, 0, and 3. 1. f(x) = 2x − 5

2. g(x) = −x − 1

3. WHAT IF? In Example 2, let f (t) be the outside temperature (°F) t hours after

9 a.m. Explain the meaning of each statement. a. f(4) = 75

122

Chapter 3

Graphing Linear Functions

b. f (m) = 70

c. f (2) = f (9)

d. f (6) > f(0)

Using Function Notation to Solve and Graph Solving for the Independent Variable For h(x) = —23 x − 5, find the value of x for which h(x) = −7.

SOLUTION h(x) = —23 x − 5

Write the function.

−7 = —23 x − 5

Substitute –7 for h(x).

+5

Add 5 to each side.

+5

−2 = —23 x

3 —2

⋅ (−2) = ⋅ 3 —2

Simplify. 2 —3 x

Multiply each side by —32 .

−3 = x

Simplify.

When x = −3, h(x) = −7.

Graphing a Linear Function in Function Notation Graph f (x) = 2x + 5.

SOLUTION Step 1 Make an input-output table to find ordered pairs. x

f (x)

−2

−1

0

1

2

1

3

5

7

9

Step 2 Plot the ordered pairs. Step 3 Draw a line through the points. y

STUDY TIP

8

The graph of y = f(x) consists of the points (x, f(x)).

6

f(x) = 2x + 5 2

−4

Monitoring Progress

2

4 x

Help in English and Spanish at BigIdeasMath.com

Find the value of x so that the function has the given value. 4. f(x) = 6x + 9; f (x) = 21

1

5. g(x) = −—2 x + 3; g(x) = −1

Graph the linear function. 6. f(x) = 3x − 2

7. g(x) = −x + 4

Section 3.3

3

8. h(x) = −—4 x − 1

Function Notation

123

Solving Real-Life Problems Modeling with Mathematics The graph shows the number of miles a helicopter is from its destination after x hours on its first flight. On its second flight, the helicopter travels 50 miles farther and increases its speed by 25 miles per hour. The function f(x) = 350 − 125x represents the second flight, where f (x) is the number of miles the helicopter is from its destination after x hours. Which flight takes less time? Explain.

Distance (miles)

First Flight f(x) 350 300 250 200 150 100 50 0

SOLUTION 0 1 2 3 4 5 6 x

Hours

1. Understand the Problem You are given a graph of the first flight and an equation of the second flight. You are asked to compare the flight times to determine which flight takes less time. 2. Make a Plan Graph the function that represents the second flight. Compare the graph to the graph of the first flight. The x-value that corresponds to f (x) = 0 represents the flight time. 3. Solve the Problem Graph f(x) = 350 − 125x. Step 1 Make an input-output table to find the ordered pairs. x

f (x)

0

1

2

3

350

225

100

−25

Step 2 Plot the ordered pairs. Step 3 Draw a line through the points. Note that the function only makes sense when x and f (x) are positive. So, only draw the line in the first quadrant.

y 300

f(x) = 350 − 125x

200 100 0

0

2

4

6

x

From the graph of the first flight, you can see that when f(x) = 0, x = 3. From the graph of the second flight, you can see that when f(x) = 0, x is slightly less than 3. So, the second flight takes less time.

4. Look Back You can check that your answer is correct by finding the value of x for which f(x) = 0. f(x) = 350 − 125x 0 = 350 − 125x −350 = −125x 2.8 = x

Write the function. Substitute 0 for f(x). Subtract 350 from each side. Divide each side by –125.

So, the second flight takes 2.8 hours, which is less than 3.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

9. WHAT IF? Let f(x) = 250 − 75x represent the second flight, where f (x) is the

number of miles the helicopter is from its destination after x hours. Which flight takes less time? Explain. 124

Chapter 3

Graphing Linear Functions

Exercises

3.3

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept p Check 1. COMPLETE THE SENTENCE When you write the function y = 2x + 10 as f(x) = 2x + 10,

you are using ______________. 2. REASONING Your height can be represented by a function h, where the input is your age.

What does h(14) represent?

Monitoring Progress and Modeling with Mathematics In Exercises 3–10, evaluate the function when x = –2, 0, and 5. (See Example 1.) 3. f(x) = x + 6 5. h(x) = −2x + 9 7. p(x) = −3 + 4x 9. v(x) = 12 − 2x − 5

4. g(x) = 3x 6. r(x) = −x − 7 8. b(x) = 18 − 0.5x 10. n(x) = −1 − x + 4

11. INTERPRETING FUNCTION NOTATION Let c(t) be

the number of customers in a restaurant t hours after 8 A.M. Explain the meaning of each statement. (See Example 2.) a. c(0) = 0

b. c(3) = c(8)

c. c(n) = 29

d. c(13) < c(12)

12. INTERPRETING FUNCTION NOTATION Let H(x) be the

percent of U.S. households with Internet use x years after 1980. Explain the meaning of each statement. a. H(23) = 55

b. H(4) = k

c. H(27) ≥ 61 d. H(17) + H(21) ≈ H(29)

In Exercises 19 and 20, find the value of x so that f(x) = 7. 19.

20.

y 6

6

y

f f

4 2 2 0

0

2

4

6

−2

x

2

x

21. MODELING WITH MATHEMATICS The function

C(x) = 17.5x − 10 represents the cost (in dollars) of buying x tickets to the orchestra with a $10 coupon. a. How much does it cost to buy five tickets? b. How many tickets can you buy with $130?

22. MODELING WITH MATHEMATICS The function

d(t) = 300,000t represents the distance (in kilometers) that light travels in t seconds. a. How far does light travel in 15 seconds? b. How long does it take light to travel 12 million kilometers?

In Exercises 13–18, find the value of x so that the function has the given value. (See Example 3.) 13. h(x) = −7x; h(x) = 63 14. t(x) = 3x; t(x) = 24 15. m(x) = 4x + 15; m(x) = 7

In Exercises 23–28, graph the linear i function. f ti (See Example 4.)

16. k(x) = 6x − 12; k(x) = 18

23. p(x) = 4x

1

17. q(x) = —2 x − 3; q(x) = −4 4

18. j(x) = −—5 x + 7; j(x) = −5

1

24. h(x) = −5 3

25. d(x) = −—2 x − 3

26. w(x) = —5 x + 2

27. g(x) = −4 + 7x

28. f (x) = 3 − 6x

Section 3.3

Function Notation

125

29. PROBLEM SOLVING The graph shows the percent

34. HOW DO YOU SEE IT? The function y = A(x)

p (in decimal form) of battery power remaining in a laptop computer after t hours of use. A tablet computer initially has 75% of its battery power remaining and loses 12.5% per hour. Which computer’s battery will last longer? Explain. (See Example 5.)

represents the attendance at a high school x weeks after a flu outbreak. The graph of the function is shown.

Number of students

Attendance

Power remaining (decimal form)

Laptop Battery p 1.2 1.0 0.8 0.6 0.4 0.2 0

0 1 2 3 4 5 6 t

A(x) 450 400 350 300 250 200 150 100 50 0 0

A

4

Hours

Hours Cost C(x) = 25x + 50 represents the labor cost (in dollars) for Certi ed 2 $130 Remodeling to build a deck, where 4 $160 x is the number of hours of labor. 6 $190 The table shows sample labor costs from its main competitor, Master Remodeling. The deck is estimated to take 8 hours of labor. Which company would you hire? Explain.

16 x

b. Estimate A(13) and explain its meaning. c. Use the graph to estimate the solution(s) of the equation A(x) = 400. Explain the meaning of the solution(s). d. What was the least attendance? When did that occur?

31. MAKING AN ARGUMENT Let P(x) be the number of

people in the U.S. who own a cell phone x years after 1990. Your friend says that P(x + 1) > P(x) for any x because x + 1 is always greater than x. Is your friend correct? Explain. 32. THOUGHT PROVOKING Let B(t) be your bank account

balance after t days. Describe a situation in which B(0) < B(4) < B(2).

e. How many students do you think are enrolled at this high school? Explain your reasoning.

35. INTERPRETING FUNCTION NOTATION Let f be a

function. Use each statement to find the coordinates of a point on the graph of f. a. f(5) is equal to 9. b. A solution of the equation f(n) = −3 is 5.

33. MATHEMATICAL CONNECTIONS Rewrite each

36. REASONING Given a function f, tell whether

geometry formula using function notation. Evaluate each function when r = 5 feet. Then explain the meaning of the result.

the statement f(a + b) = f(a) + f(b) is true or false for all inputs a and b. If it is false, explain why.

r

b. Area, A = π r 2

12

a. What happens to the school’s attendance after the flu outbreak?

30. PROBLEM SOLVING The function

a. Diameter, d = 2r

8

Week

c. Circumference, C = 2πr

Maintaining Mathematical Proficiency Solve the inequality. Graph the solution.

126

Reviewing what you learned in previous grades and lessons

(Section 2.5)

37. −2 ≤ x − 11 ≤ 6

38. 5a < −35 or a − 14 > 1

39. −16 < 6k + 2 < 0

40. 2d + 7 < −9 or 4d − 1 > −3

41. 5 ≤ 3y + 8 < 17

42. 4v + 9 ≤ 5 or −3v ≥ −6

Chapter 3

Graphing Linear Functions

3.1–3.3

What Did You Learn?

Core Vocabulary relation, p. 104 function, p. 104 domain, p. 106 range, p. 106 independent variable, p. 107 dependent variable, p. 107 linear equation in two variables, p. 112

linear function, p. 112 nonlinear function, p. 112 solution of a linear equation in two variables, p. 114 discrete domain, p. 114 continuous domain, p. 114 function notation, p. 122

Core Concepts Section 3.1 Determining Whether Relations Are Functions, p. 104 Vertical Line Test, p. 105

The Domain and Range of a Function, p. 106 Independent and Dependent Variables, p. 107

Section 3.2 Linear and Nonlinear Functions, p. 112 Representations of Functions, p. 113

Discrete and Continuous Domains, p. 114

Section 3.3 Using Function Notation, p. 122

Mathematical Practices 1.

How can you use technology to confirm your answers in Exercises 40–43 on page 110?

2.

How can you use patterns to solve Exercise 43 on page 119?

3.

How can you make sense of the quantities in the function in Exercise 21 on page 125?

Staying Focused during Class As soon as class starts, quickly review your notes from om the previous class and start thinking about math. Repeat what you are writing in your head. When a particular topic is difficult, ask for another example.

127 12 27

3.1–3.3

Quiz

Determine whether the relation is a function. Explain. (Section 3.1) 1.

Input, x

−1

0

1

2

3

0

1

4

4

8

Output, y

2. (−10, 2), (−8, 3), (−6, 5), (−8, 8), (−10, 6)

Find the domain and range of the function represented by the graph. (Section 3.1) 3.

1

4.

y 1

3

5.

y

y 2

2

5x

−2

−3

−4

1

−2

3x

2 −2

−2

Determine whether the graph, table, or equation represents a linear or nonlinear function. Explain. (Section 3.2) 6.

7.

y 2 −2

2

x

−2

x

y

−5

3

0

7

5

10

8. y = x(2 − x)

Determine whether the domain is discrete or continuous. Explain. (Section 3.2) 9.

Depth (feet), x

33

66

99

Pressure (ATM), y

2

3

4

10.

Hats, x

2

3

4

Cost (dollars), y

36

54

72

11. For w(x) = −2x + 7, find the value of x for which w(x) = −3. (Section 3.3)

Graph the linear function. (Section 3.3) 12. g(x) = x + 3

13. p(x) = −3x − 1

15. The function m = 30 − 3r represents the amount m (in dollars) of money you

have after renting r video games. (Section 3.1 and Section 3.2)

a. Identify the independent and dependent variables. b. Find the domain and range of the function. Is the domain discrete or continuous? Explain. c. Graph the function using its domain. 16. The function d(x) = 1375 − 110x represents the distance (in miles) a high-speed

train is from its destination after x hours. (Section 3.3)

a. How far is the train from its destination after 8 hours? b. How long does the train travel before reaching its destination? 128

Chapter 3

Graphing Linear Functions

2

14. m(x) = —3 x

x

3.4

Graphing Linear Equations in Standard Form Essential Question

How can you describe the graph of the equation

Ax + By = C?

Using a Table to Plot Points Work with a partner. You sold a total of $16 worth of tickets to a fundraiser. You lost track of how many of each type of ticket you sold. Adult tickets are $4 each. Child tickets are $2 each.

FINDING AN ENTRY POINT To be proficient in math, you need to find an entry point into the solution of a problem. Determining what information you know, and what you can do with that information, can help you find an entry point.

—

adult

⋅

Number of + — child adult tickets

⋅

Number of = child tickets

a. Let x represent the number of adult tickets. Let y represent the number of child tickets. Use the verbal model to write an equation that relates x and y. b. Copy and complete the table to show the different combinations of tickets you might have sold.

x y

c. Plot the points from the table. Describe the pattern formed by the points. d. If you remember how many adult tickets you sold, can you determine how many child tickets you sold? Explain your reasoning.

Rewriting and Graphing an Equation Work with a partner. You sold a total of $48 worth of cheese. You forgot how many pounds of each type of cheese you sold. Swiss cheese costs $8 per pound. Cheddar cheese costs $6 per pound. —

pound

⋅

Pounds of Swiss

+ — pound

⋅

Pounds of cheddar

=

a. Let x represent the number of pounds of Swiss cheese. Let y represent the number of pounds of cheddar cheese. Use the verbal model to write an equation that relates x and y. b. Solve the equation for y. Then use a graphing calculator to graph the equation. Given the real-life context of the problem, find the domain and range of the function. c. The x-intercept of a graph is the x-coordinate of a point where the graph crosses the x-axis. The y-intercept of a graph is the y-coordinate of a point where the graph crosses the y-axis. Use the graph to determine the x- and y-intercepts. d. How could you use the equation you found in part (a) to determine the x- and y-intercepts? Explain your reasoning. e. Explain the meaning of the intercepts in the context of the problem.

Communicate Your Answer 3. How can you describe the graph of the equation Ax + By = C? 4. Write a real-life problem that is similar to those shown in Explorations 1 and 2.

Section 3.4

Graphing Linear Equations in Standard Form

129

3.4 Lesson

What You Will Learn Graph equations of horizontal and vertical lines. Graph linear equations in standard form using intercepts.

Core Vocabul Vocabulary larry

Use linear equations in standard form to solve real-life problems.

standard form, p. 130 x-intercept, p. 131 y-intercept, p. 131

Horizontal and Vertical Lines The standard form of a linear equation is Ax + By = C, where A, B, and C are real numbers and A and B are not both zero.

Previous ordered pair quadrant

Consider what happens when A = 0 or when B = 0. When A = 0, the equation C C becomes By = C, or y = —. Because — is a constant, you can write y = b. Similarly, B B C when B = 0, the equation becomes Ax = C, or x = —, and you can write x = a. A

Core Concept Horizontal and Vertical Lines y

y

y=b

x=a

(0, b)

(a, 0) x

x

The graph of y = b is a horizontal line. The line passes through the point (0, b).

The graph of x = a is a vertical line. The line passes through the point (a, 0).

Horizontal and Vertical Lines Graph (a) y = 4 and (b) x = −2.

SOLUTION a. For every value of x, the value of y is 4. The graph of the equation y = 4 is a horizontal line 4 units above the x-axis.

STUDY TIP For every value of x, the ordered pair (x, 4) is a solution of y = 4.

6

(−2, 4)

b. For every value of y, the value of x is −2. The graph of the equation x = −2 is a vertical line 2 units to the left of the y-axis.

y

(−2, 3)

(3, 4) (0, 4)

(−2, 0) −5

−3

(−2, −2) 2

Monitoring Progress

x

130

Chapter 3

Graphing Linear Functions

−1

1 x

−2

Help in English and Spanish at BigIdeasMath.com

Graph the linear equation. 1. y = −2.5

y

2

2

−2

4

2. x = 5

Using Intercepts to Graph Linear Equations You can use the fact that two points determine a line to graph a linear equation. Two convenient points are the points where the graph crosses the axes.

Core Concept Using Intercepts to Graph Equations The x-intercept of a graph is the x-coordinate of a point where the graph crosses the x-axis. It occurs when y = 0. The y-intercept of a graph is the y-coordinate of a point where the graph crosses the y-axis. It occurs when x = 0.

y

y-intercept = b (0, b) x-intercept = a O

x

(a, 0)

To graph the linear equation Ax + By = C, find the intercepts and draw the line that passes through the two intercepts. • To find the x-intercept, let y = 0 and solve for x. • To find the y-intercept, let x = 0 and solve for y.

Using Intercepts to Graph a Linear Equation Use intercepts to graph the equation 3x + 4y = 12.

SOLUTION Step 1 Find the intercepts. To find the x-intercept, substitute 0 for y and solve for x.

STUDY TIP As a check, you can find a third solution of the equation and verify that the corresponding point is on the graph. To find a third solution, substitute any value for one of the variables and solve for the other variable.

3x + 4y = 12 3x + 4(0) = 12 x=4

Write the original equation. Substitute 0 for y. Solve for x.

To find the y-intercept, substitute 0 for x and solve for y. 3x + 4y = 12 3(0) + 4y = 12 y=3

Write the original equation. Substitute 0 for x. Solve for y.

Step 2 Plot the points and draw the line.

y

The x-intercept is 4, so plot the point (4, 0). The y-intercept is 3, so plot the point (0, 3). Draw a line through the points.

4

(0, 3)

2

(4, 0) 2

Monitoring Progress

4

x

Help in English and Spanish at BigIdeasMath.com

Use intercepts to graph the linear equation. Label the points corresponding to the intercepts. 3. 2x − y = 4

Section 3.4

4. x + 3y = −9

Graphing Linear Equations in Standard Form

131

Solving Real-Life Problems Modeling with Mathematics You are planning an awards banquet for your school. You need to rent tables to seat 180 people. There are two table sizes available. Small tables seat 6 people, and large tables seat 10 people. The equation 6x + 10y = 180 models this situation, where x is the number of small tables and y is the number of large tables. a. Graph the equation. Interpret the intercepts. b. Find four possible solutions in the context of the problem.

SOLUTION 1. Understand the Problem You know the equation that models the situation. You are asked to graph the equation, interpret the intercepts, and find four solutions. 2. Make a Plan Use intercepts to graph the equation. Then use the graph to interpret the intercepts and find other solutions. 3. Solve the Problem

STUDY TIP Although x and y represent whole numbers, it is convenient to draw a line segment that includes points whose coordinates are not whole numbers.

a. Use intercepts to graph the equation. Neither x nor y can be negative, so only graph the equation in the first quadrant. y

(0, 18)

16

6x + 10y = 180 12 8

The y-intercept is 18. So, plot (0, 18).

4 0

The x-intercept is 30. So, plot (30, 0).

(30, 0) 0

4

8

12

16

20

24

28

32

36

x

The x-intercept shows that you can rent 30 small tables when you do not rent any large tables. The y-intercept shows that you can rent 18 large tables when you do not rent any small tables.

Check 6x + 10y = 180 ? 6(10) + 10(12) = 180 180 = 180



So, four possible combinations of tables that will seat 180 people are 0 small and 18 large, 10 small and 12 large, 20 small and 6 large, and 30 small and 0 large.

6x + 10y = 180 ? 6(20) + 10(6) = 180 180 = 180

b. Only whole-number values of x and y make sense in the context of the problem. Besides the intercepts, it appears that the line passes through the points (10, 12) and (20, 6). To verify that these points are solutions, check them in the equation, as shown.



4. Look Back The graph shows that as the number x of small tables increases, the number y of large tables decreases. This makes sense in the context of the problem. So, the graph is reasonable.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

5. WHAT IF? You decide to rent tables from a different company. The situation can be

modeled by the equation 4x + 6y = 180, where x is the number of small tables and y is the number of large tables. Graph the equation and interpret the intercepts.

132

Chapter 3

Graphing Linear Functions

3.4

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING How are x-intercepts and y-intercepts alike? How are they different? 2. WHICH ONE DOESN’T BELONG? Which point does not belong with the other three?

Explain your reasoning. (0, −3)

(4, −3)

(0, 0)

(4, 0)

Monitoring Progress and Modeling with Mathematics In Exercises 3– 6, graph the linear equation. (See Example 1.) 3. x = 4

4. y = 2

5. y = −3

6. x = −1

24. MODELING WITH MATHEMATICS You are ordering

shirts for the math club at your school. Short-sleeved shirts cost $10 each. Long-sleeved shirts cost $12 each. You have a budget of $300 for the shirts. The equation 10x + 12y = 300 models the total cost, where x is the number of short-sleeved shirts and y is the number of long-sleeved shirts.

In Exercises 7–12, find the x- and y-intercepts of the graph of the linear equation. 7. 2x + 3y = 12 9. −4x + 8y = −16 11. 3x − 6y = 2

8. 3x + 6y = 24 10. −6x + 9y = −18 12. −x + 8y = 4

In Exercises 13–22, use intercepts to graph the linear equation. Label the points corresponding to the intercepts. (See Example 2.) 13. 5x + 3y = 30

14. 4x + 6y = 12

15. −12x + 3y = 24

16. −2x + 6y = 18

17. −4x + 3y = −30

18. −2x + 7y = −21

19. −x + 2y = 7

20. 3x − y = −5

5

21. −—2 x + y = 10

a. Graph the equation. Interpret the intercepts. b. Twelve students decide they want short-sleeved shirts. How many long-sleeved shirts can you order?

1

22. −—2 x + y = −4

23. MODELING WITH MATHEMATICS A football team

has an away game, and the bus breaks down. The coaches decide to drive the players to the game in cars and vans. Four players can ride in each car. Six players can ride in each van. There are 48 players on the team. The equation 4x + 6y = 48 models this situation, where x is the number of cars and y is the number of vans. (See Example 3.)

ERROR ANALYSIS In Exercises 25 and 26, describe and

correct the error in finding the intercepts of the graph of the equation. 25.



3x + 12y = 24

3x + 12y = 24

3x + 12(0) = 24

3(0) + 12y = 24

3x = 24

12y = 24

x=8

y=2

The intercept is at (8, 2). 26.



4x + 10y = 20

4x + 10y = 20

4x + 10(0) = 20

4(0) + 10y = 20

4x = 20

10y = 20

x=5

y=2

The x-intercept is at (0, 5), and the y-intercept is at (2, 0).

a. Graph the equation. Interpret the intercepts. b. Find four possible solutions in the context of the problem. Section 3.4

Graphing Linear Equations in Standard Form

133

27. MAKING AN ARGUMENT You overhear your friend

34. HOW DO YOU SEE IT? You are organizing a class trip

explaining how to find intercepts to a classmate. Your friend says, “When you want to find the x-intercept, just substitute 0 for x and continue to solve the equation.” Is your friend’s explanation correct? Explain. 28. ANALYZING RELATIONSHIPS You lose track of how

to an amusement park. The cost to enter the park is $30. The cost to enter with a meal plan is $45. You have a budget of $2700 for the trip. The equation 30x + 45y = 2700 models the total cost for the class to go on the trip, where x is the number of students who do not choose the meal plan and y is the number of students who do choose the meal plan.

many 2-point baskets and 3-point baskets a team makes in a basketball game. The team misses all the 1-point baskets and still scores 54 points. The equation 2x + 3y = 54 models the total points scored, where x is the number of 2-point baskets made and y is the number of 3-point baskets made.

Number of students who do choose the meal plan

Class Trip

a. Find and interpret the intercepts. b. Can the number of 3-point baskets made be odd? Explain your reasoning. c. Graph the equation. Find two more possible solutions in the context of the problem. MULTIPLE REPRESENTATIONS In Exercises 29–32, match 29. 5x + 3y = 30

30. 5x + 3y = −30

31. 5x − 3y = 30

32. 5x − 3y = −30

B.

y

12

4

C.

2

12 x

D.

y

−8

−8

4x

20 0

(90, 0) 0

20

40

60

80

x

x+ y = 30 so that the x-intercept of the graph is −10 and the y-intercept of the graph is 5. −10 4x

−3

1

5

form of a line whose intercepts are integers. Explain how you know the intercepts are integers. 4

8

37. WRITING Are the equations of horizontal and

x

vertical lines written in standard form? Explain your reasoning.

−4 −8

38. ABSTRACT REASONING The x- and y-intercepts of

−12

the graph of the equation 3x + 5y = k are integers. Describe the values of k. Explain your reasoning.

33. MATHEMATICAL CONNECTIONS Graph the equations

x = 5, x = 2, y = −2, and y = 1. What enclosed shape do the lines form? Explain your reasoning.

Maintainingg Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Simplify the expression. (Skills Review Handbook) 39.

134

2 − (−2) 4 − (−4)

—

Chapter 3

6

36. THOUGHT PROVOKING Write an equation in standard

y

−4

−4

40

35. REASONING Use the values to fill in the equation

y

−4

2

(0, 60)

a. Interpret the intercepts of the graph.

4 8

60

b. Describe the domain and range in the context of the problem.

8

4

80

Number of students who do not choose the meal plan

the equation with its graph.

A.

y

40.

14 − 18 0−2

—

Graphing Linear Functions

41.

−3 − 9 8 − (−7)

—

42.

12 − 17 −5 − (−2)

—

3.5

Graphing Linear Equations in Slope-Intercept Form Essential Question

How can you describe the graph of the

equation y = mx + b?

Slope is the rate of change between any two points on a line. It is the measure of the steepness of the line.

y 6

To find the slope of a line, find the ratio of the change in y (vertical change) to the change in x (horizontal change).

3

4

2

2

change in y slope = — change in x

slope = 2

4

2

4 x

6

3 2

x

Finding Slopes and y-Intercepts Work with a partner. Find the slope and y-intercept of each line. a.

6

b.

y

4 2

4

MAKING CONJECTURES To be proficient in math, you first need to collect and organize data. Then make conjectures about the patterns you observe in the data.

y = 23 x + 2 −2

y

2

−4

4 x

−2

y = −2x − 1 −4

−2

Writing a Conjecture Work with a partner. Graph each equation. Then copy and complete the table. Use the completed table to write a conjecture about the relationship between the graph of y = mx + b and the values of m and b. Equation

Description of graph

Slope of graph

y-Intercept

2 a. y = −— x + 3 3

Line

2 −— 3

3

b. y = 2x − 2 c. y = −x + 1 d. y = x − 4

Communicate Your Answer 3. How can you describe the graph of the equation y = mx + b?

a. How does the value of m affect the graph of the equation? b. How does the value of b affect the graph of the equation? c. Check your answers to parts (a) and (b) by choosing one equation from Exploration 2 and (1) varying only m and (2) varying only b. Section 3.5

Graphing Linear Equations in Slope-Intercept Form

135

3.5 Lesson

What You Will Learn Find the slope of a line. Use the slope-intercept form of a linear equation.

Core Vocabul Vocabulary larry

Use slopes and y-intercepts to solve real-life problems.

slope, p. 136 rise, p. 136 run, p. 136 slope-intercept form, p. 138 constant function, p. 138

The Slope of a Line

Core Concept Slope

Previous dependent variable independent variable

y

The slope m of a nonvertical line passing through two points (x1, y1) and (x2, y2) is the ratio of (x1, y1) the rise (change in y) to the run (change in x). y2 − y1 rise change in y slope = m = — = — = — run change in x x2 − x1

(x2, y2) rise = y2 − y1

run = x2 − x1 x

O

When the line rises from left to right, the slope is positive. When the line falls from left to right, the slope is negative.

Finding the Slope of a Line Describe the slope of each line. Then find the slope. a.

3

b.

y

(3, 2)

1 2

(−3, −2)

When finding slope, you can label either point as (x1, y1) and the other point as (x2, y2). The result is the same.

(0, 2) x

−3

6

In the slope formula, x1 is read as “x sub one” and y2 is read as “y sub two.” The numbers 1 and 2 in x1 and y2 are called subscripts.

3 x

(2, −1)

b. The line falls from left to right. So, the slope is negative. Let (x1, y1) = (0, 2) and (x2, y2) = (2, −1).

a. The line rises from left to right. So, the slope is positive. Let (x1, y1) = (−3, −2) and (x2, y2) = (3, 2).

Monitoring Progress

y2 − y1 −1 − 2 −3 3 m = — = — = — = −— 2 x2 − x1 2−0 2

Help in English and Spanish at BigIdeasMath.com

Describe the slope of the line. Then find the slope. 1.

2.

y

4

(3, 3)

y 4

(5, 4)

2

(1, 1)

4 −3

−4

3.

y

4 2

Chapter 3

−3 1

SOLUTION

(−4, 3)

136

−1

−3

y2 − y1 4 2 2 − (−2) m=—=—=—=— x2 − x1 3 − (−3) 6 3

READING

2

1

4

−2

STUDY TIP

y

−2

Graphing Linear Functions

2 x

−1

(−3, −1)

1

3 x

−4

(2, −3)

8

x

Finding Slope from a Table The points represented by each table lie on a line. How can you find the slope of each line from the table? What is the slope of each line? a.

STUDY TIP As a check, you can plot the points represented by the table to verify that the line through them has a slope of −2.

b.

y

2

−3

−3

1

2

−3

0

3

2

−3

6

5

2

−3

9

y

20

−1

7

14

10

8

13

2

y

4

c.

x

x

x

SOLUTION a. Choose any two points from the table and use the slope formula. Use the points (x1, y1) = (4, 20) and (x2, y2) = (7, 14).

y2 − y1 14 − 20 −6 m = — = — = —, or −2 x2 − x1 7−4 3 The slope is −2. b. Note that there is no change in y. Choose any two points from the table and use the slope formula. Use the points (x1, y1) = (−1, 2) and (x2, y2) = (5, 2). y2 − y1 0 2−2 m = — = — = —, or 0 x2 − x1 5 − (−1) 6

The change in y is 0.

The slope is 0. c. Note that there is no change in x. Choose any two points from the table and use the slope formula. Use the points (x1, y1) = (−3, 0) and (x2, y2) = (−3, 6). y2 − y1 6 6−0 m=—=—=— x2 − x1 −3 − (−3) 0




The change in x is 0.

Because division by zero is undefined, the slope of the line is undefined.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

The points represented by the table lie on a line. How can you find the slope of the line from the table? What is the slope of the line? 4.

x

2

4

6

8

y

10

15

20

25

5.

x

5

5

5

5

y

−12

−9

−6

−3

Concept Summary Slope Positive slope

Negative slope

y

O

x

O

Undefined slope y

y

y

The line rises from left to right. Section 3.5

Slope of 0

x

The line falls from left to right.

O

The line is horizontal.

x

O

x

The line is vertical.

Graphing Linear Equations in Slope-Intercept Form

137

Using the Slope-Intercept Form of a Linear Equation

Core Concept Slope-Intercept Form Words

y

A linear equation written in the form y = mx + b is in slope-intercept form. The slope of the line is m, and the (0, b) y-intercept of the line is b.

y = mx + b

y = mx + b

Algebra

slope

y-intercept

x

A linear equation written in the form y = 0x + b, or y = b, is a constant function. The graph of a constant function is a horizontal line.

Identifying Slopes and y-Intercepts Find the slope and the y-intercept of the graph of each linear equation. a. y = 3x − 4

b. y = 6.5

c. −5x − y = −2

SOLUTION a. y = mx + b

STUDY TIP For a constant function, every input has the same output. For instance, in Example 3b, every input has an output of 6.5.

slope

Write the slope-intercept form.

y-intercept

y = 3x + (−4)

Rewrite the original equation in slope-intercept form.

The slope is 3, and the y-intercept is −4. b. The equation represents a constant function. The equation can also be written as y = 0x + 6.5. The slope is 0, and the y-intercept is 6.5. c. Rewrite the equation in slope-intercept form by solving for y.

STUDY TIP When you rewrite a linear equation in slope-intercept form, you are expressing y as a function of x.

−5x − y = −2

Write the original equation.

+ 5x

Add 5x to each side.

+ 5x −y = 5x − 2 −y −1

5x − 2 −1

—=—

y = −5x + 2

Simplify. Divide each side by –1. Simplify.

The slope is −5, and the y-intercept is 2.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Find the slope and the y-intercept of the graph of the linear equation. 6. y = −6x + 1

138

Chapter 3

Graphing Linear Functions

7. y = 8

8. x + 4y = −10

Using Slope-Intercept Form to Graph

STUDY TIP

Graph 2x + y = 2. Identify the x-intercept.

You can use the slope to find points on a line in either direction. In Example 4, note that the slope can be written as 2 —. So, you could move –1 1 unit left and 2 units up from (0, 2) to find the point (–1, 4).

SOLUTION Step 1 Rewrite the equation in slope-intercept form. y = −2x + 2 Step 2 Find the slope and the y-intercept.

y

m = −2 and b = 2

4

Step 3 The y-intercept is 2. So, plot (0, 2). Step 4 Use the slope to find another point on the line. rise −2 slope = — = — run 1

(0, 2)

1

1 −2

−2 2

4

x

−2

Plot the point that is 1 unit right and 2 units down from (0, 2). Draw a line through the two points. The line crosses the x-axis at (1, 0). So, the x-intercept is 1.

REMEMBER Graphing from a Verbal Description

You can also find the x-intercept by substituting 0 for y in the equation 2x + y = 2 and solving for x.

A linear function g models a relationship in which the dependent variable increases 3 units for every 1 unit the independent variable increases. Graph g when g(0) = 3. Identify the slope, y-intercept, and x-intercept of the graph.

SOLUTION Because the function g is linear, it has a constant rate of change. Let x represent the independent variable and y represent the dependent variable. 5

(0, 3)

Step 1 Find the slope. When the dependent variable increases by 3, the change in y is +3. When the independent variable increases by 1, the change in x is +1. 3 So, the slope is —, or 3. 1

(−1, 0)

Step 2 Find the y-intercept. The statement g(0) = 3 indicates that when x = 0, y = 3. So, the y-intercept is 3. Plot (0, 3).

y

−1 −3 −4

−2

2 −2

x

Step 3 Use the slope to find another point on the line. A slope of 3 can be written −3 as —. Plot the point that is 1 unit left and 3 units down from (0, 3). Draw a −1 line through the two points. The line crosses the x-axis at (−1, 0). So, the x-intercept is −1.

The slope is 3, the y-intercept is 3, and the x-intercept is −1.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Graph the linear equation. Identify the x-intercept. 9. y = 4x − 4

10. 3x + y = −3

11. x + 2y = 6

12. A linear function h models a relationship in which the dependent variable

decreases 2 units for every 5 units the independent variable increases. Graph h when h(0) = 4. Identify the slope, y-intercept, and x-intercept of the graph. Section 3.5

Graphing Linear Equations in Slope-Intercept Form

139

Solving Real-Life Problems In most real-life problems, slope is interpreted as a rate, such as miles per hour, dollars per hour, or people per year.

Modeling with Mathematics A submersible that is exploring the ocean floor begins to ascend to the surface. The elevation h (in feet) of the submersible is modeled by the function h(t) = 650t − 13,000, where t is the time (in minutes) since the submersible began to ascend. a. Graph the function and identify its domain and range. b. Interpret the slope and the intercepts of the graph.

SOLUTION 1. Understand the Problem You know the function that models the elevation. You are asked to graph the function and identify its domain and range. Then you are asked to interpret the slope and intercepts of the graph. 2. Make a Plan Use the slope-intercept form of a linear equation to graph the function. Only graph values that make sense in the context of the problem. Examine the graph to interpret the slope and the intercepts. 3. Solve the Problem

Because t is the independent variable, the horizontal axis is the t-axis and the graph will have a “t-intercept.” Similarly, the vertical axis is the h-axis and the graph will have an “h-intercept.”

Elevation of a Submersible Time (minutes) 0

Elevation (feet)

STUDY TIP

a. The time t must be greater than or equal to 0. The elevation h is below sea level and must be less than or equal to 0. Use the slope of 650 and the h-intercept of −13,000 to graph the function in Quadrant IV.

0

4

8

12

16

20

t

(20, 0)

−4,000 −8,000 −12,000 h

(0, −13,000)

The domain is 0 ≤ t ≤ 20, and the range is −13,000 ≤ h ≤ 0. b. The slope is 650. So, the submersible ascends at a rate of 650 feet per minute. The h-intercept is −13,000. So, the elevation of the submersible after 0 minutes, or when the ascent begins, is −13,000 feet. The t-intercept is 20. So, the submersible takes 20 minutes to reach an elevation of 0 feet, or sea level.

4. Look Back You can check that your graph is correct by substituting the t-intercept for t in the function. If h = 0 when t = 20, the graph is correct. h = 650(20) − 13,000 h=0



Monitoring Progress

Substitute 20 for t in the original equation. Simplify.

Help in English and Spanish at BigIdeasMath.com

13. WHAT IF? The elevation of the submersible is modeled by h(t) = 500t − 10,000.

(a) Graph the function and identify its domain and range. (b) Interpret the slope and the intercepts of the graph.

140

Chapter 3

Graphing Linear Functions

Exercises

3.5

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE The ________ of a nonvertical line passing through two points is the

ratio of the rise to the run. 2. VOCABULARY What is a constant function? What is the slope of a constant function? 3. WRITING What is the slope-intercept form of a linear equation? Explain why this form is called

the slope-intercept form. 4. WHICH ONE DOESN’T BELONG? Which equation does not belong with the other three? Explain

your reasoning. y = −5x − 1

2x − y = 8

y=x+4

y = −3x + 13

Monitoring Progress and Modeling with Mathematics In Exercises 5–8, describe the slope of the line. Then find the slope. (See Example 1.) 5.

6.

y

(−3, 1)

4

12.

y

(4, 3)

2 2

2

7.

1 −2

8.

y 2

−2

x

x

(0, 3)

2

(2, −3)

(−2, −3)

−1

−5

1

3

(5, −1)

x

10.

11.

x

−9

−5

−1

3

y

−2

0

2

4

x

−1

2

5

8

y

−6

−6

−6

−6

x

0

0

0

0

y

−4

0

4

8

y

2

−5

−12 −19

Section 3.5

y 150

(2, 120)

100

(1, 60)

50 0

0

1

2

3 x

Time (hours)

In Exercises 9–12, the points represented by the table lie on a line. Find the slope of the line. (See Example 2.) 9.

−1

Bus Trip

y 4

−2

Distance (miles)

(2, −2)

4

(1, −1)

−2

−3

y (in miles) that a bus travels in x hours. Find and interpret the slope of the line.

x

−2

−4

13. ANALYZING A GRAPH The graph shows the distance

2 −2

x

14. ANALYZING A TABLE The table shows the amount

x (in hours) of time you spend at a theme park and the admission fee y (in dollars) to the park. The points represented by the table lie on a line. Find and interpret the slope of the line. Time (hours), x

Admission (dollars), y

6

54.99

7

54.99

8

54.99

Graphing Linear Equations in Slope-Intercept Form

141

In Exercises 15–22, find the slope and the y-intercept of the graph of the linear equation. (See Example 3.) 15. y = −3x + 2

16. y = 4x − 7

17. y = 6x

18. y = −1

19. −2x + y = 4

20. x + y = −6

21. −5x = 8 − y

22. 0 = 1 − 2y + 14x

ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in finding the slope and the y-intercept of the graph of the equation. 23.

function r models the growth of your right index ngernail. The length of the ngernail increases 0.7 millimeter every week. Graph r when r (0) = 12. Identify the slope and interpret the y-intercept of the graph. 36. GRAPHING FROM A VERBAL DESCRIPTION A linear

function m models the amount of milk sold by a farm per month. The amount decreases 500 gallons for every $1 increase in price. Graph m when m(0) = 3000. Identify the slope and interpret the x- and y-intercepts of the graph. 37. MODELING WITH MATHEMATICS The function shown



x = –4y The slope is –4, and the y-intercept is 0.

24.

35. GRAPHING FROM A VERBAL DESCRIPTION A linear



models the depth d (in inches) of snow on the ground during the rst 9 hours of a snowstorm, where t is the time (in hours) after the snowstorm begins. (See Example 6.)

y = 3x − 6 The slope is 3, and the y-intercept is 6.

In Exercises 25–32, graph the linear equation. Identify the x-intercept. (See Example 4.) 1

25. y = −x + 7

26. y = —2 x + 3

27. y = 2x

28. y = −x

29. 3x + y = −1

30. x + 4y = 8

31. −y + 5x = 0

32. 2x − y + 6 = 0

In Exercises 33 and 34, graph the function with the given description. Identify the slope, y-intercept, and x-intercept of the graph. (See Example 5.) 33. A linear function f models a relationship in which the

dependent variable decreases 4 units for every 2 units the independent variable increases. The value of the function at 0 is −2. 34. A linear function h models a relationship in which the

dependent variable increases 1 unit for every 5 units the independent variable decreases. The value of the function at 0 is 3.

142

Chapter 3

Graphing Linear Functions

d(t) =

1 t+6 2

a. Graph the function and identify its domain and range. b. Interpret the slope and the d-intercept of the graph. 38. MODELING WITH MATHEMATICS The function

c(x) = 0.5x + 70 represents the cost c (in dollars) of renting a truck from a moving company, where x is the number of miles you drive the truck. a. Graph the function and identify its domain and range. b. Interpret the slope and the c-intercept of the graph.

39. COMPARING FUNCTIONS A linear function models

the cost of renting a truck from a moving company. The table shows the cost y (in dollars) when you drive the truck x miles. Graph the function and compare the slope and the y-intercept of the graph with the slope and the c-intercept of the graph in Exercise 38. Miles, x

Cost (dollars), y

0

40

50

80

100

120

ERROR ANALYSIS In Exercises 40 and 41, describe and

correct the error in graphing the function. 40.



the relationship between the base length x and the side length (of the two equal sides) y of an isosceles triangle in meters. The perimeter of a second isosceles triangle is 8 meters more than the perimeter of the rst triangle.

y 2

y + 1 = 3x

44. MATHEMATICAL CONNECTIONS The graph shows

y 8

14 x

−2

3 −2 (0, −1)

4 0

41.



1

1 −3

−1

4

8

12

x

b. How does the graph in part (a) compare to the graph shown?

−2

3

−4x + y = −2

0

a. Graph the relationship between the base length and the side length of the second triangle.

y

(0, 4)

1

y = 6 − 2x

1

45. ANALYZING EQUATIONS Determine which of the

3 x

equations could be represented by each graph.

42. MATHEMATICAL CONNECTIONS Graph the four

equations in the same coordinate plane. 3y = −x − 3 2y − 14 = 4x 4x − 3 − y = 0 x − 12 = −3y a. What enclosed shape do you think the lines form? Explain.

a.

y = −3x + 8

4 y = −x − — 3

y = −7x

y = 2x − 4

7 1 y = —x − — 4 4

1 y = —x + 5 3

y = −4x − 9

y=6

b.

y

y

b. Write a conjecture about the equations of parallel lines. the relationship between the width y and the length x of a rectangle in inches. The perimeter of a second rectangle is 10 inches less than the perimeter of the rst rectangle. a. Graph the relationship between the width and length of the second rectangle. b. How does the graph in part (a) compare to the the graph shown?

x

x

43. MATHEMATICAL CONNECTIONS The graph shows

c.

d.

y

y

y 24

y = 20 − x

x

x

16 8 0

0

8

16

24 x

46. MAKING AN ARGUMENT Your friend says that you

can write the equation of any line in slope-intercept form. Is your friend correct? Explain your reasoning.

Section 3.5

Graphing Linear Equations in Slope-Intercept Form

143

47. WRITING Write the definition of the slope of a line in

50. HOW DO YOU SEE IT? You commute to school by

two different ways.

walking and by riding a bus. The graph represents your commute.

48. THOUGHT PROVOKING Your family goes on vacation

Commute to School Distance (miles)

to a beach 300 miles from your house. You reach your destination 6 hours after departing. Draw a graph that describes your trip. Explain what each part of your graph represents. 49. ANALYZING A GRAPH The graphs of the functions

g(x) = 6x + a and h(x) = 2x + b, where a and b are constants, are shown. They intersect at the point (p, q).

d 2 1 0

0

4

8

12

16

t

Time (minutes)

a. Describe your commute in words.

y

b. Calculate and interpret the slopes of the different parts of the graph. (p, q)

PROBLEM SOLVING In Exercises 51 and 52, find the value of k so that the graph of the equation has the given slope or y-intercept.

x

1 2

51. y = 4kx − 5; m = —

a. Label the graphs of g and h. b. What do a and b represent?

1 3

53. ABSTRACT REASONING To show that the slope of

a line is constant, let (x1, y1) and (x2, y2) be any two points on the line y = mx + b. Use the equation of the line to express y1 in terms of x1 and y2 in terms of x2. Then use the slope formula to show that the slope between the points is m.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Find the coordinates of the figure after the transformation. 54. Translate the rectangle

4 units left. 4

55. Dilate the triangle with

4

2

A −4

−2

y

−4

X

−2

1

Z

C

Q

4

−3

−1

y

1

S −2

4x

−2

D

the y-axis.

Y

4x

−2

56. Reflect the trapezoid in

R 2

2

B 2

−4

(Skills Review Handbook)

respect to the origin using a scale factor of 2.

y

5 6

52. y = −— x + — k; b = −10

c. Starting at the point (p, q), trace the graph of g until you get to the point with the x-coordinate p + 2. Mark this point C. Do the same with the graph of h. Mark this point D. How much greater is the y-coordinate of point C than the y-coordinate of point D?

T

−4

−4

Determine whether the equation represents a linear or nonlinear function. Explain. (Section 3.2) 57.

144

2 y−9=— x Chapter 3

58.

x = 3 + 15y

Graphing Linear Functions

59.

x 4

y 12

—+—=1

60.

y = 3x 4 − 6

3

x

3.6

Transformations of Graphs of Linear Functions Essential Question

How does the graph of the linear function f(x) = x compare to the graphs of g(x) = f (x) + c and h(x) = f (cx)? Comparing Graphs of Functions

USING TOOLS STRATEGICALLY To be proficient in math, you need to use the appropriate tools, including graphs, tables, and technology, to check your results.

Work with a partner. The graph of f(x) = x is shown. Sketch the graph of each function, along with f, on the same set of coordinate axes. Use a graphing calculator to check your results. What can you conclude?

a. g(x) = x + 4

b. g(x) = x + 2

c. g(x) = x − 2

d. g(x) = x − 4

4

−6

6

−4

Comparing Graphs of Functions Work with a partner. Sketch the graph of each function, along with f(x) = x, on the same set of coordinate axes. Use a graphing calculator to check your results. What can you conclude? a. h(x) = —12 x

b. h(x) = 2x

1

c. h(x) = −—2 x

d. h(x) = −2x

Matching Functions with Their Graphs Work with a partner. Match each function with its graph. Use a graphing calculator to check your results. Then use the results of Explorations 1 and 2 to compare the graph of k to the graph of f(x) = x. a. k(x) = 2x − 4 c. k(x) = A.

1 —2 x

b. k(x) = −2x + 2 1

+4

d. k(x) = −—2 x − 2 B.

4

−6

4

−6

6

−4

−4

C.

D.

4

−6

6

6

−4

6

−8

8

−6

Communicate Your Answer 4. How does the graph of the linear function f(x) = x compare to the graphs of

g(x) = f(x) + c and h(x) = f(cx)? Section 3.6

Transformations of Graphs of Linear Functions

145

3.6 Lesson

What You Will Learn Translate and reflect graphs of linear functions. Stretch and shrink graphs of linear functions.

Core Vocabul Vocabulary larry

Combine transformations of graphs of linear functions.

family of functions, p. 146 parent function, p. 146 transformation, p. 146 translation, p. 146 reflection, p. 147 horizontal shrink, p. 148 horizontal stretch, p. 148 vertical stretch, p. 148 vertical shrink, p. 148 Previous linear function

Translations and Reflections A family of functions is a group of functions with similar characteristics. The most basic function in a family of functions is the parent function. For nonconstant linear functions, the parent function is f(x) = x. The graphs of all other nonconstant linear functions are transformations of the graph of the parent function. A transformation changes the size, shape, position, or orientation of a graph.

Core Concept A translation is a transformation that shifts a graph horizontally or vertically but does not change the size, shape, or orientation of the graph.

CONNECTIONS TO GEOMETRY You will learn more about transforming geometric figures in Chapter 11.

Horizontal Translations

Vertical Translations

The graph of y = f(x − h) is a horizontal translation of the graph of y = f(x), where h ≠ 0.

The graph of y = f (x) + k is a vertical translation of the graph of y = f (x), where k ≠ 0.

y = f(x)

y

y = f(x) + k, k>0

y = f(x − h), h 0.

y

Adding k to the outputs shifts the graph down when k < 0 and up when k > 0.

Horizontal and Vertical Translations Let f(x) = 2x − 1. Graph (a) g(x) = f(x) + 3 and (b) t(x) = f(x + 3). Describe the transformations from the graph of f to the graphs of g and t.

SOLUTION

LOOKING FOR A PATTERN In part (a), the output of g is equal to the output of f plus 3. In part (b), the output of t is equal to the output of f when the input of f is 3 more than the input of t.

146

Chapter 3

a. The function g is of the form y = f (x) + k, where k = 3. So, the graph of g is a vertical translation 3 units up of the graph of f. 4

g(x) = f(x) + 3

b. The function t is of the form y = f(x − h), where h = −3. So, the graph of t is a horizontal translation 3 units left of the graph of f.

y

t(x) = f(x + 3)

y 5 3

2

f(x) = 2x − 1

f(x) = 2x − 1 −2

Graphing Linear Functions

2

1

x −2

2

x

Core Concept A reflection is a transformation that flips a graph over a line called the line of reflection.

STUDY TIP A reflected point is the same distance from the line of reflection as the original point but on the opposite side of the line.

Reflections in the x-axis

Reflections in the y-axis

The graph of y = −f(x) is a reflection in the x-axis of the graph of y = f(x).

The graph of y = f (−x) is a reflection in the y-axis of the graph of y = f (x).

y

y = f(x)

y

y = f(−x)

y = f(x)

x

x

y = −f(x)

Multiplying the outputs by −1 changes their signs.

Multiplying the inputs by −1 changes their signs.

Reflections in the x-axis and the y-axis Let f(x) = —12 x + 1. Graph (a) g(x) = −f(x) and (b) t(x) = f(−x). Describe the transformations from the graph of f to the graphs of g and t.

SOLUTION a. To find the outputs of g, multiply the outputs of f by −1. The graph of g consists of the points (x, −f(x)). x

−4

−2

0

f (x)

−1

0

1

1

0

−1

−f (x)

b. To find the outputs of t, multiply the inputs by −1 and then evaluate f. The graph of t consists of the points (x, f(−x)). −2

0

2

−x

2

0

−2

f (−x)

2

1

0

x

y

g(x) = −f(x)

t(x) = f(−x)

2

y 2

−4

f(x) =

−2 1 x 2

+1

2 x

x

−2

−4

−2 1

The graph of g is a reflection in the x-axis of the graph of f.

Monitoring Progress

f(x) = 2 x + 1

2 −2

The graph of t is a reflection in the y-axis of the graph of f.

Help in English and Spanish at BigIdeasMath.com

Using f, graph (a) g and (b) h. Describe the transformations from the graph of f to the graphs of g and h. 1. f(x) = 3x + 1; g(x) = f(x) − 2; h(x) = f(x − 2) 2. f(x) = −4x − 2; g(x) = −f(x); h(x) = f(−x)

Section 3.6

Transformations of Graphs of Linear Functions

147

Stretches and Shrinks You can transform a function by multiplying all the x-coordinates (inputs) by the same factor a. When a > 1, the transformation is a horizontal shrink because the graph shrinks toward the y-axis. When 0 < a < 1, the transformation is a horizontal stretch because the graph stretches away from the y-axis. In each case, the y-intercept stays the same. You can also transform a function by multiplying all the y-coordinates (outputs) by the same factor a. When a > 1, the transformation is a vertical stretch because the graph stretches away from the x-axis. When 0 < a < 1, the transformation is a vertical shrink because the graph shrinks toward the x-axis. In each case, the x-intercept stays the same.

Core Concept STUDY TIP The graphs of y = f(–ax) and y = –a f(x) represent a stretch or shrink and a reflection in the x- or y-axis of the graph of y = f(x).

⋅

Horizontal Stretches and Shrinks

Vertical Stretches and Shrinks

The graph of y = f(ax) is a horizontal 1 stretch or shrink by a factor of — of a the graph of y = f(x), where a > 0 and a ≠ 1.

The graph of y = a f(x) is a vertical stretch or shrink by a factor of a of the graph of y = f(x), where a > 0 and a ≠ 1.

y = f(ax), a>1

⋅

y = a ∙ f(x), a>1 y y = f(x)

y = f(x)

y

y = a ∙ f(x), 0 9

6. 24 ≤ −6t

8. −5z + 1 < −14

9. 4k − 16 < k + 2

7. 2a − 5 ≤ 13 10. 7w + 12 ≥ 2w − 3

11. ABSTRACT REASONING The graphs of the linear functions g and h have different slopes. The

value of both functions at x = a is b. When g and h are graphed in the same coordinate plane, what happens at the point (a, b)?

Dynamic Solutions available at BigIdeasMath.com

215

Mathematical Practices

Mathematically proficient students use technological tools to explore concepts.

Using a Graphing Calculator

Core Concept Finding the Point of Intersection You can use a graphing calculator to find the point of intersection, if it exists, of the graphs of two linear equations. 1.

Enter the equations into a graphing calculator.

2.

Graph the equations in an appropriate viewing window, so that the point of intersection is visible.

3.

Use the intersect feature of the graphing calculator to find the point of intersection.

Using a Graphing Calculator Use a graphing calculator to find the point of intersection, if it exists, of the graphs of the two linear equations. 1

y = −—2 x + 2

Equation 1

y = 3x − 5

Equation 2

SOLUTION

4

The slopes of the lines are not the same, so you know that the lines intersect. Enter the equations into a graphing calculator. Then graph the equations in an appropriate viewing window.

1

y = −2 x + 2 −6

6

y = 3x − 5 −4 4

Use the intersect feature to find the point of intersection of the lines. −6

The point of intersection is (2, 1).

6

Intersection X=2 Y=1 −4

Monitoring Progress Use a graphing calculator to find the point of intersection of the graphs of the two linear equations. 1. y = −2x − 3

y = —12 x − 3

216

Chapter 5

2. y = −x + 1

3. 3x − 2y = 2

y=x−2

Solving Systems of Linear Equations

2x − y = 2

5.1

Solving Systems of Linear Equations by Graphing Essential Question

How can you solve a system of linear

equations?

Writing a System of Linear Equations Work with a partner. Your family opens a bed-and-breakfast. They spend $600 preparing a bedroom to rent. The cost to your family for food and utilities is $15 per night. They charge $75 per night to rent the bedroom. a. Write an equation that represents the costs. $15 per Cost, C = night (in dollars)

Number of

⋅ nights, x

+ $600

b. Write an equation that represents the revenue (income).

MODELING WITH MATHEMATICS To be proficient in math, you need to identify important quantities in real-life problems and map their relationships using tools such as diagrams, tables, and graphs.

$75 per Revenue, R = night (in dollars)

Number of

⋅ nights, x

c. A set of two (or more) linear equations is called a system of linear equations. Write the system of linear equations for this problem.

Using a Table or Graph to Solve a System Work with a partner. Use the cost and revenue equations from Exploration 1 to determine how many nights your family needs to rent the bedroom before recovering the cost of preparing the bedroom. This is the break-even point.

a. Copy and complete the table. x (nights)

0

1

2

3

4

5

6

7

8

9

10

11

C (dollars) R (dollars)

b. How many nights does your family need to rent the bedroom before breaking even? c. In the same coordinate plane, graph the cost equation and the revenue equation from Exploration 1. d. Find the point of intersection of the two graphs. What does this point represent? How does this compare to the break-even point in part (b)? Explain.

Communicate Your Answer 3. How can you solve a system of linear equations? How can you check your

solution? 4. Solve each system by using a table or sketching a graph. Explain why you chose

each method. Use a graphing calculator to check each solution. a. y = −4.3x − 1.3 y = 1.7x + 4.7 Section 5.1

b. y = x y = −3x + 8

c. y = −x − 1 y = 3x + 5

Solving Systems of Linear Equations by Graphing

217

5.1

Lesson

What You Will Learn Check solutions of systems of linear equations. Solve systems of linear equations by graphing. Use systems of linear equations to solve real-life problems.

Core Vocabul Vocabulary larry system of linear equations, p. 218 solution of a system of linear equations, p. 218 Previous linear equation ordered pair

Systems of Linear Equations A system of linear equations is a set of two or more linear equations in the same variables. An example is shown below. x+y=7

Equation 1

2x − 3y = −11

Equation 2

A solution of a system of linear equations in two variables is an ordered pair that is a solution of each equation in the system.

Checking Solutions Tell whether the ordered pair is a solution of the system of linear equations. a. (2, 5); x + y = 7 2x − 3y = −11

b. (−2, 0); y = −2x − 4 y=x+4

Equation 1 Equation 2

Equation 1 Equation 2

SOLUTION a. Substitute 2 for x and 5 for y in each equation. Equation 1

Equation 2

x+y=7

2x − 3y = −11

? 2+5= 7

READING A system of linear equations is also called a linear system.

7=7

?

2(2) − 3(5) = −11



−11 = −11



Because the ordered pair (2, 5) is a solution of each equation, it is a solution of the linear system. b. Substitute −2 for x and 0 for y in each equation. Equation 1

Equation 2

y = −2x − 4

y=x+4

? 0 = −2(−2) − 4 0=0



?

0 = −2 + 4 0≠2

✗

The ordered pair (−2, 0) is a solution of the first equation, but it is not a solution of the second equation. So, (−2, 0) is not a solution of the linear system.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Tell whether the ordered pair is a solution of the system of linear equations. 1. (1, −2);

218

Chapter 5

2x + y = 0 −x + 2y = 5

Solving Systems of Linear Equations

2. (1, 4);

y = 3x + 1 y = −x + 5

Solving Systems of Linear Equations by Graphing The solution of a system of linear equations is the point of intersection of the graphs of the equations.

Core Concept Solving a System of Linear Equations by Graphing Step 1 Graph each equation in the same coordinate plane. Step 2 Estimate the point of intersection. Step 3 Check the point from Step 2 by substituting for x and y in each equation of the original system.

REMEMBER Note that the linear equations are in slope-intercept form. You can use the method presented in Section 3.5 to graph the equations.

Solving a System of Linear Equations by Graphing Solve the system of linear equations by graphing. y = −2x + 5

Equation 1

y = 4x − 1

Equation 2

SOLUTION Step 1 Graph each equation.

y

Step 2 Estimate the point of intersection. The graphs appear to intersect at (1, 3).

(1, 1 3)

Step 3 Check your point from Step 2.

2

Equation 1

Equation 2

y = −2x + 5

y = 4x − 1

? 3 = −2(1) + 5 3=3



y = 4x − 1

y = −2x + 5

−4

? 3 = 4(1) − 1

3=3

−2

−1

2

4 x



The solution is (1, 3). Check Use the table or intersect feature of a graphing calculator to check your answer. 6

X

When x = 1, the corresponding y-values are equal.

-2 -1 0 1 2 3 4

Y1

Y2

9 7 5 3 1 -1 -3

-9 -5 -1 3 7 11 15

y = −2x + 5 −6

6 Intersection X=1 Y=3 −2

X=1

Monitoring Progress

y = 4x − 1

Help in English and Spanish at BigIdeasMath.com

Solve the system of linear equations by graphing. 3. y = x − 2

y = −x + 4

Section 5.1

1

4. y = —2 x + 3

y=

3 −—2 x

−5

5. 2x + y = 5

3x − 2y = 4

Solving Systems of Linear Equations by Graphing

219

Solving Real-Life Problems Modeling with Mathematics A roofing contractor buys 30 bundles of shingles and 4 rolls of roofing paper for $1040. In a second purchase (at the same prices), the contractor buys 8 bundles of shingles for $256. Find the price per bundle of shingles and the price per roll of roofing paper.

SOLUTION 1. Understand the Problem You know the total price of each purchase and how many of each item were purchased. You are asked to find the price of each item. 2. Make a Plan Use a verbal model to write a system of linear equations that represents the problem. Then solve the system of linear equations. 3. Solve the Problem Words

30 8

per ⋅ Price bundle Price per

⋅ bundle

+4 +0

⋅ Price per roll

⋅ Price per roll

= 1040 = 256

Variables Let x be the price (in dollars) per bundle and let y be the price (in dollars) per roll. System

30x + 4y = 1040

Equation 1

8x = 256

Equation 2

Step 1 Graph each equation. Note that only the first quadrant is shown because x and y must be positive. Step 2 Estimate the point of intersection. The graphs appear to intersect at (32, 20). Step 3 Check your point from Step 2. Equation 1

30x + 4y = 1040 ? 30(32) + 4(20) = 1040 1040 = 1040

y 320

y = −7.5x + 260

240

x = 32

160 80

Equation 2



8x = 256 ? 8(32) = 256 256 = 256

0

(32, 20) 0

8

16

24

32

x



The solution is (32, 20). So, the price per bundle of shingles is $32, and the price per roll of roofing paper is $20. 4. Look Back You can use estimation to check that your solution is reasonable. A bundle of shingles costs about $30. So, 30 bundles of shingles and 4 rolls of roofing paper (at $20 per roll) cost about 30(30) + 4(20) = $980, and 8 bundles of shingles costs about 8(30) = $240. These prices are close to the given values, so the solution seems reasonable.

Monitoring Progress

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6. You have a total of 18 math and science exercises for homework. You have

six more math exercises than science exercises. How many exercises do you have in each subject?

220

Chapter 5

Solving Systems of Linear Equations

Exercises

5.1

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY Do the equations 5y − 2x = 18 and 6x = −4y − 10 form a system of linear

equations? Explain. 2. DIFFERENT WORDS, SAME QUESTION Consider the system of linear equations −4x + 2y = 4

and 4x − y = −6. Which is different? Find “both” answers. Solve the system of linear equations.

Solve each equation for y.

Find the point of intersection of the graphs of the equations.

Find an ordered pair that is a solution of each equation in the system.

Monitoring Progress and Modeling with Mathematics In Exercises 3–8, tell whether the ordered pair is a solution of the system of linear equations. (See Example 1.) x−y=6 x+y=8 3. (2, 6); 4. (8, 2); 2x − 10y = 4 3x − y = 0 5. (−1, 3);

7. (−2, 1);

13. y = −x + 7

18. 4x − 4y = 20

19. x − 4y = −4

20. 3y + 4x = 3

2x − y = −4

−3x − 4y = 12

y x

−2

21.

4 2



x 1

11. 6y + 3x = 18

2

x

The solution of the linear system x − 3y = 6 and 2x − 3y = 3 is (3, −1).

2x + 4y = 8 22.

y 4

4



The solution of the linear system y = 2x − 1 and y=x+1 is x = 2.

y 4 2

2 −2 −2

y

−1

4

y

−4

2

12. 2x − y = −2

−x + 4y = 24

−6

x + 3y = −6

and correct the error in solving the system of linear equations.

y − 2x = −4

y

y = −5

ERROR ANALYSIS In Exercises 21 and 22, describe

10. x + y = 5

4

1

y = −—2 x + 11

17. 9x + 3y = −3

In Exercises 9–12, use the graph to solve the system of linear equations. Check your solution.

2

3

16. y = —4 x − 4

y = —23 x + 5

6x + 5y = −7 6x + 3y = 12 8. (5, −6); 2x − 4y = −8 4x + y = 14

4x + y = 1

y = 2x − 8

15. y = —3 x + 2

y = 2x + 6 y = −3x − 14

9. x − y = 4

14. y = −x + 4

y=x+1 1

y = −7x − 4 y = 8x + 5

6. (−4, −2);

In Exercises 13–20, solve the system of linear equations by graphing. (See Example 2.)

2

x

x

Section 5.1

2

4

x

Solving Systems of Linear Equations by Graphing

221

USING TOOLS In Exercises 23–26, use a graphing

31. COMPARING METHODS Consider the equation

calculator to solve the system of linear equations.

x + 2 = 3x − 4.

23. 0.2x + 0.4y = 4

a. Solve the equation using algebra.

24. −1.6x − 3.2y = −24

−0.6x + 0.6y = −3

2.6x + 2.6y = 26

25. −7x + 6y = 0

b. Solve the system of linear equations y = x + 2 and y = 3x − 4 by graphing.

26. 4x − y = 1.5

0.5x + y = 2

c. How is the linear system and the solution in part (b) related to the original equation and the solution in part (a)?

2x + y = 1.5

27. MODELING WITH MATHEMATICS You have

40 minutes to exercise at the gym, and you want to burn 300 calories total using both machines. How much time should you spend on each machine? (See Example 3.) Elliptical Trainer

32. HOW DO YOU SEE IT? A teacher is purchasing

binders for students. The graph shows the total costs of ordering x binders from three different companies.

Stationary Bike Cost (dollars)

Buying Binders y 150 125

6 calories per minute

Company C

75 50

0 15 20 25 30 35 40 45 50 x

Number of binders

a. For what numbers of binders are the costs the same at two different companies? Explain.

28. MODELING WITH MATHEMATICS

You sell small and large candles at a craft fair. You collect $144 selling a total of 28 candles. How many of each type of candle did you sell?

Company A

100

0

8 calories per minute

Company B

$6 each

b. How do your answers in part (a) relate to systems of linear equations? $4 each

33. MAKING AN ARGUMENT You and a friend are going

29. MATHEMATICAL CONNECTIONS Write a linear

hiking but start at different locations. You start at the trailhead and walk 5 miles per hour. Your friend starts 3 miles from the trailhead and walks 3 miles per hour.

equation that represents the area and a linear equation that represents the perimeter of the rectangle. Solve the system of linear equations by graphing. Interpret your solution.

you

(3x − 3) cm 6 cm your friend

30. THOUGHT PROVOKING Your friend’s bank account

balance (in dollars) is represented by the equation y = 25x + 250, where x is the number of months. Graph this equation. After 6 months, you want to have the same account balance as your friend. Write a linear equation that represents your account balance. Interpret the slope and y-intercept of the line that represents your account balance.

Maintaining Mathematical Proficiency Solve the literal equation for y. 34. 10x + 5y = 5x + 20

222

Chapter 5

a. Write and graph a system of linear equations that represents this situation. b. Your friend says that after an hour of hiking you will both be at the same location on the trail. Is your friend correct? Use the graph from part (a) to explain your answer. Reviewing what you learned in previous grades and lessons

(Section 1.5) 35. 9x + 18 = 6y − 3x

Solving Systems of Linear Equations

1

36. —34 x + —4 y = 5

5.2

Solving Systems of Linear Equations by Substitution Essential Question

How can you use substitution to solve a

system of linear equations?

Using Substitution to Solve Systems Work with a partner. Solve each system of linear equations using two methods. Method 1 Solve for x first. Solve for x in one of the equations. Substitute the expression for x into the other equation to find y. Then substitute the value of y into one of the original equations to find x. Method 2 Solve for y first. Solve for y in one of the equations. Substitute the expression for y into the other equation to find x. Then substitute the value of x into one of the original equations to find y. Is the solution the same using both methods? Explain which method you would prefer to use for each system. a. x + y = −7

b. x − 6y = −11

−5x + y = 5

3x + 2y = 7

c. 4x + y = −1 3x − 5y = −18

Writing and Solving a System of Equations Work with a partner. a. Write a random ordered pair with integer coordinates. One way to do this is to use a graphing calculator. The ordered pair generated at the right is (−2, −3).

ATTENDING TO PRECISION To be proficient in math, you need to communicate precisely with others.

b. Write a system of linear equations that has your ordered pair as its solution.

Choose two random integers between −5 and 5. randInt(-5‚5‚2) {-2 -3}

c. Exchange systems with your partner and use one of the methods from Exploration 1 to solve the system. Explain your choice of method.

Communicate Your Answer 3. How can you use substitution to solve a system of linear equations? 4. Use one of the methods from Exploration 1 to solve each system of linear

equations. Explain your choice of method. Check your solutions. a. x + 2y = −7

b. x − 2y = −6

2x − y = −9

2x + y = −2

d. 3x + 2y = 13 x − 3y = −3

Section 5.2

e. 3x − 2y = 9 −x − 3y = 8

c. −3x + 2y = −10 −2x + y = −6 f. 3x − y = −6 4x + 5y = 11

Solving Systems of Linear Equations by Substitution

223

5.2 Lesson

What You Will Learn Solve systems of linear equations by substitution. Use systems of linear equations to solve real-life problems.

Core Vocabul Vocabulary larry Previous system of linear equations solution of a system of linear equations

Solving Linear Systems by Substitution Another way to solve a system of linear equations is to use substitution.

Core Concept Solving a System of Linear Equations by Substitution Step 1 Solve one of the equations for one of the variables. Step 2 Substitute the expression from Step 1 into the other equation and solve for the other variable. Step 3 Substitute the value from Step 2 into one of the original equations and solve.

Solving a System of Linear Equations by Substitution Solve the system of linear equations by substitution. y = −2x − 9

Equation 1

6x − 5y = −19

Equation 2

SOLUTION Step 1 Equation 1 is already solved for y. Step 2 Substitute −2x − 9 for y in Equation 2 and solve for x. 6x − 5y = −19

Check

Equation 2

6x − 5(−2x − 9) = −19

Equation 1

y = −2x − 9 ? −1 = −2(−4) − 9 −1 = −1

Substitute −2x − 9 for y.

6x + 10x + 45 = −19

Distributive Property

16x + 45 = −19

Combine like terms.

16x = −64



Subtract 45 from each side.

x = −4

Divide each side by 16.

Step 3 Substitute −4 for x in Equation 1 and solve for y.

Equation 2

y = −2x − 9

6x − 5y = −19 ? 6(−4) − 5(−1) = −19 −19 = −19



Equation 1

= −2(−4) − 9

Substitute −4 for x.

=8−9

Multiply.

= −1

Subtract.

The solution is (−4, −1).

Monitoring Progress

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Solve the system of linear equations by substitution. Check your solution. 1. y = 3x + 14

y = −4x 224

Chapter 5

Solving Systems of Linear Equations

2. 3x + 2y = 0

y=

1 —2 x

−1

3. x = 6y − 7

4x + y = −3

Solving a System of Linear Equations by Substitution

ANOTHER WAY You could also begin by solving for x in Equation 1, solving for y in Equation 2, or solving for x in Equation 2.

Solve the system of linear equations by substitution. −x + y = 3

Equation 1

3x + y = −1

Equation 2

SOLUTION Step 1 Solve for y in Equation 1. y=x+3

Revised Equation 1

Step 2 Substitute x + 3 for y in Equation 2 and solve for x. 3x + y = −1 3x + (x + 3) = −1 4x + 3 = −1 4x = −4 x = −1

Equation 2 Substitute x + 3 for y. Combine like terms. Subtract 3 from each side. Divide each side by 4.

Step 3 Substitute −1 for x in Equation 1 and solve for y. −x + y = 3 −(−1) + y = 3 y=2

Equation 1 Substitute −1 for x. Subtract 1 from each side.

The solution is (−1, 2). Graphical Check

Algebraic Check

4

Equation 1

−x + y = 3 ? −(−1) + 2 = 3 3=3

y=x+3



Equation 2

y = −3x − 1

−5

4 Intersection X=-1 Y=2 −2

3x + y = −1 ? 3(−1) + 2 = −1 −1 = −1



Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the system of linear equations by substitution. Check your solution. 4. x + y = −2

5. −x + y = −4

−3x + y = 6 6. 2x − y = −5

3x − y = 1

Section 5.2

4x − y = 10 7. x − 2y = 7

3x − 2y = 3

Solving Systems of Linear Equations by Substitution

225

Solving Real-Life Problems Modeling with Mathematics A drama club earns $1040 from a production. A total of 64 adult tickets and 132 student tickets are sold. An adult ticket costs twice as much as a student ticket. Write a system of linear equations that represents this situation. What is the price W oof each type of ticket?

SOLUTION S 11. Understand the Problem You know the amount earned, the total numbers of adult and student tickets sold, and the relationship between the price of an adult ticket and the price of a student ticket. You are asked to write a system of linear equations that represents the situation and find the price of each type of ticket. 22. Make a Plan Use a verbal model to write a system of linear equations that represents the problem. Then solve the system of linear equations. 33. Solve the Problem Words

64

ticket ⋅ Adult price

Adult ticket =2 price

+ 132

⋅ Student ticket price

= 1040

⋅ Student ticket price

Variables Let x be the price (in dollars) of an adult ticket and let y be the price (in dollars) of a student ticket. System

64x + 132y = 1040

Equation 1

x = 2y

Equation 2

Step 1 Equation 2 is already solved for x. Step 2 Substitute 2y for x in Equation 1 and solve for y. 64x + 132y = 1040

STUDY TIP You can use either of the original equations to solve for x. However, using Equation 2 requires fewer calculations.

Equation 1

64(2y) + 132y = 1040

Substitute 2y for x.

260y = 1040

Simplify.

y=4

Simplify.

Step 3 Substitute 4 for y in Equation 2 and solve for x. x = 2y

Equation 2

x = 2(4)

Substitute 4 for y.

x=8

Simplify.

The solution is (8, 4). So, an adult ticket costs $8 and a student ticket costs $4. 4. Look Back To check that your solution is correct, substitute the values of x and y into both of the original equations and simplify. 64(8) + 132(4) = 1040 1040 = 1040

Monitoring Progress



8 = 2(4) 8=8



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8. There are a total of 64 students in a drama club and a yearbook club. The drama

club has 10 more students than the yearbook club. Write a system of linear equations that represents this situation. How many students are in each club? 226

Chapter 5

Solving Systems of Linear Equations

5.2

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Describe how to solve a system of linear equations by substitution. 2. NUMBER SENSE When solving a system of linear equations by substitution, how do you decide

which variable to solve for in Step 1?

Monitoring Progress and Modeling with Mathematics In Exercises 3−8, tell which equation you would choose to solve for one of the variables. Explain. 3. x + 4y = 30

x − 2y = 0

4. 3x − y = 0

5x − y = 5

6. 3x − 2y = 19

4x + 3y = −5

8. 3x + 5y = 25

x − 2y = −6

In Exercises 9–16, solve the sytem of linear equations by substitution. Check your solution. (See Examples 1 and 2.) 9. x = 17 − 4y

y=x−2

3x + 4y = 8 13. 2x = 12

15. 5x + 2y = 9

x + y = −3

corn and wheat on a 180-acre farm. The farmer wants to plant three times as many acres of corn as wheat. Write a system of linear equations that represents this situation. How many acres of each crop should the farmer plant? (See Example 3.)

12. −5x + 3y = 51

y = 10x − 8

x − 9 = −1

20. MODELING WITH MATHEMATICS A company that

16. 11x − 7y = −14

x − 2y = −4

17. ERROR ANALYSIS Describe and correct the error in

solving for one of the variables in the linear system 8x + 2y = −12 and 5x − y = 4.



3x + y = 9 3x + 6 = 9 3x = 3 x=1

19. MODELING WITH MATHEMATICS A farmer plants

14. 2x − y = 23

x − 5y = −29

Step 3

10. 6x − 9 = y

y = −3x

11. x = 16 − 4y

Step 1 3x + y = 9 y = 9 − 3x Step 2 4x + 2(9 − 3x) = 6 4x + 18 − 6x = 6 −2x = −12 x=6

x+y=8

7. x − y = −3

solving for one of the variables in the linear system 4x + 2y = 6 and 3x + y = 9.



2x + y = −10

5. 5x + 3y = 11

18. ERROR ANALYSIS Describe and correct the error in

offers tubing trips down a river rents tubes for a person to use and “cooler” tubes to carry food and water. A group spends $270 to rent a total of 15 tubes. Write a system of linear equations that represents this situation. How many of each type of tube does the group rent?

Step 1 5x − y = 4 −y = −5x + 4 y = 5x − 4 Step 2 5x − (5x − 4) = 4 5x − 5x + 4 = 4 4=4

Section 5.2

Solving Systems of Linear Equations by Substitution

227

In Exercises 21–24, write a system of linear equations that has the ordered pair as its solution. 21. (3, 5)

22. (−2, 8)

23. (−4, −12)

24. (15, −25)

30. MAKING AN ARGUMENT Your friend says that given

a linear system with an equation of a horizontal line and an equation of a vertical line, you cannot solve the system by substitution. Is your friend correct? Explain. 31. OPEN-ENDED Write a system of linear equations in

25. PROBLEM SOLVING A math test is worth 100 points

which (3, −5) is a solution of Equation 1 but not a solution of Equation 2, and (−1, 7) is a solution of the system.

and has 38 problems. Each problem is worth either 5 points or 2 points. How many problems of each point value are on the test?

32. HOW DO YOU SEE IT? The graphs of two linear

26. PROBLEM SOLVING An investor owns shares of

equations are shown.

Stock A and Stock B. The investor owns a total of 200 shares with a total value of $4000. How many shares of each stock does the investor own?

y

y=x+1

6

Stock

Price

A

$9.50

B

$27.00

4 1

MATHEMATICAL CONNECTIONS In Exercises 27 and 28,

2

(a) write an equation that represents the sum of the angle measures of the triangle and (b) use your equation and the equation shown to find the values of x and y. 27.

y = 6 − 4x

2

4

6

x

a. At what point do the lines appear to intersect? b. Could you solve a system of linear equations by substitution to check your answer in part (a)? Explain.

x° x + 2 = 3y

33. REPEATED REASONING A radio station plays a total

y°

28.

of 272 pop, rock, and hip-hop songs during a day. The number of pop songs is 3 times the number of rock songs. The number of hip-hop songs is 32 more than the number of rock songs. How many of each type of song does the radio station play?

y°

(y − 18)° 3x − 5y = −22 x°

34. THOUGHT PROVOKING You have $2.65 in coins.

Write a system of equations that represents this situation. Use variables to represent the number of each type of coin.

29. REASONING Find the values of a and b so that the

solution of the linear system is (−9, 1). ax + by = −31 ax − by = −41

Equation 1 Equation 2

35. NUMBER SENSE The sum of the digits of a

two-digit number is 11. When the digits are reversed, the number increases by 27. Find the original number.

Maintaining Mathematical Proficiency Find the sum or difference.

Reviewing what you learned in previous grades and lessons

(Skills Review Handbook)

36. (x − 4) + (2x − 7)

37. (5y − 12) + (−5y − 1)

38. (t − 8) − (t + 15)

39. (6d + 2) − (3d − 3)

40. 4(m + 2) + 3(6m − 4)

41. 2(5v + 6) − 6(−9v + 2)

228

Chapter 5

Solving Systems of Linear Equations

5.3

Solving Systems of Linear Equations by Elimination Essential Question

How can you use elimination to solve a system

of linear equations?

Writing and Solving a System of Equations Work with a partner. You purchase a drink and a sandwich for $4.50. Your friend purchases a drink and five sandwiches for $16.50. You want to determine the price of a drink and the price of a sandwich. a. Let x represent the price (in dollars) of one drink. Let y represent the price (in dollars) of one sandwich. Write a system of equations for the situation. Use the following verbal model. Number of drinks

Number of Price per + ⋅ Price sandwiches ⋅ sandwich per drink

=

Total price

Label one of the equations Equation 1 and the other equation Equation 2. b. Subtract Equation 1 from Equation 2. Explain how you can use the result to solve the system of equations. Then find and interpret the solution.

CHANGING COURSE To be proficient in math, you need to monitor and evaluate your progress and change course using a different solution method, if necessary.

Using Elimination to Solve Systems Work with a partner. Solve each system of linear equations using two methods. Method 1 Subtract. Subtract Equation 2 from Equation 1. Then use the result to solve the system. Method 2 Add. Add the two equations. Then use the result to solve the system. Is the solution the same using both methods? Which method do you prefer? a. 3x − y = 6

b. 2x + y = 6

3x + y = 0

2x − y = 2

c. x − 2y = −7 x + 2y = 5

Using Elimination to Solve a System Work with a partner. 2x + y = 7 x + 5y = 17

Equation 1 Equation 2

a. Can you eliminate a variable by adding or subtracting the equations as they are? If not, what do you need to do to one or both equations so that you can? b. Solve the system individually. Then exchange solutions with your partner and compare and check the solutions.

Communicate Your Answer 4. How can you use elimination to solve a system of linear equations? 5. When can you add or subtract the equations in a system to solve the system?

When do you have to multiply first? Justify your answers with examples. 6. In Exploration 3, why can you multiply an equation in the system by a constant

and not change the solution of the system? Explain your reasoning. Section 5.3

Solving Systems of Linear Equations by Elimination

229

5.3 Lesson

What You Will Learn Solve systems of linear equations by elimination. Use systems of linear equations to solve real-life problems.

Core Vocabul Vocabulary larry

Solving Linear Systems by Elimination

Previous coefficient

Core Concept Solving a System of Linear Equations by Elimination Step 1 Multiply, if necessary, one or both equations by a constant so at least one pair of like terms has the same or opposite coefficients. Step 2 Add or subtract the equations to eliminate one of the variables. Step 3 Solve the resulting equation. Step 4 Substitute the value from Step 3 into one of the original equations and solve for the other variable. You can use elimination to solve a system of equations because replacing one equation in the system with the sum of that equation and a multiple of the other produces a system that has the same solution. Here is why.

System 1 a=b c=d

Equation 1 Equation 2

System 2 a + kc = b + kd c=d

Equation 3 Equation 2

Consider System 1. In this system, a and c are algebraic expressions, and b and d are constants. Begin by multiplying each side of Equation 2 by a constant k. By the Multiplication Property of Equality, kc = kd. You can rewrite Equation 1 as Equation 3 by adding kc on the left and kd on the right. You can rewrite Equation 3 as Equation 1 by subtracting kc on the left and kd on the right. Because you can rewrite either system as the other, System 1 and System 2 have the same solution.

Solving a System of Linear Equations by Elimination Solve the system of linear equations by elimination. 3x + 2y = 4 3x − 2y = −4

Equation 1 Equation 2

SOLUTION Step 1 Because the coefficients of the y-terms are opposites, you do not need to multiply either equation by a constant. Step 2 Add the equations. 3x + 2y = 4 3x − 2y = −4 6x =0

Check Equation 1

3x + 2y = 4 ? 3(0) + 2(2) = 4 4=4



3x − 2y = −4 ? 3(0) − 2(2) = −4 −4 = −4

Chapter 5

Equation 2 Add the equations.

Step 3 Solve for x. 6x = 0 x=0

Equation 2

230

Equation 1

Resulting equation from Step 2 Divide each side by 6.

Step 4 Substitute 0 for x in one of the original equations and solve for y.



3x + 2y = 4 3(0) + 2y = 4 y=2 The solution is (0, 2).

Solving Systems of Linear Equations

Equation 1 Substitute 0 for x. Solve for y.

Solving a System of Linear Equations by Elimination Solve the system of linear equations by elimination.

ANOTHER WAY To use subtraction to eliminate one of the variables, multiply Equation 2 by 2 and then subtract the equations.

−10x + 3y = 1

Equation 1

−5x − 6y = 23

Equation 2

SOLUTION Step 1 Multiply Equation 2 by −2 so that the coefficients of the x-terms are opposites.

− 10x + 3y = 1

−10x + 3y = 1

−(−10x − 12y = 46)

−5x − 6y = 23

15y = −45

Multiply by −2.

−10x + 3y = 1

Equation 1

10x + 12y = −46

Revised Equation 2

Step 2 Add the equations. − 10x + 3y = 1 10x + 12y = −46 15y = −45

Equation 1 Revised Equation 2 Add the equations.

Step 3 Solve for y. 15y = −45

Resulting equation from Step 2

y = −3

Divide each side by 15.

Step 4 Substitute −3 for y in one of the original equations and solve for x. Check

−5x − 6y = 23

10

Equation 2

−5x − 6(−3) = 23 Equation 1

−10

−5x + 18 = 23

Multiply.

−5x = 5

10

Subtract 18 from each side.

x = −1

Equation 2 Intersection X=-1 Y=-3 3 −10

Substitute −3 for y.

Divide each side by −5.

The solution is (−1, −3).

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the system of linear equations by elimination. Check your solution. 1. 3x + 2y = 7

2. x − 3y = 24

3. x + 4y = 22

3x + y = 12

4x + y = 13

−3x + 4y = 5

Concept Summary Methods for Solving Systems of Linear Equations Method

When to Use

Graphing (Lesson 5.1)

To estimate solutions

Substitution (Lesson 5.2) Elimination (Lesson 5.3) Elimination (Multiply First) (Lesson 5.3)

When one of the variables in one of the equations has a coefficient of 1 or −1 When at least one pair of like terms has the same or opposite coefficients When one of the variables cannot be eliminated by adding or subtracting the equations

Section 5.3

Solving Systems of Linear Equations by Elimination

231

Solving Real-Life Problems Modeling with Mathematics A business with two locations buys seven large delivery vans and five small delivery vans. Location A receives five large vans and two small vans for a total cost of $235,000. Location B receives two large vans and three small vans for a total cost of $160,000. What is the cost of each type of van?

SOLUTION 1. Understand the Problem You know how many of each type of van each location receives. You also know the total cost of the vans for each location. You are asked to find the cost of each type of van. 2. Make a Plan Use a verbal model to write a system of linear equations that represents the problem. Then solve the system of linear equations. 3. Solve the Problem Words

5

of ⋅ Cost large van

+2

of = 235,000 ⋅ Cost small van

2

of ⋅ Cost large van

+3

of = 160,000 ⋅ Cost small van

Variables Let x be the cost (in dollars) of a large van and let y be the cost (in dollars) of a small van. System

5x + 2y = 235,000

Equation 1

2x + 3y = 160,000

Equation 2

Step 1 Multiply Equation 1 by −3. Multiply Equation 2 by 2.

STUDY TIP In Example 3, both equations are multiplied by a constant so that the coefficients of the y-terms are opposites.

5x + 2y = 235,000 Multiply by −3.

−15x − 6y = −705,000

Revised Equation 1

2x + 3y = 160,000 Multiply by 2.

4x + 6y = 320,000

Revised Equation 2

Step 2 Add the equations. −15x − 6y = −705,000 4x + 6y = −11x

Revised Equation 1

320,000

Revised Equation 2

= −385,000

Add the equations.

Step 3 Solving the equation −11x = −385,000 gives x = 35,000. Step 4 Substitute 35,000 for x in one of the original equations and solve for y. 5x + 2y = 235,000 5(35,000) + 2y = 235,000 y = 30,000

Equation 1 Substitute 35,000 for x. Solve for y.

The solution is (35,000, 30,000). So, a large van costs $35,000 and a small van costs $30,000. 4. Look Back Check to make sure your solution makes sense with the given information. For Location A, the total cost is 5(35,000) + 2(30,000) = $235,000. For Location B, the total cost is 2(35,000) + 3(30,000) = $160,000. So, the solution makes sense.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

4. Solve the system in Example 3 by eliminating x.

232

Chapter 5

Solving Systems of Linear Equations

5.3

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. OPEN-ENDED Give an example of a system of linear equations that can be solved by first adding the

equations to eliminate one variable. 2x − 3y = −4 −5x + 9y = 7

2. WRITING Explain how to solve the system of linear equations

by elimination.

Equation 1 Equation 2

Monitoring Progress and Modeling with Mathematics In Exercises 3−10, solve the system of linear equations by elimination. Check your solution. (See Example 1.) 3. x + 2y = 13

−x + y = 5 5. 5x + 6y = 50

x − 6y = −26 7. −3x − 5y = −7

−4x + 5y = 14

4. 9x + y = 2

6. −x + y = 4

4x + 3y = 8

x + 3y = 4

x − 2y = −13 Multiply by −4.

7y − 6 = 3x

11y = −5 −5 y=— 11 21. MODELING WITH MATHEMATICS A service center

In Exercises 11–18, solve the system of linear equations by elimination. Check your solution. (See Examples 2 and 3.) 12. 8x − 5y = 11

3x + 4y = 36 15. 4x − 3y = 8

5x − 2y = −11 17. 9x + 2y = 39

6x + 13y = −9

charges a fee of x dollars for an oil change plus y dollars per quart of oil used. A sample of its sales record is shown. Write a system of linear equations that represents this situation. Find the fee and cost per quart of oil.

4x − 3y = 5

A

14. 10x − 9y = 46

−2x + 3y = 10 16. −2x − 5y = 9

3x + 11y = 4

8x + 11y = 30

solving for one of the variables in the linear system 5x − 7y = 16 and x + 7y = 8. 5x − 7y = 16 x + 7y = 8 4x = 24 x=6 Section 5.3

1

Customer

2 3 4

A B

B Oil Tank Size (quarts) 5 7

C Total Cost $22.45 $25.45

22. MODELING WITH MATHEMATICS A music website

18. 12x − 7y = −2

19. ERROR ANALYSIS Describe and correct the error in



−4x + 8y = −13

−4x − 3y = 9 10. 3x − 30 = y

13. 11x − 20y = 28

4x + 3y = 8

8. 4x − 9y = −21

5x + y = −10

2x + 7y = 9

solving for one of the variables in the linear system 4x + 3y = 8 and x − 2y = −13.



−4x − y = −17

9. −y − 10 = 6x

11. x + y = 2

20. ERROR ANALYSIS Describe and correct the error in

charges x dollars for individual songs and y dollars for entire albums. Person A pays $25.92 to download 6 individual songs and 2 albums. Person B pays $33.93 to download 4 individual songs and 3 albums. Write a system of linear equations that represents this situation. How much does the website charge to download a song? an entire album?

Solving Systems of Linear Equations by Elimination

233

In Exercises 23–26, solve the system of linear equations using any method. Explain why you chose the method. 23. 3x + 2y = 4

30. THOUGHT PROVOKING Write a system of linear

equations that can be added to eliminate a variable or subtracted to eliminate a variable.

24. −6y + 2 = −4x

2y = 8 − 5x

y−2=x

25. y − x = 2

26. 3x + y =

1

31. MATHEMATICAL CONNECTIONS A rectangle has a

perimeter of 18 inches. A new rectangle is formed by doubling the width w and tripling the lengthℓ, as shown. The new rectangle has a perimeter P of 46 inches.

1 —3

2x − 3y = —83

y = −—4 x + 7

27. WRITING For what values of a can you solve the

linear system ax + 3y = 2 and 4x + 5y = 6 by elimination without multiplying first? Explain.

2w

P = 46 in.

28. HOW DO YOU SEE IT? The circle graph shows the

3

results of a survey in which 50 students were asked about their favorite meal.

a. Write and solve a system of linear equations to find the length and width of the original rectangle.

Favorite Meal

b. Find the length and width of the new rectangle. 32. CRITICAL THINKING Refer to the discussion of

Breakfast

System 1 and System 2 on page 230. Without solving, explain why the two systems shown have the same solution.

Dinner 25 Lunch

System 1 3x − 2y = 8 Equation 1 x+y=6 Equation 2

System 2 5x = 20 Equation 3 x + y = 6 Equation 2

a. Estimate the numbers of students who chose breakfast and lunch.

33. PROBLEM SOLVING You are making 6 quarts of

b. The number of students who chose lunch was 5 more than the number of students who chose breakfast. Write a system of linear equations that represents the numbers of students who chose breakfast and lunch.

34. PROBLEM SOLVING A motorboat takes 40 minutes

fruit punch for a party. You have bottles of 100% fruit juice and 20% fruit juice. How many quarts of each type of juice should you mix to make 6 quarts of 80% fruit juice? to travel 20 miles downstream. The return trip takes 60 minutes. What is the speed of the current?

c. Explain how you can solve the linear system in part (b) to check your answers in part (a).

35. CRITICAL THINKING Solve for x, y, and z in the

system of equations. Explain your steps.

29. MAKING AN ARGUMENT Your friend says that any

x + 7y + 3z = 29 3z + x − 2y = −7 5y = 10 − 2x

system of equations that can be solved by elimination can be solved by substitution in an equal or fewer number of steps. Is your friend correct? Explain.

Maintaining Mathematical Proficiency

Equation 1 Equation 2 Equation 3

Reviewing what you learned in previous grades and lessons

Solve the equation. Determine whether the equation has one solution, no solution, or infinitely many solutions. (Section 1.3) 36. 5d − 8 = 1 + 5d 37. 9 + 4t = 12 − 4t 38. 3n + 2 = 2(n − 3)

39. −3(4 − 2v) = 6v − 12

Write an equation of the line that passes through the given point and is parallel to the given line. (Section 4.3) 2 40. (4, 1); y = −2x + 7 41. (0, 6); y = 5x − 3 42. (−5, −2); y = —3 x + 1 234

Chapter 5

Solving Systems of Linear Equations

5.4

Solving Special Systems of Linear Equations Essential Question

Can a system of linear equations have no solution or infinitely many solutions? Using a Table to Solve a System Work with a partner. You invest $450 for equipment to make skateboards. The materials for each skateboard cost $20. You sell each skateboard for $20. a. Write the cost and revenue equations. Then copy and complete the table for your cost C and your revenue R. 0

x (skateboards)

1

2

3

4

5

6

7

8

9

10

C (dollars) R (dollars)

b. When will your company break even? What is wrong?

MODELING WITH MATHEMATICS To be proficient in math, you need to interpret mathematical results in real-life contexts.

Writing and Analyzing a System Work with a partner. A necklace and matching bracelet have two types of beads. The necklace has 40 small beads and 6 large beads and weighs 10 grams. The bracelet has 20 small beads and 3 large beads and weighs 5 grams. The threads holding the beads have no significant weight. a. Write a system of linear equations that represents the situation. Let x be the weight (in grams) of a small bead and let y be the weight (in grams) of a large bead. b. Graph the system in the coordinate plane shown. What do you notice about the two lines? c. Can you find the weight of each type of bead? Explain your reasoning.

y 2 1.5 1 0.5

Communicate Your Answer

0

0

0.1

0.2

0.3

0.4

x

3. Can a system of linear equations have no solution or infinitely many solutions?

Give examples to support your answers. 4. Does the system of linear equations represented by each graph have no solution,

one solution, or infinitely many solutions? Explain. a.

b.

y

6 4

1 −1

4

x

−x + y = 1 2

Section 5.4

y

3

3

x+y=2

6

y=x+2

y=x+2

y=x+2

2

c.

y

4

1 x

−2x + 2y = 4 2

Solving Special Systems of Linear Equations

4

x

235

5.4 Lesson

What You Will Learn Determine the numbers of solutions of linear systems. Use linear systems to solve real-life problems.

Core Vocabul Vocabulary larry

The Numbers of Solutions of Linear Systems

Previous parallel

Core Concept Solutions of Systems of Linear Equations A system of linear equations can have one solution, no solution, or infinitely many solutions. One solution

No solution

y

y

y

ANOTHER WAY

x

x

x

You can solve some linear systems by inspection. In Example 1, notice you can rewrite the system as

Infinitely many solutions

The lines intersect.

The lines are parallel.

The lines are the same.

−2x + y = 1 −2x + y = −5. This system has no solution because −2x + y cannot be equal to both 1 and −5.

Solving a System: No Solution Solve the system of linear equations. y = 2x + 1 y = 2x − 5

Equation 1 Equation 2

SOLUTION Method 1

Solve by graphing.

Graph each equation.

y

The lines have the same slope and different y-intercepts. So, the lines are parallel. Because parallel lines do not intersect, there is no point that is a solution of both equations. So, the system of linear equations has no solution. Method 2

2

2

y = 2x + 1

1

−2 2

1

4

y = 2x − 5

−2

2

−4

1

Solve by substitution.

Substitute 2x − 5 for y in Equation 1. y = 2x + 1

STUDY TIP A linear system with no solution is called an inconsistent system.

236

Chapter 5

2x − 5 = 2x + 1 −5 = 1

✗

Equation 1 Substitute 2x − 5 for y. Subtract 2x from each side.

The equation −5 = 1 is never true. So, the system of linear equations has no solution.

Solving Systems of Linear Equations

x

Solving a System: Infinitely Many Solutions Solve the system of linear equations. −2x + y = 3

Equation 1

−4x + 2y = 6

Equation 2

SOLUTION Method 1 Solve by graphing. 6

Graph each equation. The lines have the same slope and the same y-intercept. So, the lines are the same. Because the lines are the same, all points on the line are solutions of both equations.

−4x + 2y = 6

y

4

−2x + y = 3 1 −4

So, the system of linear equations has infinitely many solutions.

2

4 x

−2

Method 2 Solve by elimination. Step 1 Multiply Equation 1 by −2. −2x + y = 3

Multiply by −2.

−4x + 2y = 6

4x − 2y = −6

Revised Equation 1

−4x + 2y = 6

Equation 2

Step 2 Add the equations.

STUDY TIP A linear system with infinitely many solutions is called a consistent dependent system.

4x − 2y = −6 −4x + 2y = 6 0=0

Revised Equation 1 Equation 2 Add the equations.

The equation 0 = 0 is always true. So, the solutions are all the points on the line −2x + y = 3. The system of linear equations has infinitely many solutions. Check X Use the table feature of -3 a graphing calculator to -2 -1 check your answer. You 0 can see that for any x-value, 1 2 the corresponding y-values 3 are equal. X=0

Monitoring Progress

Y1

Y2

-3 -1 1 3 5 7 9

-3 -1 1 3 5 7 9

Help in English and Spanish at BigIdeasMath.com

Solve the system of linear equations. 1. x + y = 3

2x + 2y = 6 3. x + y = 3

x + 2y = 4

Section 5.4

2. y = −x + 3

2x + 2y = 4 4. y = −10x + 2

10x + y = 10

Solving Special Systems of Linear Equations

237

Solving Real-Life Problems Modeling with Mathematics The perimeter of the trapezoidal piece of land is 48 kilometers. The perimeter of the rectangular piece of land is 144 kilometers. Write and solve a system of linear equations to find the values of x and y.

2x 6y

6y

SOLUTION

4x 18y

9x

9x

18y

1. Understand the Problem You know the perimeter of each piece of land and the side lengths in terms of x or y. You are asked to write and solve a system of linear equations to find the values of x and y. 2. Make a Plan Use the figures and the definition of perimeter to write a system of linear equations that represents the problem. Then solve the system of linear equations. 3. Solve the Problem Perimeter of trapezoid

Perimeter of rectangle

2x + 4x + 6y + 6y = 48 6x + 12y = 48

9x + 9x + 18y + 18y = 144

System

18x + 36y = 144

Equation 1

6x + 12y = 48

Equation 1

18x + 36y = 144

Equation 2

Method 1 Solve by graphing.

y

Graph each equation.

CONNECTIONS TO GEOMETRY When two lines have the same slope and the same y-intercept, the lines coincide. You will learn more about coincident lines in Chapter 10.

Equation 2

6

The lines have the same slope and the same y-intercept. So, the lines are the same.

18x + 36y = 144

4

In this context, x and y must be positive. Because the lines are the same, all the points on the line in Quadrant I are solutions of both equations.

6x + 12y = 48

2 0

0

2

4

6

x

So, the system of linear equations has infinitely many solutions. Method 2 Solve by elimination.

Multiply Equation 1 by −3 and add the equations. 6x + 12y = 48

Multiply by −3.

18x + 36y = 144

−18x − 36y = −144 18x + 36y = 0=0

144

Revised Equation 1 Equation 2 Add the equations.

The equation 0 = 0 is always true. In this context, x and y must be positive. So, the solutions are all the points on the line 6x + 12y = 48 in Quadrant I. The system of linear equations has infinitely many solutions. 4. Look Back Choose a few of the ordered pairs (x, y) that are solutions of Equation 1. You should find that no matter which ordered pairs you choose, they will also be solutions of Equation 2. So, infinitely many solutions seems reasonable.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

5. WHAT IF? What happens to the solution in Example 3 when the perimeter of the

trapezoidal piece of land is 96 kilometers? Explain.

238

Chapter 5

Solving Systems of Linear Equations

Exercises

5.4

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. REASONING Is it possible for a system of linear equations to have exactly two solutions? Explain. 2. WRITING Compare the graph of a system of linear equations that has infinitely many solutions and

the graph of a system of linear equations that has no solution.

Monitoring Progress and Modeling with Mathematics In Exercises 3−8, match the system of linear equations with its graph. Then determine whether the system has one solution, no solution, or infinitely many solutions. 3. −x + y = 1

6. x − y = 0

−4x − 2y = −8

8. 5x + 3y = 17

3x − 6y = 9 A.

x − 3y = −2 B.

y 4

2

C.

2

−2

2

4

x

F.



x

−2

−3

2

x

In Exercises 9–16, solve the system of linear equations. (See Examples 1 and 2.) 10. y = −6x − 8

y = 2x − 4

y = −6x + 8

11. 3x − y = 6

12. −x + 2y = 7

−3x + y = −6

x − 2y = 7 Section 5.4

2

x

−3

The lines do not intersect. So, the system has no solution.

−3

9. y = −2x − 4

−4x + y = 4 4x + y = 12

y 1

2 4

3x − 6y = 30

and correct the error in solving the system of linear equations.

y

2 2

22. 2x − 2y = 16

3x − y = −2

23.

−3

y

21. −18x + 6y = 24

ERROR ANALYSIS In Exercises 23 and 24, describe 1

4x

−2

−1

4x

2x − 2y = −18

y

2

E.

1

D.

y

20. −7x + 7y = 1

4x − 3y = −27

−2

x

12x + 2y = −6

19. 4x + 3y = 27

2

−1

18. y = −6x − 2

−21x + 3y = 39

4

2

4x + 5y = 47

17. y = 7x + 13

y

6

16. 3x − 2y = −5

In Exercises 17–22, use only the slopes and y-intercepts of the graphs of the equations to determine whether the system of linear equations has one solution, no solution, or infinitely many solutions. Explain.

5x − 2y = 6

7. −2x + 4y = 1

−3x + y = 4

6x − 10y = −16

−x + y = −2

5. 2x + y = 4

14. 15x − 5y = −20

−2x − 2y = 4 15. 9x − 15y = 24

4. 2x − 2y = 4

x−y=1

13. 4x + 4y = −8

24.



y = 3x − 8 y = 3x − 12

The lines have the same slope. So, the system has infinitely many solutions. Solving Special Systems of Linear Equations

239

25. MODELING WITH MATHEMATICS A small bag of

30. HOW DO YOU SEE IT? The graph shows information

about the last leg of a 4 × 200-meter relay for three relay teams. Team A’s runner ran about 7.8 meters per second, Team B’s runner ran about 7.8 meters per second, and Team C’s runner ran about 8.8 meters per second.

trail mix contains 3 cups of dried fruit and 4 cups of almonds. A large bag contains 4 —12 cups of dried fruit and 6 cups of almonds. Write and solve a system of linear equations to find the price of 1 cup of dried fruit and 1 cup of almonds. (See Example 3.)

Distance (meters)

Last Leg of 4 × 200-Meter Relay

$9

$6

y

Team A Team C

150

50 0

0

4

Team A is traveling 6 miles per hour and is 2 miles ahead of Team B. Team B is also traveling 6 miles per hour. The teams continue traveling at their current rates for the remainder of the race. Write a system of linear equations that represents this situation. Will Team B catch up to Team A? Explain.

Money collected (dollars)

Washington, D.C.

150

80

22,860

New York City

170

100

27,280

16

20

24

28 x

b. If the race was longer, could Team C’s runner have passed Team A’s runner? Explain. c. If the race was longer, could Team B’s runner have passed Team A’s runner? Explain.

City to Washington, D.C., and then back to New York City. The table shows the number of tickets purchased for each leg of the trip. The cost per ticket is the same for each leg of the trip. Is there enough information to determine the cost of one coach ticket? Explain. Business class tickets

12

a. Estimate the distance at which Team C’s runner passed Team B’s runner.

27. PROBLEM SOLVING A train travels from New York

Coach tickets

8

Time (seconds)

26. MODELING WITH MATHEMATICS In a canoe race,

Destination

Team B

100

31. ABSTRACT REASONING Consider the system of

linear equations y = ax + 4 and y = bx − 2, where a and b are real numbers. Determine whether each statement is always, sometimes, or never true. Explain your reasoning. a. The system has infinitely many solutions. b. The system has no solution. c. When a < b, the system has one solution.

32. MAKING AN ARGUMENT One admission to an ice

skating rink costs x dollars, and renting a pair of ice skates costs y dollars. Your friend says she can determine the exact cost of one admission and one skate rental. Is your friend correct? Explain.

28. THOUGHT PROVOKING Write a system of three

linear equations in two variables so that any two of the equations have exactly one solution, but the entire system of equations has no solution. 29. REASONING In a system of linear equations, one

equation has a slope of 2 and the other equation has 1 a slope of −—3 . How many solutions does the system have? Explain.

3 Admissions 2 Skate Rentals Total $ 38.00

Maintaining Mathematical Proficiency

240

∣ x − 7 ∣ = ∣ 2x − 8 ∣ Chapter 5

36.

Total $ 190.00

Reviewing what you learned in previous grades and lessons

Solve the equation. Check your solutions. (Section 1.4) 33. ∣ 2x + 6 ∣ = ∣ x ∣ 34. ∣ 3x − 45 ∣ = ∣ 12x ∣ 35.

15 Admissions 10 Skate Rentals

∣ 2x + 1 ∣ = ∣ 3x − 11 ∣

Solving Systems of Linear Equations

5.1–5.4

What Did You Learn?

Core Vocabulary system of linear equations, p. 218

solution of a system of linear equations, p. 218

Core Concepts Section 5.1 Solving a System of Linear Equations by Graphing, p. 219

Section 5.2 Solving a System of Linear Equations by Substitution, p. 224

Section 5.3 Solving a System of Linear Equations by Elimination, p. 230

Section 5.4 Solutions of Systems of Linear Equations, p. 236

Mathematical Practices 1.

Describe the given information in Exercise 33 on page 228 and your plan for finding the solution.

2.

Describe another real-life situation similar to Exercise 22 on page 233 and the mathematics that you can apply to solve the problem.

3.

What question(s) can you ask your friend to help her understand the error in the statement she made in Exercise 32 on page 240?

Analyzing Your Errors Study Errors What Happens: You do not study the right material or you do not learn it well enough to remember it on a test without resources such as notes. How to Avoid This Error: Take a practice test. Work with a study group. Discuss the topics on the test with your teacher. Do not try to learn a whole chapter’s worth of material in one night.

241 41 1

5.1–5.4

Quiz

Use the graph to solve the system of linear equations. Check your solution. (Section 5.1) 1

1

1. y = −—3 x + 2

2. y = —2 x − 1

y=x−2 3

3. y = 1

y = 4x + 6

y = 2x + 1

y 2

y

y 2

1 −2

−1

−4 2

−2

4 x

x −2

−2

2

x

−2 −4

Solve the system of linear equations by substitution. Check your solution. (Section 5.2) 4. y = x − 4

−2x + y = 18

5. 2y + x = −4

y − x = −5

6. 3x − 5y = 13

x + 4y = 10

Solve the system of linear equations by elimination. Check your solution. (Section 5.3) 7. x + y = 4

−3x − y = −8

8. x + 3y = 1

5x + 6y = 14

9. 2x − 3y = −5

5x + 2y = 16

Solve the system of linear equations. (Section 5.4) 10. x − y = 1

x−y=6

11. 6x + 2y = 16

2x − y = 2

12. 3x − 3y = −2

−6x + 6y = 4

13. You plant a spruce tree that grows 4 inches per year and a hemlock tree

that grows 6 inches per year. The initial heights are shown. (Section 5.1) a. Write a system of linear equations that represents this situation. b. Solve the system by graphing. Interpret your solution.

14 in. 8 in.

14. It takes you 3 hours to drive to a concert 135 miles away. You drive

55 miles per hour on highways and 40 miles per hour on the rest of the roads. (Section 5.1, Section 5.2, and Section 5.3) a. How much time do you spend driving at each speed? b. How many miles do you drive on highways? the rest of the roads? 15. In a football game, all of the home team’s points are from 7-point

touchdowns and 3-point field goals. The team scores six times. Write and solve a system of linear equations to find the numbers of touchdowns and field goals that the home team scores. (Section 5.1, Section 5.2, and Section 5.3)

242

Chapter 5

Solving Systems of Linear Equations

spruce tree

hemlock tree

5.5

Solving Equations by Graphing Essential Question

How can you use a system of linear equations to solve an equation with variables on both sides? Previously, you learned how to use algebra to solve equations with variables on both sides. Another way is to use a system of linear equations.

Solving an Equation by Graphing 1

Work with a partner. Solve 2x − 1 = −—2 x + 4 by graphing. a. Use the left side to write a linear equation. Then use the right side to write another linear equation.

USING TOOLS STRATEGICALLY To be proficient in math, you need to consider the available tools, which may include pencil and paper or a graphing calculator, when solving a mathematical problem.

b. Graph the two linear equations from part (a). Find the x-value of the point of intersection. Check that the x-value is the solution of

6

y

4

1

2

2x − 1 = −—2 x + 4. c. Explain why this “graphical method” works.

−2

2

4

6 x

−2

Solving Equations Algebraically and Graphically Work with a partner. Solve each equation using two methods. Method 1

Use an algebraic method.

Method 2

Use a graphical method.

Is the solution the same using both methods? 1

a. —12 x + 4 = −—4 x + 1

b. —23 x + 4 = —13 x + 3

2

c. −—3 x − 1 = —13 x − 4

d. —45 x + —75 = 3x − 3

e. −x + 2.5 = 2x − 0.5

f. − 3x + 1.5 = x + 1.5

Communicate Your Answer 3. How can you use a system of linear equations to solve an equation with

variables on both sides? 4. Compare the algebraic method and the graphical method for solving a

linear equation with variables on both sides. Describe the advantages and disadvantages of each method.

Section 5.5

Solving Equations by Graphing

243

5.5 Lesson

What You Will Learn Solve linear equations by graphing. Solve absolute value equations by graphing.

Core Vocabul Vocabulary larry

Use linear equations to solve real-life problems.

Previous absolute value equation

Solving Linear Equations by Graphing You can use a system of linear equations to solve an equation with variables on both sides.

Core Concept Solving Linear Equations by Graphing Step 1

To solve the equation ax + b = cx + d, write two linear equations. ax + b = cx + d y = ax + b

Step 2

y = cx + d

and

Graph the system of linear equations. The x-value of the solution of the system of linear equations is the solution of the equation ax + b = cx + d.

Solving an Equation by Graphing Solve −x + 1 = 2x − 5 by graphing. Check your solution.

SOLUTION Step 1 Write a system of linear equations using each side of the original equation. −x + 1 = 2x − 5 y = 2x − 5

y = −x + 1

Step 2 Graph the system. y = −x + 1 y = 2x − 5

1

Equation 1

−1

Equation 2

−x + 1 = 2x − 5

(2, −1)

y = −x + 1

y = 2x − 5

The graphs intersect at (2, −1). So, the solution of the equation is x = 2.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation by graphing. Check your solution. 1. —12 x − 3 = 2x

244

Chapter 5

x

−4

? −(2) + 1 = 2(2) − 5



1

−2

Check

−1 = −1

y

Solving Systems of Linear Equations

2. −4 + 9x = −3x + 2

Solving Absolute Value Equations by Graphing Solving an Absolute Value Equation by Graphing Solve ∣ x + 1 ∣ = ∣ 2x − 4 ∣ by graphing. Check your solutions.

SOLUTION Recall that an absolute value equation of the form ∣ ax + b ∣ = ∣ cx + d ∣ has two related equations. ax + b = cx + d ax + b = −(cx + d)

Equation 1 Equation 2

So, the related equations of ∣ x + 1 ∣ = ∣ 2x − 4 ∣ are as follows. x + 1 = 2x − 4 x + 1 = −(2x − 4)

Equation 1 Equation 2

Apply the steps for solving an equation by graphing to each of the related equations. Step 1 Write a system of linear equations for each related equation. Equation 1

Equation 2

x + 1 = 2x − 4 y=x+1

x + 1 = −(2x − 4) x + 1 = −2x + 4

y = 2x − 4

y = −2x + 4

y=x+1

System 1

System 2

Check

∣ x + 1 ∣ = ∣ 2x − 4 ∣

Step 2 Graph each system.

? ∣5 + 1∣ = ∣ 2(5) − 4 ∣ ? ∣6∣ = ∣6∣ 6=6



y 6

∣ x + 1 ∣ = ∣ 2x − 4 ∣

4

? ∣1 + 1∣ = ∣ 2(1) − 4 ∣

2

? ∣2∣ = ∣ −2 ∣ 2=2



System 1

System 2

y=x+1 y = 2x − 4

y=x+1 y = −2x + 4

y=x+1

(5, 6)

y 6

y=x+1

4 2

(1, 2)

y = 2x − 4 1

4

6 x

The graphs intersect at (5, 6).

1

y = −2x + 4 4

6 x

The graphs intersect at (1, 2).

So, the solutions of the equation are x = 5 and x = 1.

Monitoring Progress

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Solve the equation by graphing. Check your solutions. 3. ∣ 2x + 2 ∣ = ∣ x − 2 ∣ 4. ∣ x − 6 ∣ = ∣ −x + 4 ∣

Section 5.5

Solving Equations by Graphing

245

Solving Real-Life Problems Modeling with Mathematics You are studying two glaciers. In 2000, Glacier A had an area of about 40 square miles and Glacier B had an area of about 32 square miles. You estimate that Glacier A will melt at a rate of 2 square miles per decade and Glacier B will melt at a rate of 0.25 square mile per decade. In what year will the areas of the glaciers be the same?

SOLUTION Step 1 Use a verbal model to write an equation that represents the problem. Let x be the number of decades after 2000. Then write a system of linear equations using each side of the equation. Glacier A

Area lost Area of Glacier A − per decade after 2000 in 2000

Glacier B

Number

Area of

after 2000

in 2000

Area lost

Number

decade ⋅ of decades ⋅ of decades = Glacier B − per after 2000 after 2000

40 − 2x = 32 − 0.25x y = 32 − 0.25x

y = 40 − 2x y 40

Step 2 Graph the system. The graphs intersect between x = 4 and x = 5. Make a table using x-values between 4 and 5. Use an increment of 0.1.

y = 40 − 2x

30

y = 32 − 0.25x

20 10 0

0

2

4

6

8

45

y = 40 − 2x

4.2

4.3

4.4

4.5

4.6

4.7

y = 40 − 2x

31.8

31.6

31.4

31.2

31

30.8

30.6

y = 32 − 0.25x

30.98

30.95

30.93

30.9

30.88

30.85

30.83

x

4.51

4.52

4.53

4.54

4.55

4.56

4.57

4.58

y = 40 − 2x

30.98

30.96

30.94

30.92

30.9

30.88

30.86

30.84

y = 32 − 0.25x

30.87

30.87

30.87

30.87

30.86

30.86

30.86

30.86

When x = 4.57, the corresponding y-values are about the same. So, the graphs intersect at about (4.57, 30.86).

y = 32 − 0.25x Intersection 0 X=4.5714286 Y=30.857143 0

4.1

Notice when x = 4.5, the area of Glacier A is greater than the area of Glacier B. But when x = 4.6, the area of Glacier A is less than the area of Glacier B. So, the solution must be between x = 4.5 and x = 4.6. Make another table using x-values between 4.5 and 4.6. Use an increment of 0.01.

x

Check

x

10

So, the areas of the glaciers will be the same after about 4.57 decades, or around the year 2046.

Monitoring Progress

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5. WHAT IF? In 2000, Glacier C had an area of about 30 square miles. You estimate

that it will melt at a rate of 0.45 square mile per decade. In what year will the areas of Glacier A and Glacier C be the same?

246

Chapter 5

Solving Systems of Linear Equations

Exercises

5.5

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. REASONING The graphs of the equations y = 3x − 20 and y = −2x + 10 intersect at the

point (6, −2). Without solving, find the solution of the equation 3x − 20 = −2x + 10.

2. WRITING Explain how to rewrite the absolute value equation ∣ 2x − 4 ∣ = ∣ −5x + 1 ∣ as two systems

of linear equations.

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, use the graph to solve the equation. Check your solution. 3. −2x + 3 = x

3

20. —12 (8x + 3) = 4x + —2

4. −3 = 4x + 1 y

y

In Exercises 21 and 22, use the graphs to solve the equation. Check your solutions.

1 3

1

19. −x − 5 = −—3 (3x + 5)

−2

2

x

21.

∣ x − 4 ∣ = ∣ 3x ∣

1

y

−3

1

1

5. −x − 1 = —3 x + 3

−4

−2

−2 −4

4

x

22.

∣ 2x + 4 ∣ = ∣ x − 1 ∣

−4 1x

y −6

−6

9. x + 5 = −2x − 4

10. −2x + 6 = 5x − 1 1

12. −5 + —4 x = 3x + 6

13. 5x − 7 = 2(x + 1)

14. −6(x + 4) = −3x − 6

In Exercises 15−20, solve the equation by graphing. Determine whether the equation has one solution, no solution, or infinitely many solutions.

17. −4(2 − x) = 4x − 8 18. −2x − 3 = 2(x − 2)

−4

4

−3

−6

−1

3x

8. 4x = x + 3

11. —12 x − 2 = 9 − 5x

15. 3x − 1 = −x + 7

y

x

−4

In Exercises 7−14, solve the equation by graphing. Check your solution. (See Example 1.) 7. x + 4 = −x

2

−6

−2

2

−2

3

2

4

x

2

6. −—2 x − 2 = −4x + 3 y

y

y

x

−2

3x

16. 5x − 4 = 5x + 1

In Exercises 23−30, solve the equation by graphing. Check your solutions. (See Example 2.) 23.

∣ 2x ∣ = ∣ x + 3 ∣

25.

∣ −x + 4 ∣ = ∣ 2x − 2 ∣

26.

∣ x + 2 ∣ = ∣ −3x + 6 ∣

27.

∣x + 1∣ = ∣x − 5∣

28.

∣ 2x + 5 ∣ = ∣ −2x + 1 ∣

29.

∣ x − 3 ∣ = 2∣ x ∣

Section 5.5

24.

∣ 2x − 6 ∣ = ∣ x ∣

30. 4∣ x + 2 ∣ = ∣ 2x + 7 ∣

Solving Equations by Graphing

247

USING TOOLS In Exercises 31 and 32, use a graphing

37. OPEN-ENDED Find values for m and b so that the

solution of the equation mx + b = − 2x − 1 is x = −3.

calculator to solve the equation. 31. 0.7x + 0.5 = −0.2x − 1.3

38. HOW DO YOU SEE IT? The graph shows the total

32. 2.1x + 0.6 = −1.4x + 6.9

revenue and expenses of a company x years after it opens for business.

33. MODELING WITH MATHEMATICS There are about

34 million gallons of water in Reservoir A and about 38 million gallons in Reservoir B. During a drought, Reservoir A loses about 0.8 million gallons per month and Reservoir B loses about 1.1 million gallons per month. After how many months will the reservoirs contain the same amount of water? (See Example 3.)

Millions of dollars

Revenue and Expenses

34. MODELING WITH MATHEMATICS Your dog is

16 years old in dog years. Your cat is 28 years old in cat years. For every human year, your dog ages by 7 dog years and your cat ages by 4 cat years. In how many human years will both pets be the same age in their respective types of years?

y 6

expenses

4

revenue

2 0

0

4

2

6

8

10

x

Year

a. Estimate the point of intersection of the graphs. b. Interpret your answer in part (a). 39. MATHEMATICAL CONNECTIONS The value of the

perimeter of the triangle (in feet) is equal to the value of the area of the triangle (in square feet). Use a graph to find x. x ft

35. MODELING WITH MATHEMATICS You and a friend

race across a field to a fence and back. Your friend has a 50-meter head start. The equations shown represent you and your friend’s distances d (in meters) from the fence t seconds after the race begins. Find the time at which you catch up to your friend.

(x − 2) ft

40. THOUGHT PROVOKING A car has an initial value

of $20,000 and decreases in value at a rate of $1500 per year. Describe a different car that will be worth the same amount as this car in exactly 5 years. Specify the initial value and the rate at which the value decreases.

You: d = ∣ −5t + 100 ∣

∣

Your friend: d = −3—13 t + 50

∣

36. MAKING AN ARGUMENT The graphs of y = −x + 4

6 ft

41. ABSTRACT REASONING Use a graph to determine the

and y = 2x − 8 intersect at the point (4, 0). So, your friend says the solution of the equation −x + 4 = 2x − 8 is (4, 0). Is your friend correct? Explain.

sign of the solution of the equation ax + b = cx + d in each situation. a. 0 < b < d and a < c

Maintaining Mathematical Proficiency

b. d < b < 0 and a < c

Reviewing what you learned in previous grades and lessons

Graph the inequality. (Section 2.1) 42. y > 5

43. x ≤ −2

44. n ≥ 9

45. c < −6

Use the graphs of f and g to describe the transformation from the graph of f to the graph of g. (Section 3.6) 46. f(x) = x − 5; g(x) = f(x + 2)

47. f(x) = 6x; g(x) = −f(x)

48. f(x) = −2x + 1; g(x) = f(4x)

49. f(x) = —2 x − 2; g(x) = f(x − 1)

248

Chapter 5

Solving Systems of Linear Equations

1

5.6

Graphing Linear Inequalities in Two Variables Essential Question

How can you graph a linear inequality in

two variables?

A solution of a linear inequality in two variables is an ordered pair (x, y) that makes the inequality true. The graph of a linear inequality in two variables shows all the solutions of the inequality in a coordinate plane.

Writing a Linear Inequality in Two Variables Work with a partner.

4

a. Write an equation represented by the dashed line.

y

2

b. The solutions of an inequality are represented by the shaded region. In words, describe the solutions of the inequality.

−4

−2

2

4 x

−2

c. Write an inequality represented by the graph. Which inequality symbol did you use? Explain your reasoning.

Using a Graphing Calculator

USING TOOLS STRATEGICALLY To be proficient in math, you need to use technological tools to explore and deepen your understanding of concepts.

Work with a partner. Use a graphing calculator to graph y ≥ —14 x − 3. a. Enter the equation y = —14 x − 3 into your calculator. b. The inequality has the symbol ≥. So, the region to be shaded is above the graph of y = —14 x − 3, as shown. Verify this by testing a point in this region, such as (0, 0), to make sure it is a solution of the inequality.

10

1

y ≥ 4x − 3 −10

10

−10

Because the inequality symbol is greater than or equal to, the line is solid and not dashed. Some graphing calculators always use a solid line when graphing inequalities. In this case, you have to determine whether the line should be solid or dashed, based on the inequality symbol used in the original inequality.

Graphing Linear Inequalities in Two Variables Work with a partner. Graph each linear inequality in two variables. Explain your steps. Use a graphing calculator to check your graphs. a. y > x + 5

1

b. y ≤ −—2 x + 1

c. y ≥ −x − 5

Communicate Your Answer 4. How can you graph a linear inequality in two variables? 5. Give an example of a real-life situation that can be modeled using a linear

inequality in two variables. Section 5.6

Graphing Linear Inequalities in Two Variables

249

5.6 Lesson

What You Will Learn Check solutions of linear inequalities. Graph linear inequalities in two variables. Use linear inequalities to solve real-life problems.

Core Vocabul Vocabulary larry linear inequality in two variables, p. 250 solution of a linear inequality in two variables, p. 250 graph of a linear inequality, p. 250 half-planes, p. 250

Linear Inequalities A linear inequality in two variables, x and y, can be written as ax + by ≤ c

ax + by < c

ax + by > c

ax + by ≥ c

where a, b, and c are real numbers. A solution of a linear inequality in two variables is an ordered pair (x, y) that makes the inequality true.

Previous ordered pair

Checking Solutions Tell whether the ordered pair is a solution of the inequality. b. x − 3y ≥ 8; (2, −2)

a. 2x + y < −3; (−1, 9)

SOLUTION a.

2x + y < −3 ? 2(−1) + 9 < −3

Write the inequality. Substitute −1 for x and 9 for y.

✗

7 < −3

Simplify. 7 is not less than −3.

So, (−1, 9) is not a solution of the inequality.

b.

x − 3y ≥ 8 ? 2 − 3(−2) ≥ 8 8≥8

Write the inequality. Substitute 2 for x and −2 for y.



Simplify. 8 is equal to 8.

So, (2, −2) is a solution of the inequality.

Monitoring Progress

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Tell whether the ordered pair is a solution of the inequality. 1. x + y > 0; (−2, 2)

2. 4x − y ≥ 5; (0, 0)

3. 5x − 2y ≤ −1; (−4, −1)

4. −2x − 3y < 15; (5, −7)

Graphing Linear Inequalities in Two Variables The graph of a linear inequality in two variables shows all the solutions of the inequality in a coordinate plane.

READING A dashed boundary line means that points on the line are not solutions. A solid boundary line means that points on the line are solutions.

250

Chapter 5

4

All solutions of y < 2x lie on one side of the boundary line y = 2 x.

y

The boundary line divides the coordinate plane into two half-planes. The shaded half-plane is the graph of y < 2x .

2

−2

Solving Systems of Linear Equations

2

x

Core Concept Graphing a Linear Inequality in Two Variables Step 1 Graph the boundary line for the inequality. Use a dashed line for < or >. Use a solid line for ≤ or ≥. Step 2 Test a point that is not on the boundary line to determine whether it is a solution of the inequality. Step 3 When the test point is a solution, shade the half-plane that contains the point. When the test point is not a solution, shade the half-plane that does not contain the point.

Graphing a Linear Inequality in One Variable

STUDY TIP It is often convenient to use the origin as a test point. However, you must choose a different test point when the origin is on the boundary line.

Graph y ≤ 2 in a coordinate plane.

SOLUTION Step 1 Graph y = 2. Use a solid line because the inequality symbol is ≤.

y 3

Step 2 Test (0, 0). y≤2 0≤2

1

Write the inequality.



Substitute.

(0, 0)

−1

2

4 x

Step 3 Because (0, 0) is a solution, shade the half-plane that contains (0, 0).

Graphing a Linear Inequality in Two Variables Graph −x + 2y > 2 in a coordinate plane.

SOLUTION

Check

Step 1 Graph −x + 2y = 2, or y = —12 x + 1. Use a dashed line because the inequality symbol is >.

4

Step 2 Test (0, 0).

2

5

−x + 2y > 2

Write the inequality.

? −(0) + 2(0) > 2 −2

3 −1

0>2

Substitute.

✗

y

(0, 0) −2

2

x

Simplify.

Step 3 Because (0, 0) is not a solution, shade the half-plane that does not contain (0, 0).

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Graph the inequality in a coordinate plane. 5. y > −1

6. x ≤ −4

7. x + y ≤ −4

8. x − 2y < 0

Section 5.6

Graphing Linear Inequalities in Two Variables

251

Solving Real-Life Problems Modeling with Mathematics You can spend at most $10 on grapes and apples for a fruit salad. Grapes cost $2.50 per pound, and apples cost $1 per pound. Write and graph an inequality that represents the amounts of grapes and apples you can buy. Identify and interpret two solutions of the inequality.

SOLUTION 1. Understand the Problem You know the most that you can spend and the prices per pound for grapes and apples. You are asked to write and graph an inequality and then identify and interpret two solutions. 2. Make a Plan Use a verbal model to write an inequality that represents the problem. Then graph the inequality. Use the graph to identify two solutions. Then interpret the solutions. 3. Solve the Problem Words Fruit Salad

Pounds of apples

⋅

Amount Pounds ≤ you can of apples spend

⋅

⋅

Inequality 2.50 x + 1 y ≤ 10

9 8

Step 1 Graph 2.5x + y = 10, or y = −2.5x + 10. Use a solid line because the inequality symbol is ≤. Restrict the graph to positive values of x and y because negative values do not make sense in this real-life context.

(1, 6)

7 6

(2, 5)

5

Step 2 Test (0, 0).

4

2.5x + y ≤ 10

3

? 2.5(0) + 0 ≤ 10

2 1 0

1

2

3

4

5

6 x

Pounds of grapes

2.5x + y ≤ 10

?

2.5(1) + 6 ≤ 10 8.5 ≤ 10



? 2.5(2) + 5 ≤ 10

Chapter 5



Substitute. Simplify.

Step 3 Because (0, 0) is a solution, shade the half-plane that contains (0, 0).

4. Look Back Check your solutions by substituting them into the original inequality, as shown.

Monitoring Progress

2.5x + y ≤ 10

10 ≤ 10

0 ≤ 10

Write the inequality.

One possible solution is (1, 6) because it lies in the shaded half-plane. Another possible solution is (2, 5) because it lies on the solid line. So, you can buy 1 pound of grapes and 6 pounds of apples, or 2 pounds of grapes and 5 pounds of apples.

Check

252

⋅

Cost per Pounds + pound of of grapes apples

Variables Let x be pounds of grapes and y be pounds of apples.

y 10

0

Cost per pound of grapes



Help in English and Spanish at BigIdeasMath.com

9. You can spend at most $12 on red peppers and tomatoes for salsa. Red peppers

cost $4 per pound, and tomatoes cost $3 per pound. Write and graph an inequality that represents the amounts of red peppers and tomatoes you can buy. Identify and interpret two solutions of the inequality.

Solving Systems of Linear Equations

5.6

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY How can you tell whether an ordered pair is a solution of a linear inequality? 2. WRITING Compare the graph of a linear inequality in two variables with the graph of a linear

equation in two variables.

Monitoring Progress and Modeling with Mathematics In Exercises 3–10, tell whether the ordered pair is a solution of the inequality. (See Example 1.)

In Exercises 19–24, graph the inequality in a coordinate plane. (See Example 2.)

3. x + y < 7; (2, 3)

4. x − y ≤ 0; (5, 2)

19. y ≤ 5

20. y > 6

5. x + 3y ≥ −2; (−9, 2)

6. 8x + y > −6; (−1, 2)

21. x < 2

22. x ≥ −3

7. −6x + 4y ≤ 6; (−3, −3)

23. y > −7

24. x < 9

8. 3x − 5y ≥ 2; (−1, −1) 9. −x − 6y > 12; (−8, 2)

In Exercises 25−30, graph the inequality in a coordinate plane. (See Example 3.)

10. −4x − 8y < 15; (−6, 3)

25. y > −2x − 4

26. y ≤ 3x − 1

In Exercises 11−16, tell whether the ordered pair is a solution of the inequality whose graph is shown.

27. −4x + y < −7

28. 3x − y ≥ 5

11. (0, −1)

12. (−1, 3)

29. 5x − 2y ≤ 6

30. −x + 4y > −12

13. (1, 4)

14. (0, 0)

15. (3, 3)

16. (2, 1)

4

y

2 −2

ERROR ANALYSIS In Exercises 31 and 32, describe and 2

x

−2

17. MODELING WITH MATHEMATICS A carpenter has

at most $250 to spend on lumber. The inequality 8x + 12y ≤ 250 represents the numbers x of 2-by-8 boards and the numbers y of 4-by-4 boards the carpenter can buy. Can the carpenter buy twelve 2-by-8 boards and fourteen 4-by-4 boards? Explain.

correct the error in graphing the inequality. 31. y < −x + 1



y 3

−2

3x −2

32. y ≤ 3x − 2 4 in. x 4 in. x 8 ft $12 each

2 in. x 8 in. x 8 ft $8 each

18. MODELING WITH MATHEMATICS The inequality

3x + 2y ≥ 93 represents the numbers x of multiplechoice questions and the numbers y of matching questions you can answer correctly to receive an A on a test. You answer 20 multiple-choice questions and 18 matching questions correctly. Do you receive an A on the test? Explain. Section 5.6



4

y

2 −2

−1

2

x

Graphing Linear Inequalities in Two Variables

253

33. MODELING WITH MATHEMATICS You have at

40. HOW DO YOU SEE IT? Match each inequality with

most $20 to spend at an arcade. Arcade games cost $0.75 each, and snacks cost $2.25 each. Write and graph an inequality that represents the numbers of games you can play and snacks you can buy. Identify and interpret two solutions of the inequality. (See Example 4.)

its graph. a. 3x − 2y ≤ 6

b. 3x − 2y < 6

c. 3x − 2y > 6

d. 3x − 2y ≥ 6

A.

1 −2

34. MODELING WITH MATHEMATICS A drama club

must sell at least $1500 worth of tickets to cover the expenses of producing a play. Write and graph an inequality that represents how many adult and student tickets the club must sell. Identify and interpret two solutions of the inequality.

B.

y 1

1 −2

3x

1 −2

1

3x

1

3x

−2

−2

C.

y

D.

y 1

3x

−2

1

y

−2 −2

In Exercises 35–38, write an inequality that represents the graph. 35.

36.

y

y

4

41. REASONING When graphing a linear inequality in

4

two variables, why must you choose a test point that is not on the boundary line?

2 1 −2

2

x

−2

2

x

42. THOUGHT PROVOKING Write a linear inequality in

two variables that has the following two properties. 37.

1 −1

38.

y 2

1 −2

x

• (0, 0), (0, −1), and (0, 1) are not solutions. • (1, 1), (3, −1), and (−1, 3) are solutions.

y 2

x

−2 −3

43. WRITING Can you always use (0, 0) as a test point

when graphing an inequality? Explain.

−5

39. PROBLEM SOLVING Large boxes weigh 75 pounds,

and small boxes weigh 40 pounds. a. Write and graph an inequality that represents the numbers of large and small boxes a 200-pound delivery person can take on the elevator.

Weight limit: 2000 lb

CRITICAL THINKING In Exercises 44 and 45, write and graph an inequality whose graph is described by the given information. 44. The points (2, 5) and (−3, −5) lie on the boundary

line. The points (6, 5) and (−2, −3) are solutions of the inequality. 45. The points (−7, −16) and (1, 8) lie on the boundary

line. The points (−7, 0) and (3, 14) are not solutions of the inequality.

b. Explain why some solutions of the inequality might not be practical in real life.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Write the next three terms of the arithmetic sequence. (Section 4.6) 46. 0, 8, 16, 24, 32, . . .

254

Chapter 5

3

1 1 3 5

47. −5, −8, −11, −14, −17, . . . 48. −—2 , −—2 , —2 , —2 , —2 , . . .

Solving Systems of Linear Equations

5.7

Systems of Linear Inequalities Essential Question

How can you graph a system of

linear inequalities?

Graphing Linear Inequalities Work with a partner. Match each linear inequality with its graph. Explain your reasoning. 2x + y ≤ 4

Inequality 1

2x − y ≤ 0

Inequality 2

A.

B.

y

−4

y

4

4

2

2

−2

1

4

−4

x

−2

2

4

x

−2 −4

−4

Graphing a System of Linear Inequalities

MAKING SENSE OF PROBLEMS To be proficient in math, you need to explain to yourself the meaning of a problem.

Work with a partner. Consider the linear inequalities given in Exploration 1. 2x + y ≤ 4

Inequality 1

2x − y ≤ 0

Inequality 2

a. Use two different colors to graph the inequalities in the same coordinate plane. What is the result? b. Describe each of the shaded regions of the graph. What does the unshaded region represent?

Communicate Your Answer 3. How can you graph a system of linear inequalities? 4. When graphing a system of linear inequalities, which region represents the

solution of the system? 5. Do you think all systems of linear inequalities

have a solution? Explain your reasoning. 6. Write a system of linear inequalities

6

represented by the graph.

y

4 2 −4

−2

1

4

x

−2 −4

Section 5.7

Systems of Linear Inequalities

255

5.7

Lesson

What You Will Learn Check solutions of systems of linear inequalities. Graph systems of linear inequalities. Write systems of linear inequalities.

Core Vocabul Vocabulary larry system of linear inequalities, p. 256 solution of a system of linear inequalities, p. 256 graph of a system of linear inequalities, p. 257 Previous linear inequality in two variables

Use systems of linear inequalities to solve real-life problems.

Systems of Linear Inequalities A system of linear inequalities is a set of two or more linear inequalities in the same variables. An example is shown below. y < x+2

Inequality 1

y ≥ 2x − 1

Inequality 2

A solution of a system of linear inequalities in two variables is an ordered pair that is a solution of each inequality in the system.

Checking Solutions Tell whether each ordered pair is a solution of the system of linear inequalities. y < 2x

Inequality 1

y ≥ x+1

Inequality 2

a. (3, 5)

b. (−2, 0)

SOLUTION a. Substitute 3 for x and 5 for y in each inequality. Inequality 1

Inequality 2

y < 2x

y ≥ x+1

? 5 < 2(3) 5 x−1

Graphing Systems of Linear Inequalities The graph of a system of linear inequalities is the graph of all the solutions of the system.

Core Concept Graphing a System of Linear Inequalities Step 1 Graph each inequality in the same coordinate plane.

y

6 4

Step 2 Find the intersection of the half-planes that are solutions of the inequalities. This intersection is the graph of the system.

y x+2

? 1 > −3 + 2 1 > −1



2

4

x

Graphing a System of Linear Inequalities Graph the system of linear inequalities. y≤3

Inequality 1

y > x+2

Inequality 2

y 4

SOLUTION

(−3, 1)

Step 1 Graph each inequality.

−1

−4

Step 2 Find the intersection of the half-planes. One solution is (−3, 1).

1 2

x

−2

The solution is the purple-shaded region.

Graphing a System of Linear Inequalities: No Solution y

Graph the system of linear inequalities. 2x + y < −1

Inequality 1

2x + y > 3

Inequality 2

2 −4

−2

1

SOLUTION

−2

Step 1 Graph each inequality.

−4

3

x

Step 2 Find the intersection of the half-planes. Notice that the lines are parallel, and the half-planes do not intersect. So, the system has no solution.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Graph the system of linear inequalities. 3. y ≥ −x + 4

x+y ≤ 0

4. y > 2x − 3

y ≥ —12 x + 1 Section 5.7

5. −2x + y < 4

2x + y > 4

Systems of Linear Inequalities

257

Writing Systems of Linear Inequalities Writing a System of Linear Inequalities Write a system of linear inequalities represented by the graph.

y 2

SOLUTION

−2

−4

Inequality 1 The horizontal boundary line passes through (0, −2). So, an equation of the line is y = −2. The shaded region is above the solid boundary line, so the inequality is y ≥ −2.

2

4x

−4

Inequality 2 The slope of the other boundary line is 1, and the y-intercept is 0. So, an equation of the line is y = x. The shaded region is below the dashed boundary line, so the inequality is y < x. The system of linear inequalities represented by the graph is y ≥ −2

Inequality 1

y < x.

Inequality 2

Writing a System of Linear Inequalities Write a system of linear inequalities represented by the graph.

4

y

2

SOLUTION Inequality 1 The vertical boundary line passes through (3, 0). So, an equation of the line is x = 3. The shaded region is to the left of the solid boundary line, so the inequality is x ≤ 3.

2

4

6

x

−4

Inequality 2 The slope of the other boundary line is —23 , and the y-intercept is −1. So, an equation of the line is y = —23 x − 1. The shaded region is above the dashed boundary line, so the inequality is y > —23 x − 1. The system of linear inequalities represented by the graph is x≤3 y>

2 —3 x

Inequality 1

− 1.

Inequality 2

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Write a system of linear inequalities represented by the graph. 6.

7.

y 1

4 2

2

4

x 2

−2 −2

258

Chapter 5

y

Solving Systems of Linear Equations

4

x

Solving Real-Life Problems Modeling with Mathematics You have at most 8 hours to spend at the mall and at the beach. You want to spend at least 2 hours at the mall and more than 4 hours at the beach. Write and graph a system that represents the situation. How much time can you spend at each location?

SOLUTION 1. Understand the Problem You know the total amount of time you can spend at the mall and at the beach. You also know how much time you want to spend at each location. You are asked to write and graph a system that represents the situation and determine how much time you can spend at each location. 2. Make a Plan Use the given information to write a system of linear inequalities. Then graph the system and identify an ordered pair in the solution region. 3. Solve the Problem Let x be the number of hours at the mall and let y be the number of hours at the beach. x+y ≤ 8

at most 8 hours at the mall and at the beach

x≥2

at least 2 hours at the mall

y>4

more than 4 hours at the beach

Graph the system.

Check x+y ≤ 8

?

2.5 + 5 ≤ 8 7.5 ≤ 8



Hours at the beach

Time at the Mall and at the Beach y 8 7 6 5 4 3 2 1 0

x≥2 2.5 ≥ 2



1

2

3

4

5

6

7

8

9 x

Hours at the mall

One ordered pair in the solution region is (2.5, 5).

y>4 5>4

0



So, you can spend 2.5 hours at the mall and 5 hours at the beach. 4. Look Back Check your solution by substituting it into the inequalities in the system, as shown.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

8. Name another solution of Example 6. 9. WHAT IF? You want to spend at least 3 hours at the mall. How does this change

the system? Is (2.5, 5) still a solution? Explain.

Section 5.7

Systems of Linear Inequalities

259

Exercises

5.7

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY How can you verify that an ordered pair is a solution

of a system of linear inequalities?

y 2

2. WHICH ONE DOESN’T BELONG? Use the graph shown. Which of

−2

the ordered pairs does not belong with the other three? Explain your reasoning. (1, −2)

(0, −4)

(−1, −6)

5 x

−4

(2, −4)

−6

Monitoring Progress and Modeling with Mathematics In Exercises 3−6, tell whether the ordered pair is a solution of the system of linear inequalities. 3. (−4, 3)

y 4

4. (−3, −1) 5. (−2, 5) −3

1x

6. (1, 1)

x−y ≥ 2

y ≥ −x + 1

y>2

21.

4

22.

y

y 2

2

In Exercises 7−10, tell whether the ordered pair is a solution of the system of linear inequalities. (See Example 1.) y −2 7. (−5, 2); 8. (1, −1); y > x−5 y > x+3 9. (0, 0); y ≤ x + 7

y ≥ 2x + 3

10. (4, −3);

y ≤ −x + 1 y ≤ 5x − 2

−1 −2

11. y > −3

y ≥ 5x 13. y < −2

y>2 15. y ≥ −5

y − 1 < 3x 17. x + y > 1

−x − y < −3

Chapter 5

2

23.

x

24.

y

−2

1

3

5x

−2

y

2

In Exercises 11−20, graph the system of linear inequalities. (See Examples 2 and 3.)

260

y>1

In Exercises 21−26, write a system of linear inequalities represented by the graph. (See Examples 4 and 5.)

2 −5

20. x + y ≤ 10

19. x < 4

1 −2

2

x

4x

−1

12. y < −1

x>4 14. y < x − 1

25.

y ≥ —32 x − 9 18. 2x + y ≤ 5

y + 2 ≥ −2x

Solving Systems of Linear Equations

y

2

y ≥ x+1 16. x + y > 4

26.

y

2 x

−4

−2

2x −3

−3

2 −3

ERROR ANALYSIS In Exercises 27 and 28, describe

and correct the error in graphing the system of linear inequalities. 27.



y ≤ x−1 y ≥ x+3

y

31. MODELING WITH MATHEMATICS You are fishing Dynamic Solutions available at BigIdeasMath.com for surfperch and rockfi sh, which are species of bottomfish. Gaming laws allow you to catch no more than 15 surfperch per day, no more than 10 rockfish per day, and no more than 20 total bottomfish per day.

a. Write and graph a system of linear inequalities that represents the situation.

1 −2

2

x

b. Use the graph to determine whether you can catch 11 surfperch and 9 rockfish in 1 day.

−3

28.



y ≤ 3x + 4

y

1 x+2 — 2

4

y>

surfperch 1 −4

−2

2x

rockfish

32. REASONING Describe the intersection of the

half-planes of the system shown. x−y ≤ 4 x−y ≥ 4

29. MODELING WITH MATHEMATICS You can spend at

most $21 on fruit. Blueberries cost $4 per pound, and strawberries cost $3 per pound. You need at least 3 pounds of fruit to make muffins. (See Example 6.) a. Write and graph a system of linear inequalities that represents the situation. b. Identify and interpret a solution of the system. c. Use the graph to determine whether you can buy 4 pounds of blueberries and 1 pound of strawberries. 30. MODELING WITH MATHEMATICS You earn

$10 per hour working as a manager at a grocery store. You are required to work at the grocery store at least 8 hours per week. You also teach music lessons for $15 per hour. You need to earn at least $120 per week, but you do not want to work more than 20 hours per week. a. Write and graph a system of linear inequalities that represents the situation. b. Identify and interpret a solution of the system. c. Use the graph to determine whether you can work 8 hours at the grocery store and teach 1 hour of music lessons.

33. MATHEMATICAL CONNECTIONS The following points

are the vertices of a shaded rectangle. (−1, 1), (6, 1), (6, −3), (−1, −3) a. Write a system of linear inequalities represented by the shaded rectangle. b. Find the area of the rectangle. 34. MATHEMATICAL CONNECTIONS The following points

are the vertices of a shaded triangle. (2, 5), (6, −3), (−2, −3) a. Write a system of linear inequalities represented by the shaded triangle. b. Find the area of the triangle. 35. PROBLEM SOLVING You plan to spend less than

half of your monthly $2000 paycheck on housing and savings. You want to spend at least 10% of your paycheck on savings and at most 30% of it on housing. How much money can you spend on savings and housing? 36. PROBLEM SOLVING On a road trip with a friend, you

drive about 70 miles per hour, and your friend drives about 60 miles per hour. The plan is to drive less than 15 hours and at least 600 miles each day. Your friend will drive more hours than you. How many hours can you and your friend each drive in 1 day?

Section 5.7

Systems of Linear Inequalities

261

37. WRITING How are solving systems of linear

43. All solutions have one positive coordinate and one

inequalities and solving systems of linear equations similar? How are they different?

negative coordinate. 44. There are no solutions.

38. HOW DO YOU SEE IT? The graphs of two linear

45. OPEN-ENDED One inequality in a system is

equations are shown.

−4x + 2y > 6. Write another inequality so the system has (a) no solution and (b) infinitely many solutions.

y

C 2

A −4

−2

y = 2x + 1

1

−2

46. THOUGHT PROVOKING You receive a gift certificate

B D

3

for a clothing store and plan to use it to buy T-shirts and sweatshirts. Describe a situation in which you can buy 9 T-shirts and 1 sweatshirt, but you cannot buy 3 T-shirts and 8 sweatshirts. Write and graph a system of linear inequalities that represents the situation.

x

y = −3x + 4

−4

Replace the equal signs with inequality symbols to create a system of linear inequalities that has point C as a solution, but not points A, B, and D. Explain your reasoning. y

−3x + 4

y

2x + 1

47. CRITICAL THINKING Write a system of linear

inequalities that has exactly one solution. 48. MODELING WITH MATHEMATICS You make

necklaces and key chains to sell at a craft fair. The table shows the amounts of time and money it takes to make a necklace and a key chain, and the amounts of time and money you have available for making them.

39. USING STRUCTURE Write a system of linear

inequalities that is equivalent to ∣ y ∣ < x, where x > 0. Graph the system.

40. MAKING AN ARGUMENT Your friend says that a

system of linear inequalities in which the boundary lines are parallel must have no solution. Is your friend correct? Explain.

Necklace

Key chain

Available

Time to make (hours)

0.5

0.25

20

Cost to make (dollars)

2

3

120

a. Write and graph a system of four linear inequalities that represents the number x of necklaces and the number y of key chains that you can make.

41. CRITICAL THINKING Is it possible for the solution

set of a system of linear inequalities to be all real numbers? Explain your reasoning.

b. Find the vertices (corner points) of the graph of the system.

OPEN-ENDED In Exercises 42−44, write a system of

c. You sell each necklace for $10 and each key chain for $8. The revenue R is given by the equation R = 10x + 8y. Find the revenue corresponding to each ordered pair in part (b). Which vertex results in the maximum revenue?

linear inequalities with the given characteristic. 42. All solutions are in Quadrant I.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Write the product using exponents. (Skills Review Handbook)

⋅ ⋅ ⋅ ⋅

49. 4 4 4 4 4

⋅

⋅

⋅⋅⋅⋅⋅

50. (−13) (−13) (−13)

51. x x x x x x

Write an equation of the line with the given slope and y-intercept. 52. slope: 1

y-intercept: −6

262

Chapter 5

53. slope: −3

y-intercept: 5

54. slope:

(Section 4.1)

1 −—4

y-intercept: −1

Solving Systems of Linear Equations

4

55. slope: —3

y-intercept: 0

5.5–5.7

What Did You Learn?

Core Vocabulary linear inequality in two variables, p. 250 solution of a linear inequality in two variables, p. 250 graph of a linear inequality, p. 250

half-planes, p. 250 system of linear inequalities, p. 256 solution of a system of linear inequalities, p. 256 graph of a system of linear inequalities, p. 257

Core Concepts Section 5.5 Solving Linear Equations by Graphing, p. 244 Solving Absolute Value Equations by Graphing, p. 245

Section 5.6 Graphing a Linear Inequality in Two Variables, p. 251

Section 5.7 Graphing a System of Linear Inequalities, p. 257 Writing a System of Linear Inequalities, p. 258

Mathematical Practices 1.

Why do the equations in Exercise 35 on page 248 contain absolute value expressions?

2.

Why is it important to be precise when answering part (a) of Exercise 39 on page 254?

3.

Describe the overall step-by-step process you used to solve Exercise 35 on page 261.

Performance Task:

Fishing Limits Do oceans support unlimited numbers of fish? Can you use mathematics to set fishing limits so that this valuable food resource is not endangered? To explore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.

263

5

Chapter Review 5.1

Dynamic Solutions available at BigIdeasMath.com

Solving Systems of Linear Equations by Graphing (pp. 217–222) y=x−2 y = −3x + 2

Solve the system by graphing.

Equation 1 Equation 2

Step 1 Graph each equation.

y

Step 2 Estimate the point of intersection. The graphs appear to intersect at (1, −1).

2

Step 3 Check your point from Step 2. Equation 1

−1

Equation 2

y=x−2

y=x−2

2

? −1 = −3(1) + 2

−1 = −1

−1 = −1



x

y = −3x + 2

y = −3x + 2

? −1 = 1 − 2

4

(1, −1)



The solution is (1, −1). Solve the system of linear equations by graphing. 1. y = −3x + 1

2. y = −4x + 3

y=x−7

5.2

3. 5x + 5y = 15

4x − 2y = 6

2x − 2y = 10

Solving Systems of Linear Equations by Substitution (pp. 223–228)

Solve the system by substitution.

−2x + y = −8 7x + y = 10

Equation 1 Equation 2

Step 1 Solve for y in Equation 1. y = 2x − 8

Revised Equation 1

Step 2 Substitute 2x − 8 for y in Equation 2 and solve for x. 7x + y = 10 7x + (2x − 8) = 10

Equation 2 Substitute 2x − 8 for y.

9x − 8 = 10

Combine like terms.

9x = 18

Add 8 to each side.

x=2

Divide each side by 9.

Step 3 Substituting 2 for x in Equation 1 and solving for y gives y = −4. The solution is (2, −4). Solve the system of linear equations by substitution. Check your solution. 4. 3x + y = −9

y = 5x + 7

5. x + 4y = 6

x−y=1

6. 2x + 3y = 4

y + 3x = 6

7. You spend $20 total on tubes of paint and disposable brushes for an art project. Tubes of paint

cost $4.00 each and paintbrushes cost $0.50 each. You purchase twice as many brushes as tubes of paint. How many brushes and tubes of paint do you purchase? 264

Chapter 5

Solving Systems of Linear Equations

5.3

Solving Systems of Linear Equations by Elimination (pp. 229–234) 4x + 6y = −8 x − 2y = −2

Solve the system by elimination.

Equation 1 Equation 2

Step 1 Multiply Equation 2 by 3 so that the coefficients of the y-terms are opposites. 4x + 6y = −8 x − 2y = −2

Multiply by 3.

4x + 6y = −8

Equation 1

3x − 6y = −6

Revised Equation 2

Step 2 Add the equations. 4x + 6y = −8 3x − 6y = −6 7x = −14

Equation 1

Check

Revised Equation 2

Equation 1

Add the equations.

4x + 6y = −8 ? 4(−2) + 6(0) = −8

Step 3 Solve for x. 7x = −14 x = −2

Resulting equation from Step 2

−8 = −8

Divide each side by 7. Equation 2

Step 4 Substitute −2 for x in one of the original equations and solve for y. 4x + 6y = −8

x − 2y = −2 ? (−2) − 2(0) = −2

Equation 1

4(−2) + 6y = −8

Substitute −2 for x.

−8 + 6y = −8

−2 = −2

Multiply.

y=0





Solve for y.

The solution is (−2, 0). Solve the system of linear equations by elimination. Check your solution. 8. 9x − 2y = 34

9. x + 6y = 28

5x + 2y = −6

5.4

10. 8x − 7y = −3

2x − 3y = −19

6x − 5y = −1

Solving Special Systems of Linear Equations (pp. 235–240)

Solve the system.

4x + 2y = −14 y = −2x − 6

Equation 1 Equation 2

Solve by substitution. Substitute −2x − 6 for y in Equation 1. 4x + 2y = −14

Equation 1

4x + 2(−2x − 6) = −14

Substitute −2x − 6 for y.

4x − 4x − 12 = −14 −12 = −14

Distributive Property

✗

Combine like terms.

The equation −12 = −14 is never true. So, the system has no solution. Solve the system of linear equations. 11. x = y + 2

−3x + 3y = 6

12. 3x − 6y = −9

13. −4x + 4y = 32

−5x + 10y = 10

3x + 24 = 3y Chapter 5

Chapter Review

265

5.5

Solving Equations by Graphing (pp. 243–248)

Solve 3x − 1 = −2x + 4 by graphing. Check your solution. Step 1 Write a system of linear equations using each side of the original equation. y = 3x − 1

y = −2x + 4

3x − 1 = −2x + 4

Step 2 Graph h th the system. t y = 3x − 1 y = −2x + 4

y

Equation 1

4

Equation 2

(1, 2) y = −2x + 4

2

The graphs intersect at (1, 2). So, the solution of the equation is x = 1.

−1

Check

y = 3x − 1

1

3

3x − 1 = −2x + 4 ? 3(1) − 1 = −2(1) + 4

5 x

2=2



Solve the equation by graphing. Check your solution(s). 14. —13 x + 5 = −2x − 2

5.6

15.

∣ x + 1 ∣ = ∣ −x − 9 ∣

Graphing Linear Inequalities in Two Variables

16. ∣ 2x − 8 ∣ = ∣ x + 5 ∣

(pp. 249–254)

Graph 4x + 2y ≥ −6 in a coordinate plane. Step 1 Graph 4x + 2y = −6, or y = −2x − 3. Use a solid line because the inequality symbol is ≥.

1

y

−2

Step 2 Test (0, 0). 4x + 2y ≥ −6

? 4(0) + 2(0) ≥ −6 0 ≥ −6

Write the inequality.

2 x

−3

Substitute.



Simplify.

Step 3 Because (0, 0) is a solution, shade the half-plane that contains (0, 0). Graph the inequality in a coordinate plane. 18. −9x + 3y ≥ 3

17. y > −4

5.7

19. 5x + 10y < 40

Systems of Linear Inequalities (pp. 255–262)

Graph the system.

y < x−2 y ≥ 2x − 4

Inequality 1

1

Inequality 2

Step 1 Graph each inequality.

−2

The solution is the purple-shaded region.

Step 2 Find the intersection of the half-planes. One solution is (0, −3).

20. y ≤ x − 3

y ≥ x+1 266

Chapter 5

21. y > −2x + 3

y≥

1 —4 x

Solving Systems of Linear Equations

−1

3 x

(0, −3)

Graph the system of linear inequalities. 22. x + 3y > 6

2x + y < 7

y

−4

5

Chapter Test

Solve the system of linear equations using any method. Explain why you chose the method. 1. 8x + 3y = −9

−8x + y = 29 4. x = y − 11

x − 3y = 1

2. —12 x + y = −6

3. y = 4x + 4

5. 6x − 4y = 9

6. y = 5x − 7

y=

3 —5 x

+5

−8x + 2y = 8

9x − 6y = 15

−4x + y = −1

7. Write a system of linear inequalities so the points (1, 2) and (4, −3) are solutions of the

system, but the point (−2, 8) is not a solution of the system. 8. How is solving the equation ∣ 2x + 1 ∣ = ∣ x − 7 ∣ by graphing similar to solving the

equation 4x + 3 = −2x + 9 by graphing? How is it different?

Graph the system of linear inequalities. 1

9. y > —2 x + 4

2y ≤ x + 4

2

11. y ≥ −—3 x + 1

10. x + y < 1

5x + y > 4

−3x + y > −2

12. You pay $45.50 for 10 gallons of gasoline and 2 quarts of oil at a gas station.

Your friend pays $22.75 for 5 gallons of the same gasoline and 1 quart of the same oil. a.

Is there enough information to determine the cost of 1 gallon of gasoline and 1 quart of oil? Explain.

b. The receipt shown is for buying the same gasoline and same oil. Is there now enough information to determine the cost of 1 gallon of gasoline and 1 quart of oil? Explain. c. Determine the cost of 1 gallon of gasoline and 1 quart of oil. 13. Describe the advantages and disadvantages of solving a system of linear

equations by graphing. 14. You have at most $60 to spend on trophies and medals to give as

prizes for a contest. a. Write and graph an inequality that represents the numbers of trophies and medals you can buy. Identify and interpret a solution of the inequality. b. You want to purchase at least 6 items. Write and graph a system that represents the situation. How many of each item can you buy?

Medals $3 each Trophies $12 each

15. Compare the slopes and y-intercepts of the graphs of the equations

in the linear system 8x + 4y = 12 and 3y = −6x − 15 to determine whether the system has one solution, no solution, or infinitely many solutions. Explain.

Chapter 5

Chapter Test

267

5

Cumulative Assessment

1. The graph of which equation is shown? 2

y

(2, 0)

A 9x − 2y = −18 ○

4

B −9x − 2y = 18 ○

−4

C 9x + 2y = 18 ○

−8

(0, −9)

D −9x + 2y = −18 ○

2. A van rental company rents out 6-, 8-, 12-, and 16-passenger vans. The function

C(x) = 100 + 5x represents the cost C (in dollars) of renting an x-passenger van for a day. Choose the numbers that are in the range of the function. 130

140

150

170

160

180

190

200

3. Fill in the system of linear inequalities with , or ≥ so that the graph represents

the system. y

3x − 2

y

−x + 5

y

4 2 x 2

4

4. Your friend claims to be able to fill in each box with a constant so that when you set

each side of the equation equal to y and graph the resulting equations, the lines will intersect exactly once. Do you support your friend’s claim? Explain. 4x +

= 4x +

5. The tables represent the numbers of items sold at a concession stand on days with

different average temperatures. Determine whether the data represented by each table show a positive, a negative, or no correlation.

268

Temperature (°F), x

14

27

32

41

48

62

73

Cups of hot chocolate, y

35

28

22

9

4

2

1

Temperature (°F), x

14

27

32

41

48

62

73

Bottles of sports drink, y

8

12

13

16

19

27

29

Chapter 5

Solving Systems of Linear Equations

8 x

6. Which two equations form a system of linear equations that has no solution?

y = 3x + 2

1

y = —3 x + 2

y = 3x + —12

y = 2x + 3

7. Fill in a value for a so that each statement is true for the equation ax − 8 = 4 − x.

a. When a =

, the solution is x = −2.

b. When a =

, the solution is x = 12.

c. When a =

, the solution is x = 3.

8. Which ordered pair is a solution of the linear inequality whose graph is shown?

A (1, 1) ○

y 2

B (−1, 1) ○ C (−1, −1) ○

−2

2

x

−2

D (1, −1) ○

9. Which of the systems of linear equations are equivalent?

4x − 5y = 3

4x − 5y = 3

4x − 5y = 3

12x − 15y = 9

2x + 15y = −1

−4x − 30y = 2

4x + 30y = −1

2x + 15y = −1

10. The graph shows the amounts y (in dollars) that a referee earns

Volleyball Referee

for refereeing x high school volleyball games.

b. Describe the domain of the function. Is the domain discrete or continuous? c. Write a function that models the data. d. Can the referee earn exactly $500? Explain.

Amount earned (dollars)

y

a. Does the graph represent a linear or nonlinear function? Explain.

240 180 120 60 0

0

2

4

6

x

Games

Chapter 5

Cumulative Assessment

269

6 6.1 6.2 6.3 6.4 6.5 6.6

Exponential Functions and Sequences Exponential Functions Exponential Growth and Decay Comparing Linear and Exponential Functions Solving Exponential Equations Geometric Sequences Recursively Defined Sequences

SEE the Big Idea

Fibonacci (p. Fib Fi bonaccii and d Flowers Fl (p. 317) 317)

Soup Kitchen S oup K itchen h (p. (p. 312) 312))

Bacterial Culture (p (p. 304)

Town Population (p. 293) Coyote Population (p. C P l i ( 279)

Maintaining Mathematical Proficiency Using Order of Operations Evaluate 102 ÷ (30 ÷ 3) − 4(3 − 9) + 51.

Example 1

102 ÷ (30 ÷ 3) − 4(3 − 9) + 51 = 102 ÷ 10 − 4(−6) + 51

First:

Parentheses

Second:

Exponents

= 100 ÷ 10 − 4(−6) + 5

Third:

Multiplication and Division (from left to right)

= 10 + 24 + 5

Fourth:

Addition and Subtraction (from left to right)

= 39

Evaluate the expression.

( 142 )

1. 12 — − 33 + 15 − 92

2. 52

⋅8 ÷ 2

2

⋅

+ 20 3 − 4

3. −7 + 16 ÷ 24 + (10 − 42)

Zero and Negative Exponents Example 2

Evaluate 4−3. 1 4−3 = — 43 1 =— 64

Example 3 1 a−n = —n a

Evaluate 4 − 5 4−5

⋅3

0

⋅3 . 0

⋅

=4−5 1 =4−5

a0 = 1

= −1

Evaluate the expression. 4. 2−5

5. −5−3

6. (−2)−1

⋅6

7. 70 + 82 ÷ 4

0

Writing Equations for Arithmetic Sequences Example 4 Write an equation for the nth term of the arithmetic sequence 5, 15, 25, 35, . . .. The first term is 5, and the common difference is 10. an = a1 + (n − 1)d

Equation for an arithmetic sequence

an = 5 + (n − 1)(10)

Substitute 5 for a1 and 10 for d.

an = 10n − 5

Simplify.

Write an equation for the nth term of the arithmetic sequence. 8. 12, 14, 16, 18, . . .

9. 6, 3, 0, −3, . . .

10. 22, 15, 8, 1, . . .

11. ABSTRACT REASONING Let a be a nonzero number and let n be a positive integer. Explain

1 why — = an. a−n

Dynamic Solutions available at BigIdeasMath.com

271

Mathematical Practices

Mathematically proficient students look closely to find a pattern.

Problem-Solving Strategies

Core Concept Finding a Pattern When solving a real-life problem, look for a pattern in the data. The pattern could include repeating items, numbers, or events. After you find the pattern, describe it and use it to solve the problem.

Using a Problem-Solving Strategy The volumes of seven chambers of a chambered nautilus are given. Find the volume Chamber 7: 1.207 cm3 of Chamber 10.

SOLUTION

Chamber 6: 1.135 cm3

To find a pattern, try dividing each volume by the volume of the previous chamber.

Chamber 5: 1.068 cm3 Chamber 4: 1.005 cm3 Chamber 3: 0.945 cm3 Chamber 2: 0.889 cm3 Chamber 1: 0.836 cm3

— ≈ 1.063

0.889 0.836

— ≈ 1.063

0.945 0.889

— ≈ 1.063

1.068 1.005

— ≈ 1.063

1.135 1.068

— ≈ 1.063

— ≈ 1.063

1.005 0.945 1.207 1.135

From this, you can see that the volume of each chamber is about 6.3% greater than the volume of the previous chamber. To find the volume of Chamber 10, multiply the volume of Chamber 7 by 1.063 three times. 1.207(1.063) ≈ 1.283

1.283(1.063) ≈ 1.364

1.364(1.063) ≈ 1.450

volume of Chamber 8

volume of Chamber 9

volume of Chamber 10

The volume of Chamber 10 is about 1.450 cubic centimeters.

Monitoring Progress 1. A rabbit population over 8 consecutive years is given by 50, 80, 128, 205, 328, 524,

839, 1342. Find the population in the tenth year. 2. The sums of the numbers in the first eight rows of Pascal’s Triangle are 1, 2, 4, 8, 16,

32, 64, 128. Find the sum of the numbers in the tenth row.

272

Chapter 6

Exponential Functions and Sequences

6.1

Exponential Functions Essential Question

What are some of the characteristics of the graph of an exponential function? Exploring an Exponential Function Work with a partner. Copy and complete each table for the exponential function y = 16(2)x. In each table, what do you notice about the values of x? What do you notice about the values of y? x

JUSTIFYING CONCLUSIONS To be proficient in math, you need to justify your conclusions and communicate them to others.

y = 16(2)x

x

0

0

1

2

2

4

3

6

4

8

5

10

y = 16(2)x

Exploring an Exponential Function

()

x

Work with a partner. Repeat Exploration 1 for the exponential function y = 16 —12 . Do you think the statement below is true for any exponential function? Justify your answer. “As the independent variable x changes by a constant amount, the dependent variable y is multiplied by a constant factor.”

Graphing Exponential Functions Work with a partner. Sketch the graphs of the functions given in Explorations 1 and 2. How are the graphs similar? How are they different?

Communicate Your Answer 4. What are some of the characteristics of the graph of an exponential function? 5. Sketch the graph of each exponential function. Does each graph have the

characteristics you described in Question 4? Explain your reasoning. a. y = 2x

()

b. y = 2(3)x

d. y = —12

x

()

e. y = 3 —12

x

Section 6.1

c. y = 3(1.5)x

()

f. y = 2 —34

x

Exponential Functions

273

6.1

Lesson

What You Will Learn Identify and evaluate exponential functions. Graph exponential functions.

Core Vocabul Vocabulary larry

Solve real-life problems involving exponential functions.

exponential function, p. 274

Identifying and Evaluating Exponential Functions

Previous independent variable dependent variable parent function

An exponential function is a nonlinear function of the form y = abx, where a ≠ 0, b ≠ 1, and b > 0. As the independent variable x changes by a constant amount, the dependent variable y is multiplied by a constant factor, which means consecutive y-values form a constant ratio.

Identifying Functions Does each table represent an exponential function? Explain. a.

b.

x

0

1

2

3

y

2

4

12

48

x

0

1

2

3

y

4

8

16

32

SOLUTION a.

STUDY TIP In Example 1b, consecutive y-values form a constant ratio. 8 16 32 — = 2, — = 2, — = 2 4 8 16

+1

+1

b.

+1

+1

+1

+1

x

0

1

2

3

x

0

1

2

3

y

2

4

12

48

y

4

8

16

32

×2

×3

×4

×2

As x increases by 1, y is not multiplied by a constant factor. So, the function is not exponential.

×2

×2

As x increases by 1, y is multiplied by 2. So, the function is exponential.

Evaluating Exponential Functions Evaluate each function for the given value of x. a. y = −2(5)x; x = 3

b. y = 3(0.5)x; x = −2

SOLUTION a. y = −2(5)x

Write the function.

= −2(5)3 = −2(125)

b. y = 3(0.5)x = 3(0.5)−2

Substitute for x.

= 3(4)

Evaluate the power.

= −250

= 12

Multiply.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Does the table represent an exponential function? Explain. 1.

x

0

1

2

3

y

8

4

2

1

2.

x

−4

0

4

8

y

1

0

−1

−2

Evaluate the function when x = −2, 0, and 3. 3. y = 2(9)x

274

Chapter 6

Exponential Functions and Sequences

4. y = 1.5(2)x

Graphing Exponential Functions

The graph of a function y = abx is a vertical stretch or shrink by a factor of ∣ a ∣ of the graph of the parent function y = bx. When a < 0, the graph is also reflected in the x-axis. The y-intercept of the graph of y = abx is a.

Core Concept Graphing y = abx When b > 1 y

STUDY TIP The graph of y = abx approaches the x-axis but never intersects it.

Graphing y = abx When 0 < b < 1 y

a>0

a>0

(0, a)

(0, a) (0, a) x

(0, a)

x

a 0.

Graphing y = abx When 0 < b < 1

()

x

Graph f (x) = − —12 . Compare the graph to the graph of the parent function. Describe the domain and range of f. 4 2

−4

SOLUTION

y

g(x) =

( 12 )

−2

x

4 x

Step 1 Make a table of values. Step 2 Plot the ordered pairs. Step 3 Draw a smooth curve through the points.

−2

x

−4

f (x)

−1 −2

0

1

2

−1

−—12

−—14

()

x

( )

f(x) = − −4

1 x 2

The parent function is g(x) = —12 . The graph of f is a reflection in the x-axis of the graph of g. The y-intercept of the graph of f, −1, is below the y-intercept of the graph of g, 1. From the graph of f, you can see that the domain is all real numbers and the range is y < 0.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Graph the function. Compare the graph to the graph of the parent function. Describe the domain and range of f. 5. f (x) = −2(4)x

()

1 x

6. f (x) = 2 —4

Section 6.1

Exponential Functions

275

To graph a function of the form y = abx − h + k, begin by graphing y = abx. Then translate the graph horizontally h units and vertically k units.

Graphing y = abx − h + k Graph y = 4(2)x − 3 + 2. Describe the domain and range. y = 4(2) x − 3 + 2 16

Step 1 Graph y = 4(2)x. This is the same function that is in Example 3, which passes through (0, 4) and (1, 8).

y

Step 2 Translate the graph 3 units right and 2 units up. The graph passes through (3, 6) and (4, 10).

12

y = 4(2) x 8

Notice that the graph approaches the line y = 2 but does not intersect it. From the graph, you can see that the domain is all real numbers and the range is y > 2.

4

−8

−4

SOLUTION

4

8 x

Comparing Exponential Functions An exponential function g models a relationship in which the dependent variable is multiplied by 1.5 for every 1 unit the independent variable x increases. Graph g when g(0) = 4. Compare g and the function f from Example 3 over the interval x = 0 to x = 2.

SOLUTION You know (0, 4) is on the graph of g. To find points to the right of (0, 4), multiply g(x) by 1.5 for every 1 unit increase in x. To find points to the left of (0, 4), divide g(x) by 1.5 for every 1 unit decrease in x. Step 1 Make a table of values. x

−1

0

1

2

3

g(x)

2.7

4

6

9

13.5

Step 2 Plot the ordered pairs. 16

Step 3 Draw a smooth curve through the points.

STUDY TIP Note that f is increasing faster than g to the right of x = 0.

12

y

f g

8

Both functions have the same value when x = 0, but the value of f is greater than the value of g over the rest of the interval.

Monitoring Progress

−8

−4

4

8 x

Help in English and Spanish at BigIdeasMath.com

Graph the function. Describe the domain and range. 7. y = −2(3)x + 2 − 1

8. f (x) = (0.25)x + 3

9. WHAT IF? In Example 6, the dependent variable of g is multiplied by 3 for every

1 unit the independent variable x increases. Graph g when g(0) = 4. Compare g and the function f from Example 3 over the interval x = 0 to x = 2.

276

Chapter 6

Exponential Functions and Sequences

Solving Real-Life Problems For an exponential function of the form y = abx, the y-values change by a factor of b as x increases by 1. You can use this fact to write an exponential function when you know the y-intercept, a. The table represents the exponential function y = 2(5)x.

+1

+1

+1

x

0

1

2

y

2

10

50

×5

×5

+1

3

4

250 1250 ×5

×5

Modeling with Mathematics The graph represents a bacterial population y after x days.

Bacterial Population

a. Write an exponential function that represents the population.

y 800

(4, 768)

Population

700 600

SOLUTION

500 400 300 200

(0, 3) (1, 12)

100

0

b. Find the population after 5 days.

1. Understand the Problem You have a graph of the population that shows some data points. You are asked to write an exponential function that represents the population and find the population after a given amount of time.

(3, 192)

(2, 48) 0

1

2

3

4

Day

5

6

7 x

2. Make a Plan Use the graph to make a table of values. Use the table and the y-intercept to write an exponential function. Then evaluate the function to find the population. 3. Solve the Problem

+1

a. Use the graph to make a table of values.

+1

+1

+1

x

0

1

2

3

4

y

3

12

48

192

768

×4

×4

×4

×4

The y-intercept is 3. The y-values increase by a factor of 4 as x increases by 1. So, the population can be modeled by y = 3(4)x. b. To find the population after 5 days, evaluate the function when x = 5. y = 3(4)x

Write the function.

= 3(4)5

Substitute 5 for x.

= 3(1024)

Evaluate the power.

= 3072

Multiply.

There are 3072 bacteria after 5 days. 4. Look Back The graph resembles an exponential function of the form y = abx, where b > 1 and a > 0. So, the exponential function y = 3(4)x is reasonable.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

10. A bacterial population y after x days can be represented by an exponential

function whose graph passes through (0, 100) and (1, 200). (a) Write a function that represents the population. (b) Find the population after 6 days. (c) Does this bacterial population grow faster than the bacterial population in Example 7? Explain.

Section 6.1

Exponential Functions

277

Exercises

6.1

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. OPEN-ENDED Sketch an increasing exponential function whose graph has a y-intercept of 2. 2. REASONING Why is a the y-intercept of the graph of the function y = abx? 3. WRITING Compare the graph of y = 2(5)x with the graph of y = 5x. 4. WHICH ONE DOESN’T BELONG? Which equation does not belong with the other three? Explain

your reasoning. y = 3x

f (x) = 2(4)x

f (x) = (−3)x

y = 5(3)x

Monitoring Progress and Modeling with Mathematics In Exercises 5–10, determine whether the equation represents an exponential function. Explain. 5. y =

4(7)x

7. y = 2x3

function with its graph.

6. y = −6x

21. f (x) = 2(0.5)x

22. y = −2(0.5)x

8. y = −3x

23. y = 2(2)x

24. f (x) = −2(2)x

1

9. y = 9(−5)x

USING STRUCTURE In Exercises 21–24, match the

10. y = —2 (1)x

A. 4

In Exercises 11–14, determine whether the table represents an exponential function. Explain. (See Example 1.) 11.

13.

14.

y

−2

1

6

2

−10

2

12

3

−40

3

24

4

−120

4

48

y

1

2

C.

−1

0

1

2

3

y

0.25

1

4

16

64

x

−3

0

3

6

9

y

10

1

−8

−17 −26

x

y

−1

1

x

1

3x

y

4 2

1

−5

D.

y

−2

x

1 −3

−2

12.

x

x

B.

y

−3 2

x

−5

In Exercises 25–30, graph the function. Compare the graph to the graph of the parent function. Describe the domain and range of f. (See Examples 3 and 4.) 25. f (x) = 3(0.5)x

26. f (x) = −4x

In Exercises 15–20, evaluate the function for the given value of x. (See Example 2.)

27. f (x) = −2(7)x

28. f (x) = 6 —3

15. y = 3x; x = 2

16. f (x) = 3(2)x; x = −1

29. f (x) = —2 (8)x

17. y = −4(5)x; x = 2

18. f (x) = 0.5x; x = −3

In Exercises 31–36, graph the function. Describe the domain and range. (See Example 5.)

1

19. f (x) = —3 (6)x; x = 3

278

Chapter 6

1

20. y = —4 (4)x; x = 5

1

31. f (x) = 3x − 1

Exponential Functions and Sequences

()

1 x

3

30. f (x) = —2 (0.25)x

32. f (x) = 4x + 3

(1)x + 1 − 3

33. y = 5x − 2 + 7 35. y =

34. y = − —2

−8(0.75)x + 2

− 2 36. f (x) =

3(6)x − 1

44. An exponential function h models a relationship in

−5

In Exercises 37–40, compare the graphs. Find the value of h, k, or a. 37.

g(x) =

a(2) x

38.

g(x) =

f(x) = 0.25 x

1 −2 1

2 y

exponential function on a calculator. You zoom in repeatedly to 25% of the screen size. The function y = 0.25x represents the percent (in decimal form) of the original screen display that you see, where x is the number of times you zoom in.

+k

4

4

f(x) = 2 x

39.

45. MODELING WITH MATHEMATICS You graph an

y

y

6

0.25 x

which the dependent variable is multiplied by —12 for every 1 unit the independent variable x increases. The value of the function at 0 is 32.

g(x) = −3 x − h 2

4

−2

x

2

40.

a. Graph the function. Describe the domain and range.

4x

b. Find and interpret the y-intercept.

1

f(x) = 3 (6) x

c. You zoom in twice. What percent of the original screen do you see?

y

x

−2

46. MODELING WITH MATHEMATICS A population y of

coyotes in a national park triples every 20 years. The function y = 15(3)x represents the population, where x is the number of 20-year periods.

−4 2

f(x) = −3 x

−2 2

2

x

1

g(x) = 3 (6) x − h

41. ERROR ANALYSIS Describe and correct the error in

evaluating the function.



g(x) = 6(0.5)x ; x = −2 g(−2) = 6(0.5)−2 = 3−2

a. Graph the function. Describe the domain and range. b. Find and interpret the y-intercept.

= —19

c. How many coyotes are in the national park in 40 years?

42. ERROR ANALYSIS Describe and correct the error in

finding the domain and range of the function.

 The domain is all real numbers, and the range is y < 0.

47.

y −2

2

x

48.

−4

g(x) =

In Exercises 47–50, write an exponential function represented by the table or graph. (See Example 7.)

−(0.5) x

−1

In Exercises 43 and 44, graph the function with the given description. Compare the function to f (x) = 0.5(4) x over the interval x = 0 to x = 2. (See Example 6.) 43. An exponential function g models a relationship in

which the dependent variable is multiplied by 2.5 for every 1 unit the independent variable x increases. The value of the function at 0 is 8.

49.

x

0

1

2

3

y

2

14

98

686

x

0

1

2

3

y

−50

−10

−2

−0.4

y

2

(1, −1)

(3, −4)

−6

Section 6.1

y

4 x

(2, −2)

−2 −4

50.

(0, −0.5)

8 4

(0, 8) (1, 4) (2, 2) (3, 1) 2

Exponential Functions

4

x

279

51. MODELING WITH MATHEMATICS The graph

58. HOW DO YOU SEE IT? The exponential function

represents the number y of visitors to a new art gallery after x months.

y = V(x) represents the projected value of a stock x weeks after a corporation loses an important legal battle. The graph of the function is shown.

Art Gallery Stock

Visitors

150

Stock price (dollars)

y 175

(3, 135)

125 100

(2, 90)

75

(1, 60) (0, 40) 25 50

0

0

1

2

3

4

5

6 x

Month

y

80 70 60 50 40 30 20 10 0

0 1 2 3 4 5 6 7 8 9 x

Week

a. Write an exponential function that represents this situation.

a. After how many weeks will the stock be worth $20?

b. Approximate the number of visitors after 5 months.

b. Describe the change in the stock price from Week 1 to Week 3.

52. PROBLEM SOLVING A sales report shows that

3300 gas grills were purchased from a chain of hardware stores last year. The store expects grill sales to increase 6% each year. About how many grills does the store expect to sell in Year 6? Use an equation to justify your answer.

59. USING GRAPHS The graph

represents the exponential function f. Find f (7).

g(x) = −2x − 3. How are the y-intercept, domain, and range affected by the translation?

3

6

y

2

10

50

250

f (x + k) increased by a constant k, the quotient — is f (x) always the same regardless of the value of x.

⋅

62. PROBLEM SOLVING A function g models a

relationship in which the dependent variable is multiplied by 4 for every 2 units the independent variable increases. The value of the function at 0 is 5. Write an equation that represents the function.

56. OPEN-ENDED Write a function whose graph is a

horizontal translation of the graph of h(x) = 4x. 57. USING STRUCTURE The graph of g is a translation

4 units up and 3 units right of the graph of f (x) = 5x. Write an equation for g.

63. PROBLEM SOLVING Write an exponential function f

so that the slope from the point (0, f (0)) to the point (2, f (2)) is equal to 12.

Maintaining Mathematical Proficiency

280

Chapter 6

Reviewing what you learned in previous grades and lessons

(Skills Review Handbook)

65. 35%

(2, −6)

61. REASONING Let f (x) = ab x. Show that when x is

y = a 2x when a is positive and when a is negative.

64. 4%

(1, −3)

y = ab x that represents a real-life population. Explain the meaning of each of the constants a and b in the real-life context.

55. WRITING Describe the effect of a on the graph of

Write the percent as a decimal.

4 x

60. THOUGHT PROVOKING Write a function of the form

table represents an exponential function because y is multiplied by a constant factor. Is your friend correct? Explain. 1

−2

−6

54. MAKING AN ARGUMENT Your friend says that the

0

2

(0, −1.5)

−4

53. WRITING Graph the function f (x) = −2x. Then graph

x

y

66. 128%

Exponential Functions and Sequences

67. 250%

6.2

Exponential Growth and Decay Essential Question

What are some of the characteristics of exponential growth and exponential decay functions? Predicting a Future Event Work with a partner. It is estimated, that in 1782, there were about 100,000 nesting pairs of bald eagles in the United States. By the 1960s, this number had dropped to about 500 nesting pairs. In 1967, the bald eagle was declared an endangered species in the United States. With protection, the nesting pair population began to increase. Finally, in 2007, the bald eagle was removed from the list of endangered and threatened species.

To be proficient in math, you need to apply the mathematics you know to solve problems arising in everyday life.

Describe the pattern shown in the graph. Is it exponential growth? Assume the pattern continues. When will the population return to that of the late 1700s? Explain your reasoning. Bald Eagle Nesting Pairs in Lower 48 States y

Number of nesting pairs

MODELING WITH MATHEMATICS

9789

10,000 8000

6846

6000

5094 3399

4000 2000

1188

1875

0

1978 1982 1986 1990 1994 1998 2002 2006 x

Year

Describing a Decay Pattern Work with a partner. A forensic pathologist was called to estimate the time of death of a person. At midnight, the body temperature was 80.5°F and the room temperature was a constant 60°F. One hour later, the body temperature was 78.5°F. a. By what percent did the difference between the body temperature and the room temperature drop during the hour? b. Assume that the original body temperature was 98.6°F. Use the percent decrease found in part (a) to make a table showing the decreases in body temperature. Use the table to estimate the time of death.

Communicate Your Answer 3. What are some of the characteristics of exponential growth and exponential

decay functions? 4. Use the Internet or some other reference to find an example of each type of

function. Your examples should be different than those given in Explorations 1 and 2. a. exponential growth Section 6.2

b. exponential decay Exponential Growth and Decay

281

6.2 Lesson

What You Will Learn Use and identify exponential growth and decay functions. Solve real-life problems involving exponential growth and decay.

Core Vocabul Vocabulary larry exponential growth, p. 282 exponential growth function, p. 282 exponential decay, p. 283 exponential decay function, p. 283 compound interest, p. 284

Exponential Growth and Decay Functions Exponential growth occurs when a quantity increases by the same factor over equal intervals of time.

Core Concept Exponential Growth Functions A function of the form y = a(1 + r)t, where a > 0 and r > 0, is an exponential growth function. initial amount final amount

rate of growth (in decimal form)

y = a(1 + r)t

STUDY TIP Notice that an exponential growth function is of the form y = abx, where b is replaced by 1 + r and x is replaced by t.

time growth factor

Using an Exponential Growth Function The inaugural attendance of an annual music festival is 150,000. The attendance y increases by 8% each year. a. Write an exponential growth function that represents the attendance after t years. b. How many people will attend the festival in the fifth year? Round your answer to the nearest thousand.

SOLUTION a. The initial amount is 150,000, and the rate of growth is 8%, or 0.08. y = a(1 + r)t

Write the exponential growth function.

= 150,000(1 + 0.08)t

Substitute 150,000 for a and 0.08 for r.

= 150,000(1.08)t

Add.

The festival attendance can be represented by y = 150,000(1.08)t. b. The value t = 4 represents the fifth year because t = 0 represents the first year. y = 150,000(1.08)t

Write the exponential growth function.

= 150,000(1.08)4

Substitute 4 for t.

≈ 204,073

Use a calculator.

About 204,000 people will attend the festival in the fifth year.

Monitoring Progress

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1. A website has 500,000 members in 2010. The number y of members increases

by 15% each year. (a) Write an exponential growth function that represents the website membership t years after 2010. (b) How many members will there be in 2016? Round your answer to the nearest ten thousand.

282

Chapter 6

Exponential Functions and Sequences

Exponential decay occurs when a quantity decreases by the same factor over equal intervals of time.

Core Concept Exponential Decay Functions

STUDY TIP

A function of the form y = a(1 − r)t, where a > 0 and 0 < r < 1, is an exponential decay function.

Notice that an exponential decay function is of the form y = abx, where b is replaced by 1 − r and x is replaced by t.

initial amount final amount

rate of decay (in decimal form)

y = a(1 − r)t

time decay factor

For exponential growth, the value inside the parentheses is greater than 1 because r is added to 1. For exponential decay, the value inside the parentheses is less than 1 because r is subtracted from 1.

Identifying Exponential Growth and Decay Determine whether each table represents an exponential growth function, an exponential decay function, or neither. a.

b.

x

y

0

270

1

90

2

30

3

10

x

0

1

2

3

y

5

10

20

40

SOLUTION a. +1 +1 +1

+1

b.

x

y

0

270

1

90

2

30

3

10

× —13 × —13

0

1

2

3

y

5

10

20

40

×2

Monitoring Progress

+1

x

× —13

As x increases by 1, y is multiplied by —13 . So, the table represents an exponential decay function.

+1

×2

×2

As x increases by 1, y is multiplied by 2. So, the table represents an exponential growth function. Help in English and Spanish at BigIdeasMath.com

Determine whether the table represents an exponential growth function, an exponential decay function, or neither. Explain. 2.

x

0

1

2

3

y

64

16

4

1

Section 6.2

3.

x

1

3

5

7

y

4

11

18

25

Exponential Growth and Decay

283

Interpreting Exponential Functions Determine whether each function represents exponential growth or exponential decay. Identify the percent rate of change. a. y = 5(1.07)t

b. f (t) = 0.2(0.98)t

SOLUTION a. The function is of the form y = a(1 + r)t, where 1 + r > 1, so it represents exponential growth. Use the growth factor 1 + r to find the rate of growth. 1 + r = 1.07

Write an equation.

r = 0.07

Solve for r.

So, the function represents exponential growth and the rate of growth is 7%. b. The function is of the form y = a(1 − r)t, where 1 − r < 1, so it represents exponential decay. Use the decay factor 1 − r to find the rate of decay. 1 − r = 0.98

Write an equation.

r = 0.02

Solve for r.

So, the function represents exponential decay and the rate of decay is 2%.

Monitoring Progress

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Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change. 4. y = 2(0.92)t

5. f (t) = (1.2)t

Solving Real-Life Problems Exponential growth functions are used in real-life situations involving compound interest. Although interest earned is expressed as an annual rate, the interest is usually compounded more frequently than once per year. So, the formula y = a(1 + r)t must be modified for compound interest problems.

Core Concept

STUDY TIP For interest compounded yearly, you can substitute 1 for n in the formula to get y = P(1 + r)t.

Compound Interest Compound interest is the interest earned on the principal and on previously earned interest. The balance y of an account earning compound interest is

(

)

r nt y=P 1+— . n P = principal (initial amount) r = annual interest rate (in decimal form) t = time (in years) n = number of times interest is compounded per year

284

Chapter 6

Exponential Functions and Sequences

Writing a Function You deposit $100 in a savings account that earns 6% annual interest compounded monthly. Write a function that represents the balance after t years.

INTERPRETING EXPRESSIONS

SOLUTION

Notice that the function consists of the product of the principal, 100, and a factor independent of the principal, 1.00512t.

(

r y=P 1+— n

(

)

nt

Write the compound interest formula.

0.06 = 100 1 + — 12 = 100(1.005)12t

)

12t

Substitute 100 for P, 0.06 for r, and 12 for n. Simplify.

Solving a Real-Life Problem You have a checking account and a money market account at a local bank. Your checking account has a constant balance of $200. The table shows the total balance of the accounts over time. a. Write a function m that represents the balance of your money market account after t years. b. Write a function B that represents the total balance after t years. Compare the graph of m to the graph of B.

Year, t

Total balance

0 1 2 3 4 5

$300 $310 $321 $333.10 $346.41 $361.05

c. Compare the money market account to the savings account in Example 4.

SOLUTION a. The $200 balance of your checking account can be represented by the constant function c(t) = 200. To find the balances m(t) of the money market account, subtract $200 from each value in the table, as shown. From this table, you know the initial balance is $100, and it increases 10% each year. So, P = 100 and r = 0.1. m(t) = P(1 + r)t

Total balance (dollars)

Saving Money y 400 350 300

B(t) = 100(1.1)t + 200

250 200 150

= 100(1 + 0.1)t

Substitute 100 for P and 0.1 for r.

= 100(1.1)t

Add.

m(t)

0 1 2 3 4 5

$100 $110 $121 $133.10 $146.41 $161.05

b. To write a function that represents the total balance, find the sum of the expressions that represent the balances of the two accounts. B(t) = m(t) + c(t)

100

m(t) = 100(1.1)t

50

0

Write the compound interest formula when n = 1.

Year, t

0

1

2

3

4

5

6

7 t

Year

Year, t

Total balance

0 1 2 3 4

$1200 $1250 $1302.50 $1357.63 $1415.51

= 100(1.1)t + 200 From the graphs, you can see that the graph of B is a vertical translation of the graph of m. c. Each account has an initial balance of $100. The money market account earns 10% interest each year, and the savings account earns 6% interest each year. So, the balance of the money market account increases faster.

Monitoring Progress

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6. You deposit $500 in an account that earns 9% annual interest compounded

monthly. Write a function that represents the balance y (in dollars) after t years. 7. WHAT IF? Repeat Example 5 using the table shown.

Section 6.2

Exponential Growth and Decay

285

6.2

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE In the exponential growth function y = a(1 + r)t, the quantity r is

called the ________. 2. VOCABULARY What is the decay factor in the exponential decay function y = a(1 − r)t? 3. VOCABULARY Compare exponential growth and exponential decay. 4. WRITING When does the function y = abx represent exponential growth? exponential decay?

Monitoring Progress and Modeling with Mathematics In Exercises 5–12, identify the initial amount a and the rate of growth r (as a percent) of the exponential function. Evaluate the function when t = 5. Round your answer to the nearest tenth. 5. y = 350(1 + 0.75)t

6. y = 10(1 + 0.4)t

7. y = 25(1.2)t

8. y = 12(1.05)t

9. f (t) = 1500(1.074)t 11. g(t) = 6.72(2)t

18. MODELING WITH MATHEMATICS A young channel

catfish weighs about 0.1 pound. During the next 8 weeks, its weight increases by about 23% each week. a. Write an exponential growth function that represents the weight of the catfish after t weeks during the 8-week period. b. About how much will the catfish weigh after 4 weeks? Round your answer to the nearest thousandth.

10. h(t) = 175(1.028)t 12. p(t) = 1.8t

In Exercises 13–16, write a function that represents the situation. 13. Sales of $10,000 increase by 65% each year. 14. Your starting annual salary of $35,000 increases by

4% each year. 15. A population of 210,000 increases by 12.5% each year. 16. An item costs $4.50, and its price increases by 3.5%

each year. 17. MODELING WITH MATHEMATICS The population of

a city has been increasing by 2% annually. The sign shown is from the year 2000. (See Example 1.) a. Write an exponential growth function that represents the population t years after 2000. b. What will the population be in 2020? Round your answer to the nearest thousand. 286

Chapter 6

CITY LIMIT

BROOKFIELD POP. 315,000

In Exercises 19–26, identify the initial amount a and the rate of decay r (as a percent) of the exponential function. Evaluate the function when t = 3. Round your answer to the nearest tenth. 19. y = 575(1 − 0.6)t

20. y = 8(1 − 0.15)t

21. g(t) = 240(0.75)t

22. f (t) = 475(0.5)t

23. w(t) = 700(0.995)t

24. h(t) = 1250(0.865)t

()

7 t

25. y = —8

()

3 t

26. y = 0.5 —4

In Exercises 27– 29, write a function that represents the situation. 27. A population of 100,000 decreases by 2% each year. 28. A $900 sound system decreases in value by 9%

each year. 29. A stock valued at $100 decreases in value by 9.5%

Exponential Functions and Sequences

each year.

30. ERROR ANALYSIS You purchase a car in 2010 for

$25,000. The value of the car decreases by 14% annually. Describe and correct the error in finding the value of the car in 2015. v(t) = 25,000(1.14)t v(5) = 25,000(1.14)5 ≈ 48,135 The value of the car in 2015 is about $48,000.

In Exercises 31–36, determine whether the table represents an exponential growth function, an exponential decay function, or neither. Explain. (See Example 2.)

33.

35.

numbers of visitors to a website t days after it is online. t Visitors



31.

38. ANALYZING RELATIONSHIPS The table shows the total

x

y

−1

32.

x

y

50

0

32

0

10

1

28

1

2

2

24

2

0.4

3

20

x

y

34.

45

11,000

12,100

13,310

14,641

a. Determine whether the table represents an exponential growth function, an exponential decay function, or neither. b. How many people will have visited the website after it is online 47 days? In Exercises 39–46, determine whether each function represents exponential growth or exponential decay. Identify the percent rate of change. (See Example 3.) 39. y = 4(0.8)t

40. y = 15(1.1)t

41. y = 30(0.95)t

42. y = 5(1.08)t

43. r(t) = 0.4(1.06)t

44. s(t) = 0.65(0.48)t

5 t

()

4 t

46. m(t) = —5

In Exercises 47– 50, write a function that represents the balance after t years. (See Example 4.)

0

35

1

17

1

29

2

51

2

23

3

153

3

17

4

459

x

y

x

y

5

2

3

432

10

8

5

72

15

32

7

12

20

128

9

2

47. $2000 deposit that earns 5% annual interest

compounded quarterly 48. $1400 deposit that earns 10% annual interest

compounded semiannually 49. $6200 deposit that earns 8.4% annual interest

compounded monthly 50. $3500 deposit that earns 9.2% annual interest

compounded quarterly 51. PROBLEM SOLVING You have a checking account

37. ANALYZING RELATIONSHIPS The table shows the

value of a camper t years after it is purchased.

b. What is the value of the camper after 5 years?

44

()

y

a. Determine whether the table represents an exponential growth function, an exponential decay function, or neither.

43

45. g(t) = 2 —4

x

36.

42

t

Value

1

$37,000

2

$29,600

3

$23,680

4

$18,944

and a savings account at a credit union. Your checking account has a constant balance of $500. The table shows the total balance of the accounts over time. (See Example 5.) a. Write a function m that represents the balance of your savings account after t years. b. Write a function B that represents the total balance after t years. Compare the graph of m to the graph of B.

Year, t

Total balance

0 1 2 3 4 5

$2500 $2540 $2580.80 $2622.42 $2664.86 $2708.16

c. Compare the savings account to the account represented in Exercise 47.

Section 6.2

Exponential Growth and Decay

287

52. COMBINING FUNCTIONS You deposit $9000 in

56. THOUGHT PROVOKING Describe two account options

a savings account that earns 3.6% annual interest compounded monthly. You also save $40 per month in a safe at home. Write a function C(t) = b(t) + h(t), where b(t) represents the balance of your savings account and h(t) represents the amount in your safe after t years. Find and interpret C(5).

into which you can deposit $1000 and earn compound interest. Write a function that represents the balance of each account after t years. Which account would you rather use? Explain your reasoning.

57. MAKING AN ARGUMENT A store is having a sale on

53. NUMBER SENSE During a flu epidemic, the number

sweaters. On the first day, the prices of the sweaters are reduced by 20%. The prices will be reduced another 20% each day until the sweaters are sold. Your friend says the sweaters will be free on the fifth day. Is your friend correct? Explain.

of sick people triples every week. What is the growth rate as a percent? Explain your reasoning. 54. HOW DO YOU SEE IT? Match each situation with its

graph. Explain your reasoning. a. A bacterial population doubles each hour. b. The value of a computer decreases by 18% each year. c. A deposit earns 11% annual interest compounded yearly.

58. COMPARING FUNCTIONS The graphs of f and g

d. A radioactive element decays 5.5% each year. A.

C.

y 400

B.

200

200

0

0

0

8

16 t

D.

y 400 200 0

are shown.

y 400

8 6

g(t) = kf(t) 0

8

16 t

8

y 400

0

16 t

4 2

−4

200

0

−2

0

8

16 t

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

(Section 3.5)

59.

3

5

y

10

5

0

−5

60.

x

−6

−3

0

3

y

−2

2

6

10

Write an equation of the line that passes through the given points. 61. (0, −2), (3, −5)

288

Chapter 6

4t

c. The graph of g is the same as the graph of h(t) = f (t + r). Find the value of r. Explain your procedure.

The points represented by the table lie on a line. Find the slope of the line. 1

2

b. Describe the transformation from the graph of f to the graph of g. Determine the value of k.

y = abx that does not represent an exponential growth function or an exponential decay function. Explain your reasoning.

−1

f(t) = 2t

a. Explain why f is an exponential growth function. Identify the rate of growth.

55. WRITING Give an example of an equation in the form

x

y

62. (−4, −2), (0, 6)

Exponential Functions and Sequences

(Section 4.1 and Section 4.2) 63. (−7, −4), (−1, 5)

6.3

Comparing Linear and Exponential Functions Essential Question

How can you compare the growth rates of linear

and exponential functions?

Comparing Values Work with a partner. An art collector buys two paintings. The value of each painting after t years is y dollars. Complete each table. Compare the values of the two paintings. Which painting’s value has a constant growth rate? Which painting’s value has an increasing growth rate? Explain your reasoning. t

COMPARING PREDICTIONS To be proficient in math, you need to visualize the results of varying assumptions, explore consequences, and compare predictions with data.

y = 19t + 5

t

0

0

1

1

2

2

3

3

4

4

y = 3t

Comparing Values Work with a partner. Analyze the values of the two paintings over the given time periods. The value of each painting after t years is y dollars. Which painting’s value eventually overtakes the other? t

y = 19t + 5

t

4

4

5

5

6

6

7

7

8

8

9

9

y = 3t

Comparing Graphs Work with a partner. Use the tables in Explorations 1 and 2 to graph y = 19t + 5 and y = 3t in the same coordinate plane. Compare the graphs of the functions.

Communicate Your Answer 4. How can you compare the growth rates of linear and exponential functions? 5. Which function has a growth rate that is eventually much greater than the growth

rate of the other function? Explain your reasoning.

Section 6.3

Comparing Linear and Exponential Functions

289

6.3 Lesson

What You Will Learn Choose functions to model data. Compare functions using average rates of change.

Core Vocabul Vocabulary larry

Solve real-life problems involving different function types.

average rate of change, p. 291 Previous slope

Choosing Functions to Model Data So far, you have studied linear functions and exponential functions. You can use these functions to model data.

Core Concept Linear and Exponential Functions Linear Function y = mx + b

Exponential Function y = abx y

y

x x

Using Graphs to Identify Functions Plot the points. Tell whether the points appear to represent a linear function, an exponential function, or neither. a. (3, −1), (2, 1), (0, 1), (1, 3), (−1, −1)

b. (0, 1), (2, −2), (4, −5), (−2, 4), (−4, 7)

( )

c. (0, 2), (1, 1), 2, —12 , (−2, 8), (−1, 4)

SOLUTION a.

b.

y

8

c.

y

8

y

4 6

4 2 −4 −2

2

4 x

4

4

x

2

−4

−2 −8

neither

linear

Monitoring Progress

−2

2

exponential

Help in English and Spanish at BigIdeasMath.com

Plot the points. Tell whether the points appear to represent a linear function, an exponential function, or neither. 1. (−3, −4), (0, 2), (−2, 2), (1, −4), (−1, 4)

( )( 1

1

2. (1, 1), (2, 3), (3, 9), 0, —3 , −1, —9

)

3. (−2, −7), (0, −1), (2, 5), (−1, −4), (1, 2)

290

Chapter 6

Exponential Functions and Sequences

x

Core Concept Differences and Ratios of Functions You can use patterns between consecutive data pairs to determine which type of function models the data. • Linear Function The differences of consecutive y-values are constant.

REMEMBER

• Exponential Function Consecutive y-values have a common ratio.

Linear functions have a constant rate of change. So, for equally-spaced x-values, the differences of consecutive y-values are constant. Exponential functions do not have a constant rate of change.

In each case, the differences of consecutive x-values need to be constant.

Using Differences or Ratios to Identify Functions Tell whether each table of values represents a linear or an exponential function. Then write the function. a.

x y

−2

−1

9

0

5

1 −3

1

2

b.

−7

x

−3

−2

−1

0

1

y

1 —3

1

3

9

27

SOLUTION a.

STUDY TIP First determine that the differences of consecutive x-values are constant. Then check whether the y-values have a constant difference or a common ratio.

+1 x y

−2

+1

−1

9

0

5 −4

+1

1 −3

1 −4

−4

x

−1

0

1

2

3

y

32

16

8

4

2

2 −7

+1

+1

+1

x

−3

−2

−1

0

1

y

1 —3

1

3

9

27

×3

−4

The differences of consecutive y-values are constant. The slope −4 is — = −4 and the y-intercept 1 is 1. So, the table represents the linear function y = −4x + 1.

Monitoring Progress

+1

b.

+1

×3

×3

×3

Consecutive y-values have a common ratio of 3 and the y-intercept is 9. So, the table represents the exponential function y = 9(3)x.

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4. Tell whether the table of values represents a linear or an exponential function.

Then write the function.

Comparing Functions Using Average Rates of Change For nonlinear functions, the rate of change is not constant. You can compare two functions over the same interval using their average rates of change. The average rate of change of a function y = f (x) between x = a and x = b is the slope of the line through (a, f (a)) and (b, f (b)). y

change in y average rate of change = — change in x f (b) − f (a) =— b−a Section 6.3

f(b) f(b) − f(a) f(a)

b−a a

b

Comparing Linear and Exponential Functions

x

291

STUDY TIP You can explore this concept using a graphing calculator.

Core Concept Comparing Functions Using Average Rates of Change As a and b increase, the average rate of change between x = a and x = b of an increasing exponential function y = f(x) will eventually exceed the average rate of change between x = a and x = b of an increasing linear function y = g(x). So, as x increases, f(x) will eventually exceed g(x).

Using and Interpreting Average Rates of Change Two social media websites open their memberships to the public. (a) Compare the websites by calculating and interpreting the average rates of change from Day 10 to Day 20. (b) Predict which website will have more members after 50 days. Explain. Website A Members, y

0

650

5

1025

10

1400

15

1775

20

2150

25

y

Members

Day, x

Website B

1800 1200 600 0

0

10

20

30

x

Day

2525

SOLUTION a. Calculate the average rates of change by using the points whose x-coordinates are 10 and 20. Website A: Use (10, 1400) and (20, 2150). f (b) − f (a) 2150 − 1400 750 average rate of change = — = —— = — = 75 b−a 20 − 10 10 Website B: Use the graph to estimate the points when x = 10 and x = 20. Use (10, 950) and (20, 1850). f (b) − f (a) 1850 − 950 900 average rate of change = — ≈ — = — = 90 b−a 20 − 10 10 From Day 10 to Day 20, Website A membership increases at an average rate of 75 people per day, and Website B membership increases at an average rate of about 90 people per day. So, Website B membership is growing faster. b. Using the table, membership increases and the average rates of change are constant. So, Website A membership can be represented by an increasing linear function. Using the graph, membership increases and the average rates of change are increasing. It appears that Website B membership can be represented by an increasing exponential function. After 25 days, the memberships of both websites are about equal and the average rate of change of Website B exceeds the average rate of change of Website A. So, Website B will have more members after 50 days.

Monitoring Progress

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5. Compare the websites in Example 3 by calculating and interpreting the average

rates of change from Day 0 to Day 10. 292

Chapter 6

Exponential Functions and Sequences

Solving Real-Life Problems Comparing Different Function Types I 2000, Littleton had a population of 11,510 people. Littleton’s population increased In bby 600 people each year. In 2000, Tinyville had a population of 10,000 people. Tinyville’s population increased by 10% each year. T aa. In what year were the populations equal? b. Suppose Littleton’s initial population doubled to 23,020 and maintained a constant b rate of increase of 600 people each year. Did Tinyville’s population still catch up to Littleton’s population? c. Suppose Littleton’s rate of increase doubled to 1200 people each year, in addition to doubling the initial population. Did Tinyville’s population still catch up to Littleton’s population? Explain.

SOLUTION a. Let x represent the number of years since 2000. Write a function to model the population of each town.

25,000

T L

Intersection Y=13310 0 X=3 0

10

Littleton: L(x) = 600x + 11,510

Linear function

Tinyville: T(x) = 10,000(1.1)x

Exponential function

Use a graphing calculator to graph each function in the same viewing window. Use the intersect feature to find the value of x for which L(x) = T(x). The graphs intersect when x = 3. So, the populations were equal in 2003. b. Littleton’s new population function is f (x) = 600x + 23,020. Use a graphing calculator to graph f and T in the same viewing window. From the graph, you can see that Tinyville’s population eventually caught up to and exceeded Littleton’s population.

50,000

T f

0

20

0

c. Littleton’s new population function is g(x) = 1200x + 23,020. Use a graphing calculator to graph g and T in the same viewing window. From the graph, you can see that Tinyville’s population eventually caught up to and exceeded Littleton’s population. Because Littleton’s population shows linear growth and Tinyville’s population shows exponential growth, Tinyville’s population eventually exceeded Littleton’s regardless of Littleton’s constant rate or initial value.

Monitoring Progress

50,000

T

0

g

0

20

Help in English and Spanish at BigIdeasMath.com

6. WHAT IF? In 2000, Littleton had a population of 10,900 people and the population

increased by 600 people each year. In what year were the populations equal? Section 6.3

Comparing Linear and Exponential Functions

293

Exercises

6.3

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Name two types of functions that you can use to model data. Describe the equation and

graph of each type of function. 2. WRITING How can you decide whether to use a linear or an exponential function to model a data set? 3. VOCABULARY Describe how to find the average rate of change of a function y = f (x) between

x = a and x = b.

4. DIFFERENT WORDS, SAME QUESTION Let f (x) = 4x. Which is different? Find “both” answers.

As the input increases by 1, by what factor does the output increase?

What is the average rate of change of f between x = 1 and x = 2?

f (2) − f (1) What is —? 2−1

What is the slope of the line that passes through (1, 4) and (2, 16)?

Monitoring Progress and Modeling with Mathematics In Exercises 5–8, tell whether the points appear to represent a linear function, an exponential function, or neither. 5.

6

13. (−3, 5.5), (−1, 0.5), (0, −2), (1, −4.5), (3, −9.5)

−2

4 2 −2

2

12. (−1, −1), (−2, −6), (0, −2), (3, 1), (2, −2)

y

6.

y

11. (0, 0), (4, 6), (2, 3), (−2, 3), (−4, 6)

x

x −4

14. (0, 6), (1, 3), (2, 1.5), (3, 0.75), (−1, 12)

−8

In Exercises 15–18, tell whether the table of values represents a linear or an exponential function. Then write the function. (See Example 2.)

−12

−2

7.

4

15.

8.

y

16

2

y

2

16.

8

x

−2

4

−4

−4

−2

2x

In Exercises 9–14, plot the points. Tell whether the points appear to represent a linear function, an exponential function, or neither. (See Example 1.)

17.

18.

9. (−1, −3), (0, −2), (1, −1), (2, 0), (3, 1)

294

( −1, — ), ( 0, — ), (1, 1), (2, 2), (3, 4) 1 4

−1

0

1

2

3

y

−1

−0.5

0

0.5

1

x

−2

−1

0

1

2

y

1 —5

1

5

25

125

x

1

2

3

4

5

y

512

128

32

8

2

x

−5

−4

−3

−2

−1

y

12

9

6

3

0

12

−2

10.

x

1 2

Chapter 6

Exponential Functions and Sequences

23. ANALYZING RELATIONSHIPS Three organizations

school. The table shows the distances d (in miles) the bus travels in t minutes. Let the time t represent the independent variable. Tell whether the data can be modeled by a linear function, an exponential function, or neither. Explain. 1

2

3

4

5

0.7

1.4

2.1

2.1

2.8

Time, t Distance, d

are collecting donations for a cause. Organization A begins with one donation, and the number of donations quadruples each hour. The table shows the numbers of donations collected by Organization B. The graph shows the numbers of donations collected by Organization C. Time (hours), t

Number of donations, y

0

0

1

4

2

8

3

12

a. Plot the points. Let the number d of decades after 1980 represent the independent variable.

4

16

5

20

b. Tell whether the data can be modeled by a linear or an exponential function. Then write the function.

6

24

20. MODELING WITH MATHEMATICS The table shows the

size s (in hectares) of a glacier d decades after 1980. 0

1

2

3

300

270

243

218.7

Decades, d Size, s



+1

0

1

2

3

4

y

3

6

12

24

48

×2

×2

×2

60 40 20 0

(0, 1) (1, 3)

(3, 27)

(2, 9) 0

2

4

t

Time (hours)

c. For which function does the average rate of change increase most quickly? What does this tell you about the numbers of donations collected by the three organizations?

+1

x

(4, 81)

b. Find the average rates of change of each function for each 1-hour interval from t = 0 to t = 6.

in determining which type of function the table represents. +1

y 80

a. What type of function represents the numbers of donations collected by Organization A? B? C?

21. ERROR ANALYSIS Describe and correct the error

+1

Organization C Number of donations

19. MODELING WITH MATHEMATICS You ride a bus to

24. COMPARING FUNCTIONS The room expenses for two

different resorts are shown. (See Example 4.)

×2

Consecutive y-values change by a constant amount. So, the table represents a linear function. 22. ANALYZING RELATIONSHIPS The population of

Town A in 1970 was 3000. The population of Town A increased by 20% every decade. Let x represent the number of decades since 1970. The graph shows the population of Town B. (See Example 3.)

b. Predict which town will have a greater population after 2020. Explain.

Town B

a. For what length of vacation does each resort cost about the same?

y 5750

Population

a. Compare the populations of the towns by calculating and interpreting the average rates of change from 1990 to 2010.

Vacations available up to 14 nights. Each additional night is a 10% increase in price from the previous package.

b. Suppose Blue Water Resort charges $1450 for the first three nights and $105 for each additional night. Would Sea Breeze Resort ever be more expensive than Blue Water Resort? Explain.

5250 4750 4250 0

0

2

4

x

Decades since 1970

Section 6.3

c. Suppose Sea Breeze Resort charges $1200 for the first three nights. The charge increases 10% for each additional night. Would Blue Water Resort ever be more expensive than Sea Breeze Resort? Explain. Comparing Linear and Exponential Functions

295

25. REASONING Explain why the average rate of change

29. CRITICAL THINKING Is the graph of a set of points

of a linear function is constant and the average rate of change of an exponential function is not constant.

enough to determine whether the points represent a linear function or an exponential function? Justify your answer.

26. HOW DO YOU SEE IT? Match each graph with its

function. Explain your reasoning. a.

b.

y

30. THOUGHT PROVOKING Find four different patterns in

the figure. Determine whether each pattern represents a linear or an exponential function. Write a model for each pattern.

y

x x

c.

d.

y

n=1

y

n=2

n=3

31. MAKING AN ARGUMENT

Function p is an exponential function and function q is a linear function. Your friend says that after x = 2, function q will always have a greater y-value than function p. Is your friend correct? Explain.

x x

A. y = −—12x + 1 C. y = 2

()

3 x —4 +

B. y = 2(4)x + 1 D. y = 2x − 4

1

(m, n), (m + 1, 2n), (m + 2, 4n), (m + 3, 8n), (m + 4, 16n) 28. CRITICAL THINKING Write and graph a linear

function f and an exponential function g with the following characteristics. •

f(x) < g(x) when x < 0 and x > 4

Day, x Amount, a

Maintaining Mathematical Proficiency Solve the equation. Check your solution.

3

2

1

2

3

4

5

10.2

5.3

2.5

1.2

0.7

(Section 1.3) 34. 5 − t = 7t + 21

35. 6(r − 2) = 2r + 8

36. −6(s + 7) = 10(3 − s)

Find the slope and the y-intercept of the graph of the linear equation. 37. y = −6x + 7

38. y = —14 x + 7

39. 3y = 6x − 12

40. 2y + x = 8

Chapter 6

(2, 9)

8

Reviewing what you learned in previous grades and lessons

33. 8x + 12 = 4x

296

p

12

x

(in milligrams) of venom remaining in an animal’s body x days after being bitten by a venomous snake. Let the number x of days represent the independent variable. Using technology, find a function that models the data. How did you choose the model? About how much venom was initially injected into the animal’s body? Explain your reasoning.

an integer and n ≠ 0. Tell whether the ordered pairs represent a linear or an exponential function. Explain.

f(x) > g(x) when 0 < x < 4

q

y

32. USING TOOLS The table shows the amount a

27. USING STRUCTURE In the ordered pairs below, m is

•

n=4

Exponential Functions and Sequences

(Section 3.5)

6.1–6.3

What Did You Learn?

Core Vocabulary exponential function, p. 274 exponential growth, p. 282 exponential growth function, p. 282 exponential decay, p. 283

exponential decay function, p. 283 compound interest, p. 284 average rate of change, p. 291

Core Concepts Section 6.1 Graphing y = abx When b > 1, p. 275

Graphing y = abx When 0 < b < 1, p. 275

Section 6.2 Exponential Growth Functions, p. 282 Exponential Decay Functions, p. 283

Compound Interest, p. 284

Section 6.3 Linear and Exponential Functions, p. 290 Differences and Ratios of Functions, p. 291

Comparing Functions Using Average Rates of Change, p. 292

Mathematical Practices 1.

How can you use a definition to construct an argument in Exercise 54 on page 280?

2.

How is the form of the function you wrote in Exercise 52 on page 288 related to the forms of other types of functions you have learned about in this course?

3.

What patterns did you use to solve Exercise 27 on page 296?

Analyzing Your Errors Misreading Directions • What Happens: You incorrectly read or do not understand directions. • How to Avoid This Error: Read the instructions for exercises at least twice and make sure you understand what they mean. Make this a habit and use it when taking tests.

297 29 97

6.1–6.3

Quiz

Graph the function. Describe the domain and range. (Section 6.1)

() 1 6

1. y = 5x

2. y = −2 —

x

3. y = 6(2)x − 4 − 1

Determine whether the table represents an exponential growth function, an exponential decay function, or neither. Explain. (Section 6.2) 4.

0

1

2

3

y

7

21

63

189

5.

x

x

1

2

3

4

y

14,641

1331

121

11

Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change. (Section 6.2)

( 35 )

1 3

6. y = 3(1.88)t

7. f (t) = — (1.26)t

8. f (t) = 80 —

t

Tell whether the table of values represents a linear or an exponential function. Then write the function. (Section 6.3) 9.

x

−2

−1

0

1

2

y

48

12

3

—4

3

— 16

10.

3

x

−2

0

2

4

6

y

8

2

−4

−10

−16

11. The function f (t) = 5(4)t represents the number of frogs in a pond after t years.

(Section 6.1 and Section 6.2) a. Does the function represent exponential growth or exponential decay? Explain. b. Graph the function. Describe the domain and range. c. What is the yearly percent change? d. How many frogs are in the pond after 4 years?

12. You deposit $500 in a savings account that earns 3% annual interest compounded

monthly. (Section 6.2) a. Write a function that represents the balance after t years. b. What is the balance of the account after 36 months? 10 years? 13. The table shows the amount y (in grams) of an element remaining in a jar after x

centuries. (Section 6.1 and Section 6.3) Centuries, x

0

1

2

3

4

Amount remaining, y

50

25

12.5

6.25

3.125

a. Plot the points. Let the number x of centuries represent the independent variable. b. Tell whether the data can be modeled by a linear or an exponential function. Then write the function. c. How much of the element remains after 600 years? 298

Chapter 6

Exponential Functions and Sequences

6.4

Solving Exponential Equations Essential Question

How can you solve an exponential equation

graphically?

Solving an Exponential Equation Graphically Work with a partner. Use a graphing calculator to solve the exponential equation 2.5x − 3 = 6.25 graphically. Describe your process and explain how you determined the solution.

The Number of Solutions of an Exponential Equation Work with a partner. a. Use a graphing calculator to graph the equation y = 2x. 6

−6

6 −2

USING APPROPRIATE TOOLS To be proficient in math, you need to use technological tools to explore and deepen your understanding of concepts.

b. In the same viewing window, graph a linear equation (if possible) that does not intersect the graph of y = 2x. c. In the same viewing window, graph a linear equation (if possible) that intersects the graph of y = 2x in more than one point. d. Is it possible for an exponential equation to have no solution? more than one solution? Explain your reasoning.

Solving Exponential Equations Graphically Work with a partner. Use a graphing calculator to solve each equation. a. 2x = —12

b. 2x + 1 = 0

c. 2x = 1

d. 3x = 9

e. 3x − 1 = 0

1 f. 42x = — 16

g. 23x = —18

h. 3x + 2 = —19

i. 2x − 2 = —32 x − 2

Communicate Your Answer 4. How can you solve an exponential equation graphically? 5. A population of 30 mice is expected to double each year. The number p of mice in

the population each year is given by p = 30(2n). In how many years will there be 960 mice in the population?

Section 6.4

Solving Exponential Equations

299

6.4 Lesson

What You Will Learn Solve exponential equations with the same base. Solve exponential equations with unlike bases.

Core Vocabul Vocabulary larry

Solve exponential equations by graphing.

exponential equation, p. 300

Solving Exponential Equations with the Same Base Exponential equations are equations in which variable expressions occur as exponents.

Core Concept Property of Equality for Exponential Equations Two powers with the same positive base b, where b ≠ 1, are equal if and only if their exponents are equal.

Words

If 2x = 25, then x = 5. If x = 5, then 2x = 25.

Numbers

Algebra If b > 0 and b ≠ 1, then bx = by if and only if x = y.

Solving Exponential Equations with the Same Base Solve each equation. a. 3x + 1 = 35

b. 6 = 62x − 3

c. 103x = 102x + 3

SOLUTION a.

3x + 1 =

35

Write the equation.

x+1=

5

Equate the exponents.

−1

−1

x=4 b.

6 = 62x − 3

?

6 = 62(2) − 3 6=6



c.

Simplify.

6 = 62x − 3

Write the equation.

1 = 2x − 3

Equate the exponents.

+3

Check

Subtract 1 from each side.

+3

Add 3 to each side.

4 = 2x

Simplify.

—=—

4 2

Divide each side by 2.

2=x

Simplify.

2x 2

103x =

102x + 3

Write the equation.

3x =

2x + 3

Equate the exponents.

− 2x

− 2x

x=3

Subtract 2x from each side. Simplify.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solution. 1. 22x = 26

300

Chapter 6

2. 52x = 5x + 1

Exponential Functions and Sequences

3. 73x + 5 = 7x + 1

Solving Exponential Equations with Unlike Bases To solve some exponential equations, you must first rewrite each side of the equation using the same base.

Solving Exponential Equations with Unlike Bases Solve (a) 5x = 125, (b) 4x = 2x − 3, and (c) 9x + 2 = 27x.

SOLUTION a. 5x = 125

Write the equation.

5x = 53 x=3

REMEMBER Power of a Power Property (am )n

=

Rewrite 125 as 53.

b.

4x (22)x

amn

Equate the exponents.

=

2x − 3

Write the equation.

=

2x − 3

Rewrite 4 as 22.

22x = 2x − 3

Power of a Power Property

2x = x − 3

Equate the exponents.

x = −3 c.

Check

9x + 2 = 27x (32)x + 2 = (33)x

9x + 2 = 27x

32x + 4 = 33x

?

94 + 2 = 274 531,441 = 531,441

2x + 4 = 3x



4=x

Check

Solve for x.

4x = 2x − 3

?

4−3 = 2−3 − 3 1

1

=— — 64 64



Write the equation. Rewrite 9 as 32 and 27 as 33. Power of a Power Property Equate the exponents. Solve for x.

Solving Exponential Equations When 0 < b < 1

()

x

1 Solve (a) —12 = 4 and (b) 4x + 1 = — . 64

SOLUTION a.

(—)

1 x 2

(2−1)x

=4

Write the equation.

=

Rewrite —12 as 2−1 and 4 as 22.

22

2−x = 22

Power of a Power Property

−x = 2

Equate the exponents.

x = −2 b.

Check 1 4x + 1 = — 64

?

1 4−4 + 1 = — 64 1

1

=— — 64 64



1 4x + 1 = — 64 1 4x + 1 = —3 4

Solve for x. Write the equation. Rewrite 64 as 43.

4x + 1 = 4−3

Definition of negative exponent

x + 1 = −3

Equate the exponents.

x = −4

Solve for x.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solution. 4. 4x = 256

5. 92x = 3x − 6

Section 6.4

6. 43x = 8x + 1

()

7. —13

x−1

Solving Exponential Equations

= 27 301

Solving Exponential Equations by Graphing Sometimes, it is difficult or impossible to rewrite each side of an exponential equation using the same base. You can solve these types of equations by graphing each side and finding the point(s) of intersection. Exponential equations can have no solution, one solution, or more than one solution depending on the number of points of intersection.

Solving Exponential Equations by Graphing Use a graphing calculator to solve (a) 2.4 x − 1 = 5.76 and (b) 3x + 2 = x + 1.

SOLUTION a. Step 1 Write a system of equations using each side of the equation.

ANOTHER WAY It may be difficult to recognize, but knowing that 242 = 576 helps you reason that 2.42 = 5.76. This can be used to solve part (a) algebraically. 2.4x − 1 = 5.76

y = 2.4 x − 1

Equation 1

y = 5.76

Equation 2 7

Step 2 Enter the equations into a calculator. Then graph the equations in a viewing window that shows where the graphs could intersect. −6

6 −1

Step 3 Use the intersect feature to find the point of intersection. The graphs intersect at (3, 5.76).

7

2.4x − 1 = 2.42 x−1=2 x=3

−6 Intersection X=3 Y=5.76 −1

So, the solution is x = 3. b. Step 1

Step 2

6

Write a system of equations using each side of the equation. y = 3x + 2

Equation 1

y=x+1

Equation 2

Enter the equations into a calculator. Then graph the equations in a viewing window that shows where the graphs could intersect.

10

−10

10

−10

The graphs do not intersect. So, the equation has no solution.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use a graphing calculator to solve the equation. 8. 3.1x + 2 = 9.61

302

Chapter 6

9. 4x − 3 = 3x − 8

Exponential Functions and Sequences

()

10. —14

x

= −2x − 3

Exercises

6.4

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Describe how to solve an exponential equation with unlike bases. 2. WHICH ONE DOESN’T BELONG? Which equation does not belong with the other three?

Explain your reasoning. 2x = 4x + 6

53x + 8 = 52x

2x − 7 = 27

34 = x + 42

Monitoring Progress and Modeling with Mathematics In Exercises 3–12, solve the equation. Check your solution. (See Examples 1 and 2.)

In Exercises 21–24, match the equation with the graph that can be used to solve it. Then solve the equation.

3. 45x = 410

4. 7x − 4 = 78

21. 2x = 4

22. 42x − 5 = 4

5. 39x = 37x + 8

6. 24x = 2x + 9

23. 2x + 2 = 4

24. 2−x − 2 = 4

7. 2x = 64

8. 3x = 243

A.

9. 7x − 5 = 49x

12. 27x = 9x − 2

In Exercises 13–18, solve the equation. Check your solution. (See Example 3.)

( 15 ) = 125

14.

1 128

16. 34x − 9 = —

x

−8

—

( ) 1 216

−2

C.

−2

D.

8

x+1

18.

( ) 1 27

—

4−x

= 92x − 1

−5

8

53x + 2 = 25x − 8 3x + 2 = x − 8 x = −5

( )

1 5x = 32x + 8 — 8

(23)5x = (25)x + 8 215x = 25x + 40 15x = 5x + 40

5

Intersection X=2 Y=4

−4

6

Intersection X=3 Y=4

−2

−2

In Exercises 25–36, use a graphing calculator to solve the equation. (See Example 4.) 25. 0.25x + 2 = 16

correct the error in solving the exponential equation.



3

Intersection X=0 Y=4

—

ERROR ANALYSIS In Exercises 19 and 20, describe and

20.

−7

( 14 ) = 256 1 243

17. 36−3x + 3 = —



2

Intersection X=-4 Y=4

x

15. — = 25x + 3

19.

8

10. 216x = 6x + 10

11. 642x + 4 = 165x

13.

B.

8

27.

( 12 ) —

7x + 1

= −9

29. 2x + 3 = 3x + 8

4 3

( 13 )

31. — x − 1 = —

2x − 1

26. 1.9x − 4 = 3.61 28.

( 13 ) —

x+3

=9

30. 4x − 3 = 5x − 1

19 − 15x 4

32. 2−x + 1 = —

33. 5x = −4−x + 4

34. 7x − 2 = 2x − 2

35. 3x − 3 = 2−x + 3

36. 5−2x + 3 = −6 x + 5

x=4 Section 6.4

Solving Exponential Equations

303

In Exercises 37– 40, solve the equation by using the Property of Equality for Exponential Equations. 37. 30

⋅5

x+3

= 150

39. 4(3−2x − 4) = 36

38. 12

⋅2

x−7

49. PROBLEM SOLVING You deposit $500 in an account

that earns 6% annual interest compounded yearly. Write and solve an exponential equation to determine when the balance of the account will be $561.80.

= 24

40. 2(42x + 1) = 128 50. HOW DO YOU SEE IT? The graph shows the annual

41. MODELING WITH MATHEMATICS You scan a photo

attendance at two events. Each event began in 2004.

into a computer at four times its original size. You continue to increase its size repeatedly by 100% using the computer. The new size of the photo y in comparison to its original size after x enlargements on the computer is represented by y = 2x + 2. How many times must the photo be enlarged on the computer so the new photo is 32 times the original size?

Number of people

Event Attendance

42. MODELING WITH MATHEMATICS A bacterial culture

quadruples in size every hour. You begin observing the number of bacteria 3 hours after the culture is prepared. The amount y of bacteria x hours after the culture is prepared is represented by y = 192(4x − 3). When will there be 196,608 bacteria?

y

y = 4000(1.25) x

12,000

y = 12,000(0.87) x

8000 4000 0

Event 1 Event 2 0

2

4

6

8

x

Year (0 ↔ 2004)

a. Estimate when the events will have about the same attendance. b. Explain how you can verify your answer in part (a). 51. REASONING Explain why the Property of Equality

for Exponential Equations does not work when b = 1. Give an example to justify your answer.

52. THOUGHT PROVOKING Is it possible for an

exponential equation to have two different solutions? If not, explain your reasoning. If so, give an example.

In Exercises 43–46, solve the equation. 43. 33x + 6 = 27x + 2

44. 34x + 3 = 81x

45. 4 x + 3 = 22(x + 1)

46. 58(x − 1) = 625 2x − 2

In Exercises 53 – 56, use a graphing calculator to solve the equation. —

—

54. √ 7 = 7−x

53. 2x = √ 2

47. NUMBER SENSE Explain how you can use mental

math to solve the equation 8x − 4 = 1.

—

55. 8x − 1/2 = √ 8

48. PROBLEM SOLVING There are a total of 128 teams at

the start of a citywide 3-on-3 basketball tournament. Half the teams are eliminated after each round. Write and solve an exponential equation to determine after which round there are 16 teams left.

—

56. √ 5 = 5x + 1

57. USING STRUCTURE Use the results of Exercises —

53–56 to find the value of x for which ax = √a .

58. MAKING AN ARGUMENT Consider the equation

()

x

1 = b, where a > 1 and b > 1. Your friend says a the value of x will always be negative. Is your friend correct? Explain. —

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Determine whether the sequence is arithmetic. If so, find the common difference. 59. −20, −26, −32, −38, . . .

60. 9, 18, 36, 72, . . .

61. −5, −8, −12, −17, . . .

62. 10, 20, 30, 40, . . .

304

Chapter 6

Exponential Functions and Sequences

(Section 4.6)

6.5

Geometric Sequences Essential Question

How can you use a geometric sequence to

describe a pattern?

In a geometric sequence, the ratio between each pair of consecutive terms is the same. This ratio is called the common ratio.

Describing Calculator Patterns Work with a partner. Enter the keystrokes on a calculator and record the results in the table. Describe the pattern. a. Step 1

2

=

b. Step 1

6

4

=

Step 2

×

2

=

Step 2

×

.

5

=

Step 3

×

2

=

Step 3

×

.

5

=

Step 4

×

2

=

Step 4

×

.

5

=

Step 5

×

2

=

Step 5

×

.

5

=

Step

1

2

3

4

5

Calculator display

Step

1

2

3

4

5

1

2

3

4

5

Calculator display

c. Use a calculator to make your own sequence. Start with any number and multiply by 3 each time. Record your results in the table.

Step Calculator display

d. Part (a) involves a geometric sequence with a common ratio of 2. What is the common ratio in part (b)? part (c)?

LOOKING FOR REGULARITY IN REPEATED REASONING To be proficient in math, you need to notice when calculations are repeated and look both for general methods and for shortcuts.

Folding a Sheet of Paper Work with a partner. A sheet of paper is about 0.1 millimeter thick. a. How thick will it be when you fold it in half once? twice? three times? b. What is the greatest number of times you can fold a piece of paper in half? How thick is the result? c. Do you agree with the statement below? Explain your reasoning. “If it were possible to fold the paper in half 15 times, it would be taller than you.”

Communicate Your Answer 3. How can you use a geometric sequence to describe a pattern? 4. Give an example of a geometric sequence from real life other than paper folding.

Section 6.5

Geometric Sequences

305

6.5 Lesson

What You Will Learn Identify geometric sequences. Extend and graph geometric sequences.

Core Vocabul Vocabulary larry

Write geometric sequences as functions.

geometric sequence, p. 306 common ratio, p. 306 Previous arithmetic sequence common difference

Identifying Geometric Sequences

Core Concept Geometric Sequence In a geometric sequence, the ratio between each pair of consecutive terms is the same. This ratio is called the common ratio. Each term is found by multiplying the previous term by the common ratio. 1,

5,

25,

125, . . .

Terms of a geometric sequence

×5 ×5 ×5

common ratio

Identifying Geometric Sequences Decide whether each sequence is arithmetic, geometric, or neither. Explain your reasoning. a. 120, 60, 30, 15, . . .

b. 2, 6, 11, 17, . . .

SOLUTION a. Find the ratio between each pair of consecutive terms. 120

60 60

1

30 30

= —2 — 120

1

15 15

= —2 — 60

The ratios are the same. The common ratio is —12.

1

= —2 — 30

So, the sequence is geometric. b. Find the ratio between each pair of consecutive terms. 2

6 6

11

—2 = 3

11

17

5

6

17

= 1—6 — = 1— — 6 11 11

There is no common ratio, so the sequence is not geometric.

Find the difference between each pair of consecutive terms. 2

6

11

6 − 2 = 4 11 − 6 = 5

17 17 − 11 = 6

There is no common difference, so the sequence is not arithmetic.

So, the sequence is neither geometric nor arithmetic.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Decide whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. 1. 5, 1, −3, −7, . . .

306

Chapter 6

2. 1024, 128, 16, 2, . . .

Exponential Functions and Sequences

3. 2, 6, 10, 16, . . .

Extending and Graphing Geometric Sequences Extending Geometric Sequences Write the next three terms of each geometric sequence. b. 64, −16, 4, −1, . . .

a. 3, 6, 12, 24, . . .

SOLUTION Use tables to organize the terms and extend each sequence. a.

Position

1

2

3

4

5

6

7

Term

3

6

12

24

48

96

192

×2

Each term is twice the previous term. So, the common ratio is 2.

×2

×2

×2

×2

Multiply a term by 2 to find the next term.

×2

The next three terms are 48, 96, and 192. b.

Position

1

2

3

4

5

6

7

Term

64

−16

4

−1

—

1 4

1 −— 16

—

LOOKING FOR STRUCTURE When the terms of a geometric sequence alternate between positive and negative terms, or vice versa, the common ratio is negative.

1 64

Multiply a term by −—14 to find the next term.

( ) × ( −14 ) × ( −14 ) × ( −14 ) × ( −14 ) × ( −14 )

1 × −— 4

—

—

—

—

—

1 1 1 The next three terms are —, −—, and —. 4 16 64

Graphing a Geometric Sequence Graph the geometric sequence 32, 16, 8, 4, 2, . . .. What do you notice?

SOLUTION

STUDY TIP The points of any geometric sequence with a positive common ratio lie on an exponential curve.

an

Make a table. Then plot the ordered pairs (n, an). Position, n

1

2

3

4

5

Term, an

32

16

8

4

2

(1, 32)

32 24

(2, 16)

16

The points appear to lie on an exponential curve.

0

Monitoring Progress

(3, 8) (4, 4) (5, 2)

8

0

2

4

n

Help in English and Spanish at BigIdeasMath.com

Write the next three terms of the geometric sequence. Then graph the sequence. 4. 1, 3, 9, 27, . . .

5. 2500, 500, 100, 20, . . .

6. 80, −40, 20, −10, . . .

7. −2, 4, −8, 16, . . .

Section 6.5

Geometric Sequences

307

Writing Geometric Sequences as Functions Because consecutive terms of a geometric sequence have a common ratio, you can use the first term a1 and the common ratio r to write an exponential function that describes a geometric sequence. Let a1 = 1 and r = 5. Position, n

Term, an

1

first term, a1

a1

1

2

second term, a2

a1r

1 5=5

3

Written using a1 and r

r2

third term, a3

4

fourth term, a4

⋮

⋮

a1

r3

a1

⋮ rn − 1

nth term, an

n

a1

Numbers

⋅ 1 ⋅5 1 ⋅5 1

⋅

2

= 25

3

= 125 ⋮

5n − 1

Core Concept

STUDY TIP Notice that the equation an = a1 r n − 1 is of the form y = ab x.

Equation for a Geometric Sequence Let an be the nth term of a geometric sequence with first term a1 and common ratio r. The nth term is given by an = a1r n − 1.

Finding the nth Term of a Geometric Sequence Write an equation for the nth term of the geometric sequence 2, 12, 72, 432, . . .. Then find a10.

SOLUTION The first term is 2, and the common ratio is 6. an = a1r n − 1 an =

2(6)n − 1

Equation for a geometric sequence Substitute 2 for a1 and 6 for r.

Use the equation to find the 10th term. an = 2(6)n − 1 a10 =

2(6)10 − 1

= 20,155,392

Write the equation. Substitute 10 for n. Simplify.

The 10th term of the geometric sequence is 20,155,392.

Monitoring Progress

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Write an equation for the nth term of the geometric sequence. Then find a7. 8. 1, −5, 25, −125, . . . 9. 13, 26, 52, 104, . . . 10. 432, 72, 12, 2, . . . 11. 4, 10, 25, 62.5, . . .

308

Chapter 6

Exponential Functions and Sequences

You can rewrite the equation for a geometric sequence with first term a1 and common ratio r in function notation by replacing an with f (n). f (n) = a1r n − 1 The domain of the function is the set of positive integers.

Modeling with Mathematics Clicking the zoom-out button on a mapping website doubles the side length of the square map. After how many clicks on the zoom-out button is the side length of the map 640 miles?

Zoom-out clicks

1

2

3

Map side length (miles)

5

10

20

SOLUTION 1. Understand the Problem You know that the side length of the square map doubles after each click on the zoom-out button. So, the side lengths of the map represent the terms of a geometric sequence. You need to find the number of clicks it takes for the side length of the map to be 640 miles. 2. Make a Plan Begin by writing a function f for the nth term of the geometric sequence. Then find the value of n for which f (n) = 640. 3. Solve the Problem The first term is 5, and the common ratio is 2. f (n) = a1r n − 1

Function for a geometric sequence

f (n) = 5(2)n − 1

Substitute 5 for a1 and 2 for r.

The function f (n) = 5(2)n − 1 represents the geometric sequence. Use this function to find the value of n for which f (n) = 640. So, use the equation 640 = 5(2)n − 1 to write a system of equations. 1000

y=

USING APPROPRIATE TOOLS STRATEGICALLY You can also use the table feature of a graphing calculator to find the value of n for which f (n) = 640. X 3 4 5 6 7 9

X=8

Y1 20 40 80 160 320 640 1280

Y2 640 640 640 640 640 640 640

5(2)n − 1

y = 640

Equation 1

y = 640

Equation 2

Then use a graphing calculator to graph the equations and find the point of intersection. The point of intersection is (8, 640).

y = 5(2) n − 1 Intersection Y=640 0 X=8 0

12

So, after eight clicks, the side length of the map is 640 miles. 4. Look Back Find the value of n for which f (n) = 640 algebraically. 640 = 5(2)n − 1

Write the equation.

128 = (2)n − 1

Divide each side by 5.

27 = (2)n − 1

Rewrite 128 as 27.

7=n−1

Equate the exponents.

8=n

Add 1 to each side.



Monitoring Progress

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12. WHAT IF? After how many clicks on the zoom-out button is the side length of

the map 2560 miles?

Section 6.5

Geometric Sequences

309

Exercises

6.5

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Compare the two sequences.

2, 4, 6, 8, 10, . . .

2, 4, 8, 16, 32, . . .

2. CRITICAL THINKING Why do the points of a geometric sequence lie on an exponential curve only

when the common ratio is positive?

Monitoring Progress and Modeling with Mathematics In Exercises 3– 8, find the common ratio of the geometric sequence.

In Exercises 19–24, write the next three terms of the geometric sequence. Then graph the sequence. (See Examples 2 and 3.)

1

3. 4, 12, 36, 108, . . .

4. 36, 6, 1, —6 , . . .

5. —38 , −3, 24, −192, . . .

6. 0.1, 1, 10, 100, . . .

7. 128, 96, 72, 54, . . .

8. −162, 54, −18, 6, . . .

3 —8 ,

13. 192, 24, 3,

22. −375, −75, −15, −3, . . .

1

an 240

1

14. −25, −18, −11, −4, . . .

29.

16.

(4, 250)

80

17.

(1, 2) 0

an 120

2

310

18.

2

Chapter 6

4

n

28. 0.1, 0.9, 8.1, 72.9, . . .

1

2

3

4

an

7640

764

76.4

7.64

n

1

2

3

4

an

−192

48

−12

3

an 0

0

an 24

2

4

n −50

0

−100

(3, 15)

2

(1, −3)

(1, 0.5) (3, 18) 1

3

(2, −3)

5 n

32.

an 240

(1, 224)

160

(2, 112)

80

(4, 24)

(2, 6)

(3, 20) (4, 5) 0

31.

(1, 4)

12

(2, 60)

40

0

n

(1, 120)

80

0

4

1

(2, 11)

6

(3, 50) (2, 10)

1

n

30.

(4, 19) (3, 16)

12

160

0

an 18

26. 0.6, −3, 15, −75, . . .

27. −—8 , −—4 , −—2 , −1, . . .

In Exercises 15–18, determine whether the graph represents an arithmetic sequence, a geometric sequence, or neither. Explain your reasoning. 15.

8

16 24. — , —, 4, 6, . . . 9 3

25. 2, 8, 32, 128, . . .

3

3 12. — , —, 3, 21, . . . 49 7

...

21. 81, −27, 9, −3, . . .

In Exercises 25–32, write an equation for the nth term of the geometric sequence. Then find a6. (See Example 4.)

10. −1, 4, −7, 10, . . .

11. 9, 14, 20, 27, . . .

20. −3, 12, −48, 192, . . .

23. 32, 8, 2, —2 , . . .

In Exercises 9–14, determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. (See Example 1.) 9. −8, 0, 8, 16, . . .

19. 5, 20, 80, 320, . . .

4

(4, −108)

0

(3, 56) 0

2

(4, 28) 4

n

33. PROBLEM SOLVING A badminton tournament begins n

−12

Exponential Functions and Sequences

with 128 teams. After the first round, 64 teams remain. After the second round, 32 teams remain. How many teams remain after the third, fourth, and fifth rounds?

34. PROBLEM SOLVING The graphing calculator screen

38. MODELING WITH MATHEMATICS You start a chain

displays an area of 96 square units. After you zoom out once, the area is 384 square units. After you zoom out a second time, the area is 1536 square units. What is the screen area after you zoom out four times?

email and send it to six friends. The next day, each of your friends forwards the email to six people. The process continues for a few days. a. Write a function that represents the number of people who have received the email after n days.

6

−6

b. After how many days will 1296 people have received the email?

6 −2

35. ERROR ANALYSIS Describe and correct the

error in writing the next three terms of the geometric sequence.



−8,

4,

−2,

×(−2) ×(−2)

MATHEMATICAL CONNECTIONS In Exercises 39 and

1, . . .

40, (a) write a function that represents the sequence of figures and (b) describe the 10th figure in the sequence.

×(−2)

39.

The next three terms are −2, 4, and −8. 36. ERROR ANALYSIS Describe and correct the error

in writing an equation for the nth term of the geometric sequence.



40.

−2, −12, −72, −432, . . . The first term is −2, and the common ratio is −6. an = a1r n − 1 an = −2(−6)n − 1

41. REASONING Write a sequence that represents the

37. MODELING WITH MATHEMATICS The distance

(in millimeters) traveled by a swinging pendulum decreases after each swing, as shown in the table. (See Example 5.) Swing Distance (in millimeters)

1

2

3

625

500

400

number of teams that have been eliminated after n rounds of the badminton tournament in Exercise 33. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. 42. REASONING Write a sequence that represents

the perimeter of the graphing calculator screen in Exercise 34 after you zoom out n times. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. 43. WRITING Compare the graphs of arithmetic

sequences to the graphs of geometric sequences. 44. MAKING AN ARGUMENT You are given two

consecutive terms of a sequence. distance

a. Write a function that represents the distance the pendulum swings on its nth swing. b. After how many swings is the distance 256 millimeters?

. . . , −8, 0, . . . Your friend says that the sequence is not geometric. A classmate says that is impossible to know given only two terms. Who is correct? Explain.

Section 6.5

Geometric Sequences

311

45. CRITICAL THINKING Is the sequence shown an

51. REPEATED REASONING A soup kitchen makes

arithmetic sequence? a geometric sequence? Explain your reasoning.

16 gallons of soup. Each day, a quarter of the soup is served and the rest is saved for the next day. a. Write the first five terms of the sequence of the number of fluid ounces of soup left each day.

3, 3, 3, 3, . . . 46. HOW DO YOU SEE IT? Without performing any

calculations, match each equation with its graph. Explain your reasoning.

() = 20( — )

a. an = 20 —43 b. an A.

n−1

c. When is all the soup gone? Explain.

3 n−1 4

an

B.

an

16

200

8

100

0

b. Write an equation that represents the nth term of the sequence.

0

4

8

0

n

52. THOUGHT PROVOKING Find the sum of the terms of

the geometric sequence. 1 1 1 1 1, —, —, —, . . . , — ,... 2 4 8 2n − 1 0

4

8

Explain your reasoning. Write a different infinite geometric sequence that has the same sum.

n

47. REASONING What is the 9th term of the geometric

53. OPEN-ENDED Write a geometric sequence in which

sequence where a3 = 81 and r = 3?

a2  <  a1  <  a3.

48. OPEN-ENDED Write a sequence that has a pattern but

54. NUMBER SENSE Write an equation that represents the

is not arithmetic or geometric. Describe the pattern.

nth term of each geometric sequence shown.

49. ATTENDING TO PRECISION Are the terms of a

geometric sequence independent or dependent? Explain your reasoning.

n

1

2

3

4

an

2

6

18

54

50. DRAWING CONCLUSIONS A college student makes a

n

1

2

3

4

bn

1

5

25

125

deal with her parents to live at home instead of living on campus. She will pay her parents $0.01 for the first day of the month, $0.02 for the second day, $0.04 for the third day, and so on.

a. Do the terms a1 − b1, a2 − b2, a3 − b3, . . . form a geometric sequence? If so, how does the common ratio relate to the common ratios of the sequences above?

a. Write an equation that represents the nth term of the geometric sequence. b. What will she pay on the 25th day?

a a a b. Do the terms —1 , —2 , —3 , . . . form a geometric b1 b2 b3 sequence? If so, how does the common ratio relate to the common ratios of the sequences above?

c. Did the student make a good choice or should she have chosen to live on campus? Explain.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Use residuals to determine whether the model is a good fit for the data in the table. Explain. 55. y = 3x − 8

312

(Section 4.5)

56. y = −5x + 1

x

0

1

2

3

4

5

6

x

−3

−2

−1

0

1

2

3

y

−10

−2

−1

2

1

7

10

y

6

4

6

1

2

−4

−3

Chapter 6

Exponential Functions and Sequences

6.6

Recursively Defined Sequences Essential Question

How can you define a sequence recursively?

A recursive rule gives the beginning term(s) of a sequence and a recursive equation that tells how an is related to one or more preceding terms.

Describing a Pattern Work with a partner. Consider a hypothetical population of rabbits. Start with one breeding pair. After each month, each breeding pair produces another breeding pair. The total number of rabbits each month follows the exponential pattern 2, 4, 8, 16, 32, . . .. Now suppose that in the first month after each pair is born, the pair is too young to reproduce. Each pair produces another pair after it is 2 months old. Find the total number of pairs in months 6, 7, and 8. Number of pairs

Month Red pair is too young to produce.

1

Red pair produces blue pair.

2

3

1

1

Red pair produces green pair.

2 Red pair produces orange pair.

4

Blue pair produces purple pair.

3

5

5

RECOGNIZING PATTERNS To be proficient in math, you need to look closely to discern a pattern or structure.

Using a Recursive Equation Work with a partner. Consider the following recursive equation. an = an − 1 + an − 2 Each term in the sequence is the sum of the two preceding terms. Copy and complete the table. Compare the results with the sequence of the number of pairs in Exploration 1. a1

a2

1

1

a3

a4

a5

a6

a7

a8

Communicate Your Answer 3. How can you define a sequence recursively? 4. Use the Internet or some other reference to determine the mathematician who first

described the sequences in Explorations 1 and 2. Section 6.6

Recursively Defined Sequences

313

6.6 Lesson

What You Will Learn Write terms of recursively defined sequences.

Core Vocabul Vocabulary larry

Write recursive rules for sequences.

explicit rule, p. 314 recursive rule, p. 314

Write recursive rules for special sequences.

Translate between recursive rules and explicit rules.

Writing Terms of Recursively Defined Sequences

Previous arithmetic sequence geometric sequence

So far in this book, you have defined arithmetic and geometric sequences explicitly. An explicit rule gives an as a function of the term’s position number n in the sequence. For example, an explicit rule for the arithmetic sequence 3, 5, 7, 9, . . . is an = 3 + 2(n − 1), or an = 2n + 1. Now, you will define arithmetic and geometric sequences recursively. A recursive rule gives the beginning term(s) of a sequence and a recursive equation that tells how an is related to one or more preceding terms.

Core Concept Recursive Equation for an Arithmetic Sequence an = an − 1 + d, where d is the common difference

Recursive Equation for a Geometric Sequence

⋅

an = r an − 1, where r is the common ratio

Writing Terms of Recursively Defined Sequences Write the first six terms of each sequence. Then graph each sequence. a. a1 = 2, an = an − 1 + 3

b. a1 = 1, an = 3an − 1

SOLUTION You are given the first term. Use the recursive equation to find the next five terms. a. a1 = 2

STUDY TIP A sequence is a discrete function. So, the points on the graph are not connected.

a2 = a1 + 3 = 2 + 3 = 5

a2 = 3a1 = 3(1) = 3

a3 = a2 + 3 = 5 + 3 = 8

a3 = 3a2 = 3(3) = 9

a4 = a3 + 3 = 8 + 3 = 11

a4 = 3a3 = 3(9) = 27

a5 = a4 + 3 = 11 + 3 = 14

a5 = 3a4 = 3(27) = 81

a6 = a5 + 3 = 14 + 3 = 17

a6 = 3a5 = 3(81) = 243

an

an

18

240

12

160

6

80

0

314

Chapter 6

b. a1 = 1

0

2

4

6 n

Exponential Functions and Sequences

0

0

2

4

6 n

Monitoring Progress

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Write the first six terms of the sequence. Then graph the sequence. 1. a1 = 0, an = an − 1 − 8

2. a1 = −7.5, an = an − 1 + 2.5

1

3. a1 = −36, an = —2 an − 1

4. a1 = 0.7, an = 10an − 1

Writing Recursive Rules Writing Recursive Rules Write a recursive rule for each sequence. a. −30, −18, −6, 6, 18, . . .

b. 500, 100, 20, 4, 0.8, . . .

SOLUTION Use a table to organize the terms and find the pattern. a.

1

2

3

4

5

−30

−18

−6

6

18

Position, n Term, an

COMMON ERROR

+ 12

+ 12

+ 12

+ 12

The sequence is arithmetic, with first term a1 = −30 and common difference d = 12.

When writing a recursive rule for a sequence, you need to write both the beginning term(s) and the recursive equation.

an = an − 1 + d

Recursive equation for an arithmetic sequence

an = an − 1 + 12

Substitute 12 for d.

So, a recursive rule for the sequence is a1 = −30, an = an − 1 + 12. b.

1

2

3

4

5

500

100

20

4

0.8

Position, n Term, an

× —15

× —15

× —15

× —15

The sequence is geometric, with first term a1 = 500 and common ratio r = —15.

⋅

an = r an − 1

Recursive equation for a geometric sequence

an = —15 an − 1

Substitute —15 for r.

So, a recursive rule for the sequence is a1 = 500, an = —15 an − 1.

Monitoring Progress

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Write a recursive rule for the sequence. 5. 8, 3, −2, −7, −12, . . .

6. 1.3, 2.6, 3.9, 5.2, 6.5, . . .

7. 4, 20, 100, 500, 2500, . . .

8. 128, −32, 8, −2, 0.5, . . .

9. Write a recursive rule for the height of the sunflower over time.

1 month: 2 feet

2 months: 3.5 feet

Section 6.6

3 months: 5 feet

4 months: 6.5 feet

Recursively Defined Sequences

315

Translating between Recursive and Explicit Rules Translating from Recursive Rules to Explicit Rules Write an explicit rule for each recursive rule. a. a1 = 25, an = an − 1 − 10

b. a1 = 19.6, an = −0.5an − 1

SOLUTION a. The recursive rule represents an arithmetic sequence, with first term a1 = 25 and common difference d = −10. an = a1 + (n − 1)d

Explicit rule for an arithmetic sequence

an = 25 + (n − 1)(−10)

Substitute 25 for a1 and −10 for d.

an = −10n + 35

Simplify.

An explicit rule for the sequence is an = −10n + 35. b. The recursive rule represents a geometric sequence, with first term a1 = 19.6 and common ratio r = −0.5. an = a1r n − 1 an =

Explicit rule for a geometric sequence

19.6(−0.5)n − 1

Substitute 19.6 for a1 and −0.5 for r.

An explicit rule for the sequence is an = 19.6(−0.5)n − 1.

Translating from Explicit Rules to Recursive Rules Write a recursive rule for each explicit rule. a. an = −2n + 3

b. an = −3(2)n − 1

SOLUTION a. The explicit rule represents an arithmetic sequence, with first term a1 = −2(1) + 3 = 1 and common difference d = −2. an = an − 1 + d

Recursive equation for an arithmetic sequence

an = an − 1 + (−2)

Substitute −2 for d.

So, a recursive rule for the sequence is a1 = 1, an = an − 1 − 2. b. The explicit rule represents a geometric sequence, with first term a1 = −3 and common ratio r = 2.

⋅

an = r an − 1

Recursive equation for a geometric sequence

an = 2an − 1

Substitute 2 for r.

So, a recursive rule for the sequence is a1 = −3, an = 2an − 1.

Monitoring Progress

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Write an explicit rule for the recursive rule. 10. a1 = −45, an = an − 1 + 20

11. a1 = 13, an = −3an − 1

Write a recursive rule for the explicit rule. 12. an = −n + 1

316

Chapter 6

Exponential Functions and Sequences

13. an = −2.5(4)n − 1

Writing Recursive Rules for Special Sequences You can write recursive rules for sequences that are neither arithmetic nor geometric. One way is to look for patterns in the sums of consecutive terms.

Writing Recursive Rules for Other Sequences Use the sequence shown. U 1, 1, 2, 3, 5, 8, . . . aa. Write a recursive rule for the sequence. b. Write the next three terms of the sequence. b

SOLUTION S aa. Find the difference and ratio between each pair of consecutive terms. 1

1

2

3

1−1=0 2−1=1 3−2=1

Th sequence in The i Example E l 5 is i called the Fibonacci sequence. This pattern is naturally occurring in many objects, such as flowers.

1

1 1 —=1 1

2 2 —=2 1

There is no common difference, so the sequence is not arithmetic.

3 3 1 — = 1— 2 2

There is no common ratio, so the sequence is not geometric.

Find the sum of each pair of consecutive terms. a1 + a2 = 1 + 1 = 2

2 is the third term.

a2 + a3 = 1 + 2 = 3

3 is the fourth term.

a3 + a4 = 2 + 3 = 5

5 is the fifth term.

a4 + a5 = 3 + 5 = 8

8 is the sixth term.

Beginning with the third term, each term is the sum of the two previous terms. A recursive equation for the sequence is an = an − 2 + an − 1. So, a recursive rule for the sequence is a1 = 1, a2 = 1, an = an − 2 + an − 1. b. Use the recursive equation an = an − 2 + an − 1 to find the next three terms. a7 = a5 + a6

a8 = a6 + a7

a9 = a7 + a8

=5+8

= 8 + 13

= 13 + 21

= 13

= 21

= 34

The next three terms are 13, 21, and 34.

Monitoring Progress

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Write a recursive rule for the sequence. Then write the next three terms of the sequence. 14. 5, 6, 11, 17, 28, . . .

15. −3, −4, −7, −11, −18, . . .

16. 1, 1, 0, −1, −1, 0, 1, 1, . . .

17. 4, 3, 1, 2, −1, 3, −4, . . .

Section 6.6

Recursively Defined Sequences

317

Exercises

6.6

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE A recursive rule gives the beginning term(s) of a sequence and

a(n) _____________ that tells how an is related to one or more preceding terms. 2. WHICH ONE DOESN’T BELONG? Which rule does not belong with the other three? Explain

your reasoning. a1 = −1, an = 5an − 1

an = 6n − 2

a1 = −3, an = an − 1 + 1

a1 = 9, an = 4an − 1

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, determine whether the recursive rule represents an arithmetic sequence or a geometric sequence. 3. a1 = 2, an = 7an − 1

4. a1 = 18, an = an − 1 + 1

17. 0, −3, −6, −9, −12, . . . 18. 5, −20, 80, −320, 1280, . . . 19.

5. a1 = 5, an = an − 1 − 4 6. a1 = 3, an = −6an − 1

an 0

(2, −4) −24

In Exercises 7–12, write the first six terms of the sequence. Then graph the sequence. (See Example 1.)

20.

(1, −1) 3

5 n

(3, −16)

−48

7. a1 = 0, an = an − 1 + 2

an 36

(1, 35) (2, 24)

24

(3, 13)

12

(4, −64)

8. a1 = 10, an = an − 1 − 5

(4, 2) 0

0

2

4

n

21. MODELING WITH MATHEMATICS Write a recursive

9. a1 = 2, an = 3an − 1

rule for the number of bacterial cells over time.

10. a1 = 8, an = 1.5an − 1

1 hour

1

11. a1 = 80, an = −—2 an − 1 2 hours

12. a1 = −7, an = −4an − 1

3 hours

In Exercises 13–20, write a recursive rule for the sequence. (See Example 2.) 13.

14.

n

1

2

3

4

an

7

16

25

34

n

1

2

3

4

an

8

24

72

216

4 hours

22. MODELING WITH MATHEMATICS Write a recursive

rule for the length of the deer antler over time.

15. 243, 81, 27, 9, 3, . . .

1 day: 1 4 2 in.

16. 3, 11, 19, 27, 35, . . .

318

Chapter 6

Exponential Functions and Sequences

2 days: 3 4 4 in.

3 days: 5 in.

4 days: 1 5 4 in.

In Exercises 23–28, write an explicit rule for the recursive rule. (See Example 3.)

45. ERROR ANALYSIS Describe and correct the error in

writing an explicit rule for the recursive rule a1 = 6, an = an − 1 − 12.

23. a1 = −3, an = an − 1 + 3



24. a1 = 8, an = an − 1 − 12 25. a1 = 16, an = 0.5an − 1 26. a1 = −2, an = 9an − 1 27. a1 = 4, an = an − 1 + 17

an = a1 + (n − 1)d an = 6 + (n − 1)(12) an = 6 + 12n − 12 an = −6 + 12n

46. ERROR ANALYSIS Describe and correct the error

in writing a recursive rule for the sequence 2, 4, 6, 10, 16, . . ..

28. a1 = 5, an = −5an − 1

In Exercises 29–34, write a recursive rule for the explicit rule. (See Example 4.) 29. an = 7(3)n − 1

30. an = −4n + 2

31. an = 1.5n + 3

32. an = 6n − 20

33. an = (−5)n − 1

34. an = −81 —



2,

4,

6, . . .

+2 +2 The sequence is arithmetic, with first term a1 = 2 and common difference d = 2. an = an − 1 + d a1 = 2, an = an − 1 + 2

( 23 )

n−1

In Exercises 35–38, graph the first four terms of the sequence with the given description. Write a recursive rule and an explicit rule for the sequence.

In Exercises 47–51, the function f represents a sequence. Find the 2nd, 5th, and 10th terms of the sequence.

35. The first term of a sequence is 5. Each term of the

47. f (1) = 3, f (n) = f (n − 1) + 7

sequence is 15 more than the preceding term. 36. The first term of a sequence is 16. Each term of the

sequence is half the preceding term.

48. f (1) = −1, f (n) = 6f (n − 1) 49. f (1) = 8, f (n) = −f (n − 1)

37. The first term of a sequence is −1. Each term of the

sequence is −3 times the preceding term.

38. The first term of a sequence is 19. Each term of the

sequence is 13 less than the preceding term.

50. f (1) = 4, f (2) = 5, f (n) = f (n − 2) + f (n − 1) 51. f (1) = 10, f (2) = 15, f (n) = f (n − 1) − f (n − 2) 52. MODELING WITH MATHEMATICS The X-ray shows

the lengths (in centimeters) of bones in a human hand.

In Exercises 39– 44, write a recursive rule for the sequence. Then write the next two terms of the sequence. (See Example 5.) 40. 10, 9, 1, 8, −7, 15, . . .

39. 1, 3, 4, 7, 11, . . .

41. 2, 4, 2, −2, −4, −2, . . .

9.5

an 30

44. (5, 27)

10 0

an 60

(1, 64)

a. Write a recursive rule for the lengths of the bones.

40

20

(2, 3) (1, 1) 0

2

(4, 9) (3, 3) 4

n

3.5 2.5

42. 6, 1, 7, 8, 15, 23, . . . 43.

6

(2, 16) (4, 4) (5, 1) (3, 4)

20 0

0

2

4

b. Measure the lengths of different sections of your hand. Can the lengths be represented by a recursively defined sequence? Explain.

n

Section 6.6

Recursively Defined Sequences

319

53. USING TOOLS You can use a spreadsheet to generate

55. REASONING Write the first 5 terms of the sequence

a1 = 5, an = 3an − 1 + 4. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.

the terms of a sequence. = =A1+2

A2

A 1 2 3 4

B

C

3 5

56. THOUGHT PROVOKING Describe the pattern for

the numbers in Pascal’s Triangle, shown below. Write a recursive rule that gives the mth number in the nth row.

1 1

a. To generate the terms of the sequence a1 = 3, an = an − 1 + 2, enter the value of a1, 3, into cell A1. Then enter “=A1+2” into cell A2, as shown. Use the fill down feature to generate the first 10 terms of the sequence.

1 1 1 1

b. Use a spreadsheet to generate the first 10 terms of the sequence a1 = 3, an = 4an − 1. (Hint: Enter “=4*A1” into cell A2.)

1 1

2 3

4

1

3 6

4

1

5 10 10 5

1

57. REASONING The explicit rule an = a1 + (n − 1)d

defines an arithmetic sequence.

c. Use a spreadsheet to generate the first 10 terms of the sequence a1 = 4, a2 = 7, an = an − 1 − an − 2. (Hint: Enter “=A2-A1” into cell A3.)

a. Explain why an − 1 = a1 + [(n − 1) − 1]d. b. Justify each step in showing that a recursive equation for the sequence is an = an − 1 + d. an = a1 + (n − 1)d

54. HOW DO YOU SEE IT? Consider Squares 1–6 in

the diagram.

= a1 + [(n − 1) + 0]d = a1 + [(n − 1) − 1 + 1]d = a1 + [((n − 1) − 1) + 1]d

5

= a1 + [(n − 1) − 1]d + d

6

= an − 1 + d

2 1 3

4

58. MAKING AN ARGUMENT Your friend claims that

the sequence a. Write a sequence in which each term an is the side length of square n.

−5, 5, −5, 5, −5, . . . cannot be represented by a recursive rule. Is your friend correct? Explain.

b. What is the name of this sequence? What is the next term of this sequence?

59. PROBLEM SOLVING Write a recursive rule for

c. Use the term in part (b) to add another square to the diagram and extend the spiral.

the sequence. 3, 7, 15, 31, 63, . . .

Maintaining Mathematical Proficiency Simplify the expression. 60. 5x + 12x

Reviewing what you learned in previous grades and lessons

(Skills Review Handbook) 61. 9 − 6y −14

Write a linear function f with the given values.

62. 2d − 7 − 8d

63. 3 − 3m + 11m

(Section 4.2)

64. f (2) = 6, f (−1) = −3

65. f (−2) = 0, f (6) = −4

66. f (−3) = 5, f (−1) = 5

67. f (3) = −1, f (−4) = −15

320

Chapter 6

Exponential Functions and Sequences

6.4–6.6

What Did You Learn?

Core Vocabulary exponential equation, p. 300 geometric sequence, p. 306

common ratio, p. 306 explicit rule, p. 314

recursive rule, p. 314

Core Concepts Section 6.4 Property of Equality for Exponential Equations, p. 300 Solving Exponential Equations by Graphing, p. 302

Section 6.5 Geometric Sequence, p. 306 Equation for a Geometric Sequence, p. 308

Section 6.6 Recursive Equation for an Arithmetic Sequence, p. 314 Recursive Equation for a Geometric Sequence, p. 314

Mathematical Practices 1.

How did you decide on an appropriate level of precision for your answer in Exercise 50 part (a) on page 304?

2.

Explain how writing a function in Exercise 39 part (a) on page 311 created a shortcut for answering part (b).

3.

How did you choose an appropriate tool in Exercise 52 part (b) on page 319?

Performance Task:

Mathematical Recursion Most people think of the Fibonacci sequence when they think about mathematical recursion. How are recursive sequences used in language, art, music, nature, and games? To explore the answer to this question and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.

321

6

Chapter Review 6.1

Dynamic Solutions available at BigIdeasMath.com

Exponential Functions (pp. 273–280)

Graph f (x) = 9(3)x. Step 1

Make a table of values. x

−2

−1

0

1

2

1

3

9

27

81

f (x)

60 40

Step 2

Plot the ordered pairs.

Step 3

Draw a smooth curve through the points.

20

−4

Graph the function. Describe the domain and range.

()

1 x

2. f (x) = 3x + 2

1. f (x) = −4 —4 4.

Write and graph an exponential function f represented by the table. Then compare the graph to the graph x of g(x) = —12 .

Exponential Growth and Decay

−2

2

4 x

3. f (x) = 2x − 4 − 3

()

6.2

y

80

x

0

1

2

3

y

2

1

0.5

0.25

(pp. 281–288)

Determine whether y = 2(1.13)t represents exponential growth or exponential decay. Identify the percent rate of change. The function is of the form y = a(1 + r)t, where 1 + r > 1, so it represents exponential growth. Use the growth factor 1 + r to find the rate of growth. 1 + r = 1.13

Write an equation.

r = 0.13

Solve for r.

So, the function represents exponential growth and the rate of growth is 13%. Determine whether the table represents an exponential growth function, an exponential decay function, or neither. Explain. 5.

x

0

1

2

3

y

3

6

12

24

6.

x

1

2

3

4

y

162

108

72

48

Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change. 7. y = 0.99t

8. f (t) = 6(0.84)t

9. f (t) = 4(1.05)t

10. You deposit $750 in a savings account that earns 5% annual interest compounded quarterly.

(a) Write a function that represents the balance after t years. (b) What is the balance of the account after 4 years? 11. The value of a TV is $1500. Its value decreases by 14% each year. (a) Write a function that

represents the value y (in dollars) of the TV after t years. (b) What is the value of the TV after 3 years? Round your answer to the nearest dollar. 322

Chapter 6

Exponential Functions and Sequences

6.3

Comparing Linear and Exponential Functions (pp. 289−296)

Tell whether each table of values represents a linear or an exponential function. Then write the function. a.

x

−1

0

1

2

3

y

−6

−1

4

9

14

+1

+1

+1

b.

x

−2

−1

0

1

2

y

5

10

20

40

80

+1

+1

+1

+1

+1

x

−1

0

1

2

3

x

−2

−1

0

1

2

y

−6

−1

4

9

14

y

5

10

20

40

80

+5

+5

+5

×2

+5

The differences of consecutive y-values are constant. The slope is —51 = 5 and the y-intercept is −1. So, the table represents the linear function y = 5x − 1.

×2

×2

×2

Consecutive y-values have a common ratio of 2 and the y-intercept is 20. So, the table represents the exponential function y = 20(2)x.

Tell whether the table represents a linear or an exponential function. Then write the function. 12.

x

−1

0

1

2

3

y

384

96

24

6

1.5

13.

x

−3

−2

−1

0

1

y

64

48

32

16

0

14. The balance y (in dollars) of your savings account after t years is represented by y = 300(1.05)t.

The beginning balance of your friend’s account is $340, and the balance increases by $10 each year. (a) Compare the account balances by calculating and interpreting the average rates of change from t = 2 to t = 6. (b) Predict which account will have a greater balance after 10 years. Explain.

6.4

Solving Exponential Equations (pp. 299–304)

Solve —19 = 3x + 6. 1

—9 = 3x + 6

Write the equation.

3−2 = 3x + 6

Rewrite —19 as 3−2.

−2 = x + 6

Equate the exponents.

x = −8

Solve for x.

Solve the equation. 15. 5x = 53x − 2 18.

(—)

1 2x + 3 3

=x+5

16. 3x − 2 = 1 19.

(—)

1 3x 16

= 642(x + 8)

17. −4 = 64x − 3 20. 272x + 2 = 81x + 4

Chapter 6

Chapter Review

323

6.5

Geometric Sequences (pp. 305–312)

Write the next three terms of the geometric sequence 2, 6, 18, 54, . . .. Use a table to organize the terms and extend the sequence. Position

1

2

3

4

5

6

7

Term

2

6

18

54

162

486

1458

Each term is 3 times the previous term. So, the common ratio is 3.

×3

×3

×3

×3

×3

Multiply a term by 3 to find the next term.

×3

The next three terms are 162, 486, and 1458. Decide whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. If the sequence is geometric, write the next three terms and graph the sequence. 21. 3, 12, 48, 192, . . .

22. 9, −18, 27, −36, . . .

23. 375, −75, 15, −3, . . .

Write an equation for the nth term of the geometric sequence. Then find a9. 24. 1, 4, 16, 64, . . .

6.6

25. 5, −10, 20, −40, . . .

Recursively Defined Sequences

26. 486, 162, 54, 18, . . .

(pp. 313–320)

Write a recursive rule for the sequence 5, 12, 19, 26, 33, . . .. Use a table to organize the terms and find the pattern.

Position, n

1

2

3

4

5

Term, an

5

12

19

26

33

+7

+7

+7

+7

The sequence is arithmetic, with first term a1 = 5 and common difference d = 7. an = an − 1 + d

Recursive equation for an arithmetic sequence

an = an − 1 + 7

Substitute 7 for d.

So, a recursive rule for the sequence is a1 = 5, an = an − 1 + 7. Write the first six terms of the sequence. Then graph the sequence. 27. a1 = 4, an = an − 1 + 5

28. a1 = −4, an = −3an − 1

1

29. a1 = 32, an = —4 an − 1

Write a recursive rule for the sequence. 30. 3, 8, 13, 18, 23, . . .

31. 3, 6, 12, 24, 48, . . .

32. 7, 6, 13, 19, 32, . . .

33. The first term of a sequence is 8. Each term of the sequence is 5 times the preceding term. Graph

the first four terms of the sequence. Write a recursive rule and an explicit rule for the sequence.

324

Chapter 6

Exponential Functions and Sequences

6

Chapter Test

Write and graph a function that represents the situation. 1. Your starting annual salary of $42,500 increases by 3% each year. 2. You deposit $500 in an account that earns 6.5% annual interest compounded yearly.

Tell whether the table of values represents a linear or an exponential function. Then write the function. 3.

x

0

1

2

3

4

y

−8

−4

0

4

8

4.

x

−2

−1

0

1

2

y

0.2

0.6

1.8

5.4

16.2

n

1

2

3

4

an

400

100

25

6.25

Write an explicit rule and a recursive rule for the sequence. 5.

n

1

2

3

4

an

−6

8

22

36

6.

Solve the equation. Check your solution. 1 128

7. 2x = —

8. 256 x + 2 = 163x − 1

9. Graph f (x) = 2(6)x. Compare the graph to the graph of g(x) = 6x. Describe the domain

and range of f. Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change. 10. y = 1.7(1.09)t

()

1 t

11. r(t) = 4 —3

12. The first two terms of a sequence are a1 = 3 and a2 = −12. Let a3 be the third term when

the sequence is arithmetic and let b3 be the third term when the sequence is geometric. Find a3 − b3.

13. At sea level, Earth’s atmosphere exerts a pressure of 1 atmosphere. Atmospheric pressure

P (in atmospheres) decreases with altitude. It can be modeled by P = (0.99988)a, where a is the altitude (in meters). a. Identify the initial amount, decay factor, and decay rate. b. Use a graphing calculator to graph the function. Use the graph to estimate the atmospheric pressure at an altitude of 5000 feet.

14. You follow the training schedule from your coach.

a. Write an explicit rule and a recursive rule for the geometric sequence. b. What is the first day that you run more than 2 kilometers? more than 3 kilometers? c. What is the total distance that you run in the first week? Round your answer to the nearest kilometer.

Training On Your Own Day 1: Run 1 km. Each day after Day 1: Run 20% farther

than the previous day.

Chapter 6

Chapter Test

325

6

Cumulative Assessment

1. Fill in values to write a linear function g so that f (x) = g (x) has two solutions,

x = 2 and x = 4.

f (x) = —14(2)x

g (x) =

x−

2. The graph of the exponential function f is shown. Find f (−7). y 6 4 2 −4

−2

2

4x

3. Student A claims he can form a linear system from the equations shown that has

infinitely many solutions. Student B claims she can form a linear system from the equations shown that has one solution. Student C claims he can form a linear system from the equations shown that has no solution. 3x + y = 12

3x + 2y = 12

6x + 2y = 6

3y + 9x = 36

2y − 6x = 12

9x − 3y = −18

a. Select two equations to support Student A’s claim. b. Select two equations to support Student B’s claim. c. Select two equations to support Student C’s claim. 4. Fill in the inequality with < , ≤ , > , or ≥ so that the system of linear inequalities has

no solution. Inequality 1 y − 2x ≤ 4 Inequality 2 6x − 3y

− 12

5. The second term of a sequence is 7. Each term of the sequence is 10 more than the preceding

term. Fill in values to write a recursive rule and an explicit rule for the sequence.

326

a1 =

, an = an − 1 +

an =

n−

Chapter 6

Exponential Functions and Sequences

6. A data set consists of the heights y (in feet) of a hot-air balloon t minutes after it

begins its descent. An equation of the line of best fit is y = 870 − 14.8t. Which of the following is a correct interpretation of the line of best fit?

A The initial height of the hot-air balloon is 870 feet. The slope has no ○ meaning in this context.

B The initial height of the hot-air balloon is 870 feet, and it descends ○ 14.8 feet per minute.

C The initial height of the hot-air balloon is 870 feet, and it ascends ○ 14.8 feet per minute.

D The hot-air balloon descends 14.8 feet per minute. The y-intercept has no ○ meaning in this context.

7. Select all the functions whose x-value is an integer when f (x) = 10.

f (x) = 3x − 2

f (x) = −2x + 4

f (x) = —32 x + 4

f (x) = −3x + 5

f (x) = —12 x − 6

f (x) = 4x + 14

8. Place each function into one of the three categories. For exponential functions, state

whether the function represents exponential growth, exponential decay, or neither.

Exponential

Linear

Neither

f (x) = −2(8)x

f (x) = 15 − x

f (x) = —12 (3)x

f (x) = 6x2 + 9

f (x) = 4(1.6)2x

f (x) = x(18 − x)

f (x) = −3(4x + 1 − x)

f (x) = 2

()

f (x) = 3

1 x —6

9. How does the graph shown compare to the graph of f (x) = 2x? y 6 4 2 −4

−2

2

4x

Chapter 6

Cumulative Assessment

327

7

Data Analysis and Displays

7.1 7.2 7.3 7.4 7.5

Measures of Center and Variation Box-and-Whisker Plots Shapes of Distributions Two-Way Tables Choosing a Data Display

SEE the Big Idea

Watching Sports on TV (p (p. 362)

Sh ( 349) Shoes (p.

Backpacking B k ki (p. ( 340)

Bowling B li Scores S (p. ( 337)

Altitudes of Airplanes (p. (p 335)

Maintaining Mathematical Proficiency Displaying Data Example 1 The frequency table shows the numbers of books that 12 people read last month. Display the data in a histogram. Number of books

Frequency

0–1

6

2–3

4

4–5

0

6–7

2

Books Read Last Month 7

Frequency

6 5 4 3 2 1 0

0–1

2–3

4–5

6–7

Number of books

Example 2 The table shows the results of a survey. Display the data in a circle graph. Class trip location Students

Water park

Museum

Zoo

Other

25

11

5

4 Class Trip Locations

A total of 45 students took the survey. Water park: 25

— 45

⋅

Museum: 11

360° = 200°

— 45

Zoo: 5

— 45

⋅

Museum Zoo

360° = 88°

Other

Other:

⋅ 360° = 40°

4

— 45

Water park

⋅ 360° = 32°

The table shows the results of a survey. Display the data in a histogram. 1.

After-school activities

Frequency

0–1 2–3 4–5 6–7

11 8 6 1

2.

Pets

Frequency

0–1

10

2–3

18

4–5

2

The table shows the results of a survey. Display the data in a circle graph. 3.

Favorite subject Students

Math

Science

English

History

8

5

7

4

4. ABSTRACT REASONING Twenty people respond “yes” or “no” to a survey question. Let a and

b represent the frequencies of the responses. What must be true about the sum of a and b? What must be true about the sum when “maybe” is an option for the response? Dynamic Solutions available at BigIdeasMath.com

329

Mathematical Practices

Mathematically proficient students use diagrams and graphs to show relationships between data. They also analyze data to draw conclusions.

Using Data Displays

Core Concept Displaying Data Graphically When solving a problem involving data, it is helpful to display the data graphically. This can be done in a variety of ways. A

Data Display

What does it do?

B

Pictograph

shows data using pictures

D

Bar Graph

shows data in specific categories

Circle Graph

shows data as parts of a whole

Line Graph

shows how data change over time

Histogram

shows frequencies of data values in intervals of the same size

Stem-and-Leaf Plot

orders numerical data and shows how they are distributed

Box-and-Whisker Plot

shows the variability of a data set using quartiles

Dot Plot

shows the number of times each value occurs in a data set

Scatter Plot

shows the relationship between two data sets using ordered pairs in a coordinate plane

Monitoring Progress 1. The table shows the estimated

populations of males and females by age in the United States in 2012. Use a spreadsheet, graphing calculator, or some other form of technology to make two different displays for the data. 2. Explain why you chose each type of

data display in Monitoring Progress Question 1. What conclusions can you draw from your data displays?

330

Chapter 7

Data Analysis and Displays

C

1 2 3 4

0 2 3 6 1 1 5 9 0 6

U.S. Population by Age and Gender Ages (years)

Males

Females

0–14

31,242,542

29,901,556

15–29

33,357,203

31,985,028

30–44

30,667,513

30,759,902

45–59

31,875,279

33,165,976

60 –74

19,737,347

22,061,730

75–89

6,999,292

10,028,195

7.1

Measures of Center and Variation Essential Question

How can you describe the variation of

a data set?

Describing the Variation of Data Work with a partner. The graphs show the weights of the players on a professional football team and a professional baseball team. Weights of Players on a Football Team Tackles = Guards/Centers = DE/TE = Linebackers =

155

175

195

215

235

Quarterbacks = Running backs = Wide receivers = Other players =

255

275

295

315

335

Weight (pounds)

Weights of Players on a Baseball Team Pitchers = Catchers = Infielders = Outfielders = Designated hitters =

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to reason inductively about data, making plausible arguments that take into account the context from which the data arose.

155

175

195

215

235

255

275

295

315

335

Weight (pounds)

a. Describe the data in each graph in terms of how much the weights vary from the mean. Explain your reasoning. b. Compare how much the weights of the players on the football team vary from the mean to how much the weights of the players on the baseball team vary from the mean. c. Does there appear to be a correlation between the body weights and the positions of players in professional football? in professional baseball? Explain.

Describing the Variation of Data Work with a partner. The weights (in pounds) of the players on a professional basketball team by position are as follows. Power forwards: 235, 255, 295, 245; small forwards: 235, 235; centers: 255, 245, 325; point guards: 205, 185, 205; shooting guards: 205, 215, 185 Make a graph that represents the weights and positions of the players. Does there appear to be a correlation between the body weights and the positions of players in professional basketball? Explain your reasoning.

Communicate Your Answer 3. How can you describe the variation of a data set?

Section 7.1

Measures of Center and Variation

331

Lesson

7.1

What You Will Learn Compare the mean, median, and mode of a data set. Find the range and standard deviation of a data set.

Core Vocabul Vocabulary larry

Identify the effects of transformations on data.

measure of center, p. 332 mean, p. 332 median, p. 332 mode, p. 332 outlier, p. 333 measure of variation, p. 333 range, p. 333 standard deviation, p. 334 data transformation, p. 335

Comparing the Mean, Median, and Mode A measure of center is a measure that represents the center, or typical value, of a data set. The mean, median, and mode are measures of center.

Core Concept Mean The mean of a numerical data set is the sum of the data divided by the number of data values. The symbol — x represents the mean. It is read as “x-bar.”

CONNECTIONS TO GEOMETRY In a future course, you will study medians of triangles and trapezoids, which are segments that use the midpoint, or “middle point” of a side of a figure.

Median The median of a numerical data set is the middle number when the values are written in numerical order. When a data set has an even number of values, the median is the mean of the two middle values.

Mode The mode of a data set is the value or values that occur most often. There may be one mode, no mode, or more than one mode.

Comparing Measures of Center Students’ Hourly Wages

An amusement park hires students for the summer. The students’ hourly wages are shown in the table.

$16.50

$8.25

a. Find the mean, median, and mode of the hourly wages.

$8.75

$8.45

b. Which measure of center best represents the data? Explain.

$8.65

$8.25

$9.10

$9.25

SOLUTION

16.5 + 8.75 + 8.65 + 9.1 + 8.25 + 8.45 + 8.25 + 9.25 — x = ————— = 9.65 8 Median 8.25, 8.25, 8.45, 8.65, 8.75, 9.10, 9.25, 16.50 Order the data.

a. Mean

}

17.4 2 8.25, 8.25, 8.45, 8.65, 8.75, 9.10, 9.25, 16.50 — = 8.7

STUDY TIP Mode is the only measure of center that can represent a nonnumerical data set.

Mode

Mean of two middle values 8.25 occurs most often.

The mean is $9.65, the median is $8.70, and the mode is $8.25. b. The median best represents the data. The mode is less than most of the data, and the mean is greater than most of the data.

Monitoring Progress

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1. WHAT IF? The park hires another student at an hourly wage of $8.45.

(a) How does this additional value affect the mean, median, and mode? Explain. (b) Which measure of center best represents the data? Explain. 332

Chapter 7

Data Analysis and Displays

An outlier is a data value that is much greater than or much less than the other values in a data set.

Removing an Outlier Consider the data in Example 1. (a) Identify the outlier. How does the outlier affect the mean, median, and mode? (b) Describe one possible explanation for the outlier.

SOLUTION a. The value $16.50 is much greater than the other wages. It is the outlier. Find the mean, median, and mode without the outlier. 8.75 + 8.65 + 9.1 + 8.25 + 8.45 + 8.25 + 9.25 — x = ———— ≈ 8.67 7 Median 8.25, 8.25, 8.45, 8.65, 8.75, 9.10, 9.25 The middle value is 8.65. Mean

STUDY TIP Outliers usually have the greatest effect on the mean.

Mode

8.25, 8.25, 8.45, 8.65, 8.75, 9.10, 9.25

The mode is 8.25.

When you remove the outlier, the mean decreases $9.65 − $8.67 = $0.98, the median decreases $8.70 − $8.65 = $0.05, and the mode is the same. b. The outlier could be a student who is hired to maintain the park’s website, while the other students could be game attendants.

Annual Salaries

$32,000

$42,000

$41,000

$38,000

$38,000

$45,000

$72,000

$35,000

Monitoring Progress

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2. The table shows the annual salaries of the employees of an auto repair service.

(a) Identify the outlier. How does the outlier affect the mean, median, and mode? (b) Describe one possible explanation for the outlier.

Finding the Range and Standard Deviation A measure of variation is a measure that describes the spread, or distribution, of a data set. One measure of variation is the range. The range of a data set is the difference of the greatest value and the least value.

Finding a Range Show A

Show B

Ages

Ages

20

29

25

19

19

22

20

27

25

27

22

25

27

29

27

22

30

20

48

21

21

31

32

24

Two reality cooking shows select 12 contestants each. The ages of the contestants are shown in the tables. Find the range of the ages for each show. Compare your results.

SOLUTION Show A 19, 20, 20, 21, 22, 25, 27, 27, 29, 29, 30, 31

Order the data.

So, the range is 31 − 19, or 12 years. Show B 19, 20, 21, 22, 22, 24, 25, 25, 27, 27, 32, 48 So, the range is 48 − 19, or 29 years.

Order the data.

The range of the ages for Show A is 12 years, and the range of the ages for Show B is 29 years. So, the ages for Show B are more spread out.

Monitoring Progress

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3. After the first week, the 25-year-old is voted off Show A and the 48-year-old is

voted off Show B. How does this affect the range of the ages of the remaining contestants on each show in Example 3? Explain. Section 7.1

Measures of Center and Variation

333

A disadvantage of using the range to describe the spread of a data set is that it uses only two data values. A measure of variation that uses all the values of a data set is the standard deviation.

Core Concept Standard Deviation

REMEMBER An ellipsis “ . . . ” indicates that a pattern continues.

The standard deviation of a numerical data set is a measure of how much a typical value in the data set differs from the mean. The symbol σ represents the standard deviation. It is read as “sigma.” It is given by ———— (x1 − — x )2 + (x2 − — x )2 + . . . + (xn − — x )2 σ = ———— n

√

where n is the number of values in the data set. The deviation of a data value x is the difference of the data value and the mean of the data set, x − — x. — Step 1 Find the mean, x . Step 2 Find the deviation of each data value, x − — x. Step 3 Square each deviation, (x − — x )2.

Step 4 Find the mean of the squared deviations. This is called the variance. Step 5 Take the square root of the variance.

A small standard deviation means that the data are clustered around the mean. A large standard deviation means that the data are more spread out.

Finding a Standard Deviation Find the standard deviation of the ages for Show A in Example 3. Use a table to organize your work. Interpret your result.

SOLUTION x

— x

x−— x

(x − — x )2

20

25

−5

25

29

25

4

16

19

25

−6

36

22

25

−3

9

25

25

0

0

27

25

2

4

27

25

2

4

29

25

4

16

30

25

5

25

20

25

−5

25

21

25

−4

16

31

25

6

36

Step 1 Find the mean, — x. 300 — x = — = 25 12 Step 2 Find the deviation of each data value, x − — x , as shown. Step 3 Square each deviation, (x − — x )2, as shown. Step 4 Find the mean of the squared deviations, or variance. x )2 + (x2 − — x )2 + . . . + (xn − — x )2 25 + 16 + . . . + 36 212 (x1 − — = —— = — ≈ 17.7 ———— n 12 12 Step 5 Use a calculator to take the square root of the variance.

√

————

√

— x )2 + (x2 − — x )2 + . . . + (xn − — x )2 (x1 − — 212 ———— = — ≈ 4.2 n 12

The standard deviation is about 4.2. This means that the typical age of a contestant on Show A differs from the mean by about 4.2 years.

Monitoring Progress

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4. Find the standard deviation of the ages for Show B in Example 3. Interpret

your result. 5. Compare the standard deviations for Show A and Show B. What can you conclude?

334

Chapter 7

Data Analysis and Displays

Effects of Data Transformations A data transformation is a procedure that uses a mathematical operation to change a data set into a different data set.

Core Concept

STUDY TIP

Data Transformations Using Addition

The standard deviation stays the same because the amount by which each data value deviates from the mean stays the same.

When a real number k is added to each value in a numerical data set • the measures of center of the new data set can be found by adding k to the original measures of center. • the measures of variation of the new data set are the same as the original measures of variation.

Data Transformations Using Multiplication When each value in a numerical data set is multiplied by a real number k, where k > 0, the measures of center and variation can be found by multiplying the original measures by k.

Real-Life Application Consider the data in Example 1. (a) Find the mean, median, mode, range, and standard deviation when each hourly wage increases by $0.50. (b) Find the mean, median, mode, range, and standard deviation when each hourly wage increases by 10%.

SOLUTION a. Method 1 Make a new table by adding $0.50 to each hourly wage. Find the mean, median, mode, range, and standard deviation of the new data set.

Students’ Hourly Wages

$17.00

$8.75

$9.25

$8.95

$9.15

$8.75

$9.60

$9.75

Mean: $10.15 Range: $8.25

Median: $9.20 Mode: $8.75 Standard deviation: $2.61

Method 2 Find the mean, median, mode, range, and standard deviation of the original data set. Mean: $9.65 Range: $8.25

Median: $8.70 Mode: $8.25 Standard deviation: $2.61

From Example 1

Add $0.50 to the mean, median, and mode. The range and standard deviation are the same as the original range and standard deviation. Mean: $10.15 Range: $8.25

1

4 5 mi

b. Increasing by 10% means to multiply by 1.1. So, multiply the original mean, median, mode, range, and standard deviation from Method 2 of part (a) by 1.1. Mean: $10.62 Range: $9.08

9

1 10 mi

2

1 5 mi

Median: $9.20 Mode: $8.75 Standard deviation: $2.61

Median: $9.57 Mode: $9.08 Standard deviation: $2.87

1

1 2 mi

1

1 5 mi

9 10

mi

Monitoring Progress

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6. Find the mean, median, mode, range, and standard deviation of the altitudes of the

airplanes when each altitude increases by 1—12 miles. Section 7.1

Measures of Center and Variation

335

7.1

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY In a data set, what does a measure of center represent? What does a measure of

variation describe? 2. WRITING Describe how removing an outlier from a data set affects the mean of the data set. 3. OPEN-ENDED Create a data set that has more than one mode. 4. REASONING What is an advantage of using the range to describe a data set? Why do you think the

standard deviation is considered a more reliable measure of variation than the range?

Monitoring Progress and Modeling with Mathematics In Exercises 5– 8, (a) find the mean, median, and mode of the data set and (b) determine which measure of center best represents the data. Explain. (See Example 1.) 5. 3, 5, 1, 5, 1, 1, 2, 3, 15

In Exercises 11–14, find the value of x. 11. 2, 8, 9, 7, 6, x; The mean is 6. 12. 12.5, −10, −7.5, x; The mean is 11.5.

6. 12, 9, 17, 15, 10

7. 13, 30, 16, 19, 20, 22, 25, 31

13. 9, 10, 12, x, 20, 25; The median is 14.

8. 14, 15, 3, 15, 14, 14, 18, 15, 8, 16

14. 30, 45, x, 100; The median is 51.

9. ANALYZING DATA

15. ANALYZING DATA The table shows the masses of

The table shows the lengths of nine movies. a. Find the mean, median, and mode of the lengths.

Movie Lengths (hours)

1—13

1—23

2

3

2—13

1—23

2

2

1—23

b. Which measure of center best represents the data? Explain. 10. ANALYZING DATA The table shows the daily changes

in the value of a stock over 12 days. Changes in Stock Value (dollars)

Masses (kilograms)

455

262

471

358

364

553

62

351

a. Identify the outlier. How does the outlier affect the mean, median, and mode? b. Describe one possible explanation for the outlier. 16. ANALYZING DATA The sizes of emails (in kilobytes)

in your inbox are 2, 3, 5, 2, 1, 46, 3, 7, 2, and 1.

1.05

2.03

−13.78

−2.41

2.64

0.67

4.02

1.39

a. Identify the outlier. How does the outlier affect the mean, median, and mode?

0.66

−0.28

−3.01

2.20

b. Describe one possible explanation for the outlier.

a. Find the mean, median, and mode of the changes in stock value. b. Which measure of center best represents the data? Explain. c. On the 13th day, the value of the stock increases by $4.28. How does this additional value affect the mean, median, and mode? Explain. 336

eight polar bears. (See Example 2.)

Chapter 7

Data Analysis and Displays

17. ANALYZING DATA The

scores of two golfers are shown. Find the range of the scores for each golfer. Compare your results. (See Example 3.)

Golfer A

Golfer B

83

88

89

87

84

95

93

95

91

89

92

94

90

87

88

91

98

95

89

92

18. ANALYZING DATA The graph shows a player’s

27. TRANSFORMING DATA Find the values of the

monthly home run totals in two seasons. Find the range of the number of home runs for each season. Compare your results.

measures shown when each value in the data set increases by 14. (See Example 5.) Mean: 62 Range: 46

16 14

28. TRANSFORMING DATA Find the values of the

Rookie season This season

12

measures shown when each value in the data set is multiplied by 0.5.

10

Mean: 320 Range: 210

8 6

Median: 300 Mode: none Standard deviation: 70.6

4 2

29. ERROR ANALYSIS Describe and correct the error in r

t

finding the median of the data set.

be em

pt

✗

Se

A

ug

us

ly Ju

Ju

ay M

pr A

ne

0 il

Number of home runs

Home Run Statistics

Median: 55 Mode: 49 Standard deviation: 15.5

Month

In Exercises 19–22, find (a) the range and (b) the standard deviation of the data set.

7, 4, 6, 2, 4, 6, 8, 8, 3 The median is 4.

30. ERROR ANALYSIS Describe and correct the error

in finding the range of the data set after the given transformation.

19. 40, 35, 45, 55, 60

✗

20. 141, 116, 117, 135, 126, 121 21. 0.5, 2.0, 2.5, 1.5, 1.0, 1.5

−13, −12, −7, 2, 10, 13 Add 10 to each value. The range is 26 + 10 = 36.

22. 8.2, 10.1, 2.6, 4.8, 2.4, 5.6, 7.0, 3.3 31. PROBLEM SOLVING In a bowling match, the team

23. ANALYZING DATA Consider the data in Exercise 17.

with the greater mean score wins. The scores of the members of two bowling teams are shown.

(See Example 4.) a. Find the standard deviation of the scores of Golfer A. Interpret your result. b. Find the standard deviation of the scores of Golfer B. Interpret your result. c. Compare the standard deviations for Golfer A and Golfer B. What can you conclude? 24. ANALYZING DATA Consider the data in Exercise 18.

a. Find the standard deviation of the monthly home run totals in the player’s rookie season. Interpret your result. b. Find the standard deviation of the monthly home run totals in this season. Interpret your result. c. Compare the standard deviations for the rookie season and this season. What can you conclude? In Exercises 25 and 26, find the mean, median, and mode of the data set after the given transformation. 25. In Exercise 5, each data value increases by 4.

Team A: 172, 130, 173, 212 Team B: 136, 184, 168, 192 a. Which team wins the match? If the team with the greater median score wins, is the result the same? Explain. b. Which team is more consistent? Explain. c. In another match between the two teams, all the members of Team A increase their scores by 15 and all the members of Team B increase their scores by 12.5%. Which team wins this match? Explain. 32. MAKING AN ARGUMENT Your friend says that

when two data sets have the same range, you can assume the data sets have the same standard deviation, because both range and standard deviation are measures of variation. Is your friend correct? Explain.

26. In Exercise 6, each data value increases by 20%.

Section 7.1

Measures of Center and Variation

337

33. ANALYZING DATA The table shows the results of

36. CRITICAL THINKING Can the standard deviation of a

a survey that asked 12 students about their favorite meal. Which measure of center (mean, median, or mode) can be used to describe the data? Explain.

data set be 0? Can it be negative? Explain. 37. USING TOOLS Measure the heights (in inches) of the

students in your class.

Favorite Meal

spaghetti

pizza

steak

hamburger

steak

taco

pizza

chili

pizza

chicken

fish

spaghetti

a. Find the mean, median, mode, range, and standard deviation of the heights. b. A new student who is 7 feet tall joins your class. How would you expect this student’s height to affect the measures in part (a)? Verify your answer.

34. HOW DO YOU SEE IT? The dot plots show the ages

38. THOUGHT PROVOKING To find the p% trimmed

of the members of three different adventure clubs. Without performing calculations, which data set has the greatest standard deviation? Which has the least standard deviation? Explain your reasoning.

mean of a data set, first order the data and remove the lowest p% of the data values and the highest p% of the data values. Then find the mean of the remaining data values. Find the 10% trimmed mean for the data set in Exercise 8. Compare the arithmetic mean and the trimmed mean. Why do you think it is useful to calculate the trimmed mean of a data set?

A ○

Age 12

13

14

15

16

17

39. PROBLEM SOLVING The circle graph shows the

18

distribution of the ages of 200 students in a college Psychology I class.

B ○

College Student Ages Age 12

13

14

15

16

17

18

20 yr: 14%

19 yr: 30% 21 yr: College Student 20% Ages

C ○

18 yr: 35%

Age 12

13

14

15

16

17

18

a. Find the mean, median, and mode of the students’ ages.

35. REASONING A data set is described by the measures

b. Identify the outliers. How do the outliers affect the mean, median, and mode?

shown. Mean: 27 Range: 41

Median: 32 Mode: 18 Standard deviation: 9

c. Suppose all 200 students take the same Psychology II class exactly 1 year later. Draw a new circle graph that shows the distribution of the ages of this class and find the mean, median, and mode of the students’ ages.

Find the mean, median, mode, range, and standard deviation of the data set when each data value is multiplied by 3 and then increased by 8.

Maintaining Mathematical Proficiency Solve the inequality.

41. −3(3y − 2) < 1 − 9y

Evaluate the function for the given value of x.

338

Chapter 7

Reviewing what you learned in previous grades and lessons

(Section 2.4)

40. 6x + 1 ≤ 4x − 9

44. f (x) = 4x; x = 3

37 yr: 1%

42. 2(5c − 4) ≥ 5(2c + 8)

43. 4(3 − w) > 3(4w − 4)

(Section 6.1)

45. f (x) = 7x; x = −2

Data Analysis and Displays

46. f (x) = 5(2)x; x = 6

47. f (x) = −2(3)x; x = 4

7.2

Box-and-Whisker Plots Essential Question

How can you use a box-and-whisker plot to

describe a data set?

Drawing a Box-and-Whisker Plot Numbers of First Cousins 3 9 23 6 12 13

10 3 19 3 45 24

18 0 13 3 1 16

8 32 8 10 5 14

Work with a partner. The numbers of first cousins of the students in a ninth-grade class are shown. A box-and-whisker plot is one way to represent the data visually. a. Order the data on a strip of grid paper with 24 equally spaced boxes.

Fold the paper in half to find the median.

b. Fold the paper in half again to divide the data into four groups. Because there are 24 numbers in the data set, each group should have 6 numbers. Find the least value, the greatest value, the first quartile, and the third quartile.

least value

first quartile

median

third quartile

greatest value

c. Explain how the box-and-whisker plot shown represents the data set. least value

first quartile

0

MODELING WITH MATHEMATICS To be proficient in math, you need to identify important quantities in a practical situation.

0

4

median

10 5

10

third qquartile

greatest value

17

45

15

20

25

30

35

40

45

Number of first cousins

50

Communicate Your Answer 2. How can you use a box-and-whisker plot to describe a data set? 3. Interpret each box-and-whisker plot.

a. body mass indices (BMI) of students in a ninth-grade class

BMI 17

18

19

20

21

22

23

24

25

26

27

28

b. heights of roller coasters at an amusement park

80

100

120

140

160

180

Section 7.2

200

220

240

260

Height (feet)

Box-and-Whisker Plots

339

7.2

Lesson

What You Will Learn Use box-and-whisker plots to represent data sets. Interpret box-and-whisker plots.

Core Vocabul Vocabulary larry

Use box-and-whisker plots to compare data sets.

box-and-whisker plot, p. 340 quartile, p. 340 five-number summary, p. 340 interquartile range, p. 341

Using Box-and-Whisker Plots to Represent Data Sets

Core Concept Box-and-Whisker Plot A box-and-whisker plot shows the variability of a data set along a number line using the least value, the greatest value, and the quartiles of the data. Quartiles divide the data set into four equal parts. The median (second quartile, Q2) divides the data set into two halves. The median of the lower half is the first quartile, Q1. The median of the upper half is the third quartile, Q3.

STUDY TIP Sometimes, the first quartile is called the lower quartile and the third quartile is called the upper quartile.

median, Q2 box

first quartile, Q1 least value

whisker

third quartile, Q3 whisker

greatest value

The five numbers that make up a box-and-whisker plot are called the five-number summary of the data set.

Making a Box-and-Whisker Plot Make a box-and-whisker plot that represents the ages of the members of a backpacking expedition in the mountains. 24, 30, 30, 22, 25, 22, 18, 25, 28, 30, 25, 27

SOLUTION Step 1 Order the data. Find the median and the quartiles. least value

lower half 18

22

22

24

upper half 25

first quartile, 23

25

25

27

median, 25

28

30

30

30

greatest value

third quartile, 29

Step 2 Draw a number line that includes the least and greatest values. Graph points above the number line for the five-number summary. Step 3 Draw a box using Q1 and Q3. Draw a line through the median. Draw whiskers from the box to the least and greatest values.

Age 18

19

20

21

22

Monitoring Progress

23

24

25

26

27

28

29

30

Help in English and Spanish at BigIdeasMath.com

1. A basketball player scores 14, 16, 20, 5, 22, 30, 16, and 28 points during a

tournament. Make a box-and-whisker plot that represents the data. 340

Chapter 7

Data Analysis and Displays

Interpreting Box-and-Whisker Plots The figure shows how data are distributed in a box-and-whisker plot.

STUDY TIP

1 4

A long whisker or box indicates that the data are more spread out.

1 2

of the data are

in each whisker.

1 4

of the data

are in the box.

first quartile, Q1

of the data are

in each whisker.

median, third Q2 quartile, Q3

Another measure of variation for a data set is the interquartile range (IQR), which is the difference of the third quartile, Q3, and the first quartile, Q1. It represents the range of the middle half of the data.

Interpreting a Box-and-Whisker Plot T box-and-whisker plot represents the lengths (in seconds) of the songs played by a The rrock band at a concert.

140

160

180

200

220

240

260

280

300

320

Song length (seconds)

aa. Find and interpret the range of the data. b. Describe the distribution of the data. b cc. Find and interpret the interquartile range of the data. d. Are the data more spread out below Q1 or above Q3? Explain. d

SOLUTION S aa. The least value is 160. The greatest value is 300. So, the range is 300 − 160 = 140 seconds. This means that the song lengths vary by no more than 140 seconds. b. Each whisker represents 25% of the data. The box represents 50% of the data. So, • 25% of the song lengths are between 160 and 220 seconds. • 50% of the song lengths are between 220 and 280 seconds. • 25% of the song lengths are between 280 and 300 seconds. c. IQR = Q3 − Q1 = 280 − 220 = 60 So, the interquartile range is 60 seconds. This means that the middle half of the song lengths vary by no more than 60 seconds. d. The left whisker is longer than the right whisker. So, the data below Q1 are more spread out than data above Q3.

Monitoring Progress

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Use the box-and-whisker plot in Example 1. 2. Find and interpret the range and interquartile range of the data. 3. Describe the distribution of the data.

Section 7.2

Box-and-Whisker Plots

341

Using Box-and-Whisker Plots to Compare Data Sets

STUDY TIP If you can draw a line through the median of a box-and-whisker plot, and each side is approximately a mirror image of the other, then the distribution is symmetric.

A box-and-whisker plot shows the shape of a distribution.

Core Concept Shapes of Box-and-Whisker Plots

Skewed left

Symmetric

• The left whisker is longer than the right whisker.

Skewed right

• The whiskers are about the same length. • The median is in the middle of the plot.

• Most of the data are on the right side of the plot.

• The right whisker is longer than the left whisker. • Most of the data are on the left side of the plot.

Comparing Box-and-Whisker Plots The double box-and-whisker plot represents the test scores for your class and your friend’s class. Your class Friend’s class 55

60

65

70

75

80

85

90

95

100

Test score

a. Identify the shape of each distribution. b. Which test scores are more spread out? Explain.

SOLUTION a. For your class, the left whisker is longer than the right whisker, and most of the data are on the right side of the plot. For your friend’s class, the whisker lengths are equal, and the median is in the middle of the plot. So, the distribution for your class is skewed left, and the distribution for your friend’s class is symmetric. b. The range and interquartile range of the test scores in your friend’s class are greater than the range and interquartile range in your class. So, the test scores in your friend’s class are more spread out.

Monitoring Progress

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4. The double box-and-whisker plot represents the surfboard prices at Shop A and

Shop B. Identify the shape of each distribution. Which shop’s prices are more spread out? Explain. Shop A Shop B 0

342

Chapter 7

100

Data Analysis and Displays

200

300

400

500

600

700

800

900

Surfboard price (dollars)

Exercises

7.2

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Describe how to find the first quartile of a data set. 2. DIFFERENT WORDS, SAME QUESTION Consider the box-and-whisker plot shown.

Which is different? Find “both” answers. Find the interquartile range of the data.

1

Find the range of the middle half of the data.

0

11 5

10

15

20

15

20

24 25

Find the difference of the greatest value and the least value of the data set. Find the difference of the third quartile and the first quartile.

Monitoring Progress and Modeling with Mathematics In Exercises 3– 8, use the box-and-whisker plot to find the given measure.

0

2

4

6

8

10

12

14

3. least value

4. greatest value

5. third quartile

6. first quartile

7. median

8. range

16

14. ANALYZING DATA The stem-and-leaf plot represents

the lengths (in inches) of the fish caught on a fishing trip. Make a box-and-whisker plot that represents the data. Stem 0 1 2

Leaf 67889 00223447 12

Key: 1| 0 = 10 inches

In Exercises 9–12, make a box-and-whisker plot that represents the data. (See Example 1.) 9. Hours of television watched: 0, 3, 4, 5, 2, 4, 6, 5 10. Cat lengths (in inches): 16, 18, 20, 25, 17, 22, 23, 21 11. Elevations (in feet): −2, 0, 5, −4, 1, −3, 2, 0, 2, −3, 6 12. MP3 player prices (in dollars): 124, 95, 105, 110, 95,

124, 300, 190, 114

15. ANALYZING DATA The box-and-whisker plot

represents the prices (in dollars) of the entrées at a restaurant. (See Example 2.)

8.75 10.5 8

13.25

14.75 18.25

9 10 11 12 13 14 15 16 17 18 19

Price (dollars)

a. Find and interpret the range of the data. b. Describe the distribution of the data.

13. ANALYZING DATA The dot plot represents the

numbers of hours students spent studying for an exam. Make a box-and-whisker plot that represents the data.

c. Find and interpret the interquartile range of the data. d. Are the data more spread out below Q1 or above Q3? Explain.

Hours 0

1

2

3

4

5

6

7

Section 7.2

Box-and-Whisker Plots

343

16. ANALYZING DATA A baseball player scores 101 runs

20. HOW DO YOU SEE IT? The box-and-whisker plot

in a season. The box-and-whisker plot represents the numbers of runs the player scores against different opposing teams.

represents a data set. Determine whether each statement is always true. Explain your reasoning.

0

Runs 0

2

4

6

8

10 12 14 16 18 20

2

4

6

8

10

12

14

a. The data set contains the value 11.

a. Find and interpret the range and interquartile range of the data. b. Describe the distribution of the data.

b. The data set contains the value 6. c. The distribution is skewed right. d. The mean of the data is 5.

c. Are the data more spread out between Q1 and Q2 or between Q2 and Q3? Explain. 21. ANALYZING DATA The double box-and-whisker plot 17. ANALYZING DATA The double box-and-whisker plot

represents the battery lives (in hours) of two brands of cell phones.

represents the monthly car sales for a year for two sales representatives. (See Example 3.)

Brand A

Sales Rep A

Brand B

Sales Rep B 0

4

8

12

16

20

24

28

Cars sold

2

3

4

5

6

7

Battery life (hours)

a. Identify the shape of each distribution.

a. Identify the shape of each distribution.

b. What is the range of the upper 75% of each brand?

b. Which representative’s sales are more spread out? Explain.

c. Compare the interquartile ranges of the two data sets.

c. Which representative had the single worst sales month during the year? Explain.

d. Which brand do you think has a greater standard deviation? Explain. e. You need a cell phone that has a battery life of more than 3.5 hours most of the time. Which brand should you buy? Explain.

18. ERROR ANALYSIS Describe and correct the error in

describing the box-and-whisker plot.

22. THOUGHT PROVOKING Create a data set that can 1.8

✗

2.0

2.2

2.4

2.6

2.8

3.0

be represented by the box-and-whisker plot shown. Justify your answer.

Length (cm)

The distribution is skewed left. So, most of the data are on the left side of the plot.

0

2

4

6

Maintaining Mathematical Proficiency

14

16

Reviewing what you learned in previous grades and lessons

Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. (Section 6.5) 24. 3, 11, 21, 33, 47, . . .

25. −2, −6, −18, −54, . . .

26. 26, 18, 10, 2, −6, . . .

27. 4, 5, 9, 14, 23, . . .

Data Analysis and Displays

12

18

median, the same interquartile range, and the same range. Is it possible for the box-and-whisker plots of the data sets to be different? Justify your answer.

which number is the range and which number is the interquartile range of a data set. Explain.

Chapter 7

10

23. CRITICAL THINKING Two data sets have the same

19. WRITING Given the numbers 36 and 12, identify

344

8

7.3

Shapes of Distributions Essential Question

How can you use a histogram to characterize the basic shape of a distribution? Analyzing a Famous Symmetric Distribution Work with a partner. A famous data set was collected in Scotland in the mid-1800s. It contains the chest sizes, measured in inches, of 5738 men in the Scottish Militia. Estimate the percent of the chest sizes that lie within (a) 1 standard deviation of the mean, (b) 2 standard deviations of the mean, and (c) 3 standard deviations of the mean. Explain your reasoning.

Scottish Militiamen Mean = 40 in. Standard deviation = 2 in.

1200

Frequency

1000 800 600 400 200 0

33

35

37

39

41

43

45

47

Chest size (inches)

Comparing Two Symmetric Distributions Work with a partner. The graphs show the distributions of the heights of 250 adult American males and 250 adult American females. Adult Male Heights Mean = 70 in. Standard deviation = 3 in.

40 32 24 16 8 0

55

60

65

70

75

80

85

48

Frequency

Frequency

48

Adult Female Heights Mean = 64 in. Standard deviation = 2.5 in.

40 32 24 16 8 0

55

Height (inches)

60

65

70

75

80

85

Height (inches)

a. Which data set has a smaller standard deviation? Explain what this means in the context of the problem. b. Estimate the percent of male heights between 67 inches and 73 inches.

ATTENDING TO PRECISION To be proficient in math, you need to express numerical answers with a level of precision appropriate for the problem’s context.

Communicate Your Answer 3. How can you use a histogram to characterize the basic shape of a distribution? 4. All three distributions in Explorations 1 and 2 are roughly symmetric. The

histograms are called “bell-shaped.” a. What are the characteristics of a symmetric distribution? b. Why is a symmetric distribution called “bell-shaped?” c. Give two other real-life examples of symmetric distributions. Section 7.3

Shapes of Distributions

345

Lesson

7.3

What You Will Learn Describe the shapes of data distributions. Use the shapes of data distributions to choose appropriate measures.

Core Vocabul Vocabulary larry

Compare data distributions.

Previous histogram frequency table

Describing the Shapes of Data Distributions Recall that a histogram is a bar graph that shows the frequency of data values in intervals of the same size. A histogram is another useful data display that shows the shape of a distribution.

Core Concept Symmetric and Skewed Distributions

STUDY TIP If all the bars of a histogram are about the same height, then the distribution is a flat, or uniform, distribution. A uniform distribution is also symmetric.

tail

tail

Symmetric

Skewed left

• The “tail” of the graph extends to the left. • Most of the data are on the right.

• The data on the right of the distribution are approximately a mirror image of the data on the left of the distribution.

Frequency

1–8

5

9–16

9

17–24

16

SOLUTION

25–32

25

Step 1 Draw and label the axes.

33–40

20

41–48

8

Step 2 Draw a bar to represent the frequency of each interval.

49–56

7

• Most of the data are on the left.

The frequency table shows the numbers of raffle tickets sold by students in your grade. Display the data in a histogram. Describe the shape of the distribution. Raffle Tickets

1–10

7

11–20

8

21–30

10

31–40

16

41–50

34

51–60

15

So, the distribution is symmetric.

Monitoring Progress

25 20 15 10 5 0 1– 8 9– 16 17 –2 4 25 –3 2 33 –4 0 41 –4 8 49 –5 6

Frequency

Frequency

30

The data on the right of the distribution are approximately a mirror image of the data on the left of the distribution.

Number of pounds

Chapter 7

• The “tail” of the graph extends to the right.

Describing the Shape of a Distribution

Number of tickets sold

346

Skewed right

Number of tickets sold

Help in English and Spanish at BigIdeasMath.com

1. The frequency table shows the numbers of pounds of aluminum cans collected

by classes for a fundraiser. Display the data in a histogram. Describe the shape of the distribution. Data Analysis and Displays

Choosing Appropriate Measures Use the shape of a distribution to choose the most appropriate measure of center and measure of variation to describe the data set.

Core Concept

STUDY TIP When a distribution is symmetric, the mean and median are about the same. When a distribution is skewed, the mean will be in the direction in which the distribution is skewed while the median will be less affected.

Choosing Appropriate Measures When a data distribution is symmetric, • use the mean to describe the center and • use the standard deviation to describe the variation. mean

When a data distribution is skewed, • use the median to describe the center and • use the five-number summary to describe the variation. mean

median

median

mean

Choosing Appropriate Measures

32

44

39

53

38

48

A police officer measures the speeds (in miles per hour) of 30 motorists. The results are shown in the table at the left. (a) Display the data in a histogram using six intervals beginning with 31–35. (b) Which measures of center and variation best represent the data? (c) The speed limit is 45 miles per hour. How would you interpret these results?

56

41

42

SOLUTION

50

50

55

55

45

49

a. Make a frequency table using the described intervals. Then use the frequency table to make a histogram.

51

53

52

54

60

55

Speed (mi/h)

Frequency

52

50

52

31–35

1

55

40

60

36– 40

3

45

58

47

41–45

5

46–50

6

51–55

11

56–60

4

Speeds of Motorists 12 10 8 6 4 2 0 31 –3 5 36 –4 0 41 –4 5 46 –5 0 51 –5 5 56 –6 0

Frequency

Speeds (mi/h)

Speed (miles per hour)

b. Because most of the data are on the right and the tail of the graph extends to the left, the distribution is skewed left. So, use the median to describe the center and the five-number summary to describe the variation. Email Attachments Sent

74

105

98

68

64

85

75

60

48

51

65

55

58

45

38

64

52

65

30

70

72

5

45

77

83

42

25

95

16

120

c. Using the frequency table and the histogram, you can see that most of the speeds are more than 45 miles per hour. So, most of the motorists were speeding.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

2. You record the numbers of email attachments sent by 30 employees of a company

in 1 week. Your results are shown in the table. (a) Display the data in a histogram using six intervals beginning with 1–20. (b) Which measures of center and variation best represent the data? Explain. Section 7.3

Shapes of Distributions

347

Comparing Data Distributions Comparing Data Distributions Emoticons are graphic symbols that represent facial expressions. They are used to convey a person’s mood in a text message. The double histogram shows the distributions of emoticon messages sent by a group of female students and a group of male students during 1 week. Compare the distributions using their shapes and appropriate measures of center and variation.

Text Messaging

SOLUTION Because the data on the right of the distribution for the female students are approximately a mirror image of the data on the left of the distribution, the distribution is symmetric. So, the mean and standard deviation best represent the distribution for female students.

Female students

14 12 10 8 6 4 2 0 14

Male students

Frequency

16

12 10 8 6 4 2 9

9

–5

9

–4

50

40

9

–3 30

–2

9 –1

20

10

0–

9

0

Number of emoticon messages

Because most of the data are on the left of the distribution for the male students and the tail of the graph extends to the right, the distribution is skewed right. So, the median and five-number summary best represent the distribution for male students. The mean of the female data set is probably in the 30–39 interval, while the median of the male data set is in the 10–19 interval. So, a typical female student is much more likely to use emoticons than a typical male student. The data for the female students is more variable than the data for the male students. This means that the use of emoticons tends to differ more from one female student to the next.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

3. Compare the distributions using their shapes and appropriate measures of center

and variation.

Years of experience

348

Chapter 7

Company Employees

Data Analysis and Displays

2 3– 5 6– 8 9– 1 12 1 –1 15 4 –1 18 7 –2 21 0 –2 3

40 32 24 16 8 0 0–

Frequency

25 20 15 10 5 0 0– 2 3– 5 6– 8 9– 11 12 –1 15 4 –1 18 7 –2 0

Frequency

Professional Football Players

Years of experience

Many real-life data sets have distributions that are bell-shaped and approximately symmetric about the mean. In a future course, you will study this type of distribution in detail. For now, the following rules can help you see how valuable the standard deviation can be as a measure of variation. • About 68% of the data lie within 1 standard deviation of the mean. −2σ −σ

x

68% 95%

+σ +2σ

• About 95% of the data lie within 2 standard deviations of the mean. • Data values that are more than 2 standard deviations from the mean are considered unusual. Because the data are symmetric, you can deduce that 34% of the data lie within 1 standard deviation to the left of the mean, and 34% of the data lie within 1 standard deviation to the right of the mean.

Comparing Data Distributions T table shows the results of a survey that asked men and women how many pairs The oof shoes they own. aa. Make a double box-and-whisker plot that represents the data. Describe the shape of each distribution. b. Compare the number of pairs b of shoes owned by men to the number of pairs of shoes owned by women. cc. About how many of the women surveyed would you expect to own between 10 and 18 pairs of shoes?

Men

Women

Survey size

35

40

Minimum

2

5

Maximum

17

24

1st Quartile

5

12

Median

7

14

3rd Quartile

10

17

Mean

8

14

Standard deviation

3

4

SOLUTION S aa.

Women Men 0

2

4

6

8

10

12

14

16

18

20

22

24

Pairs of shoes

The distribution for men is skewed right, and the distribution for women is symmetric. b. The centers and spreads of the two data sets are quite different from each other. The mean for women is twice the median for men, and there is more variability in the number of pairs of shoes owned by women. c. Assuming the symmetric distribution is bell-shaped, you know about 68% of the data lie within 1 standard deviation of the mean. Because the mean is 14 and the standard deviation is 4, the interval from 10 to 18 represents about 68% of the data. So, you would expect about 0.68 40 ≈ 27 of the women surveyed to own between 10 and 18 pairs of shoes.

⋅

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

4. Why is the mean greater than the median for the men? 5. If 50 more women are surveyed, about how many more would you expect to own

between 10 and 18 pairs of shoes? Section 7.3

Shapes of Distributions

349

7.3

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY Describe how data are distributed in a symmetric distribution, a distribution that is

skewed left, and a distribution that is skewed right. 2. WRITING How does the shape of a distribution help you decide which measures of center and

variation best describe the data?

Monitoring Progress and Modeling with Mathematics 3. DESCRIBING DISTRIBUTIONS The frequency table

shows the numbers of hours that students volunteer per month. Display the data in a histogram. Describe the shape of the distribution. (See Example 1.)

In Exercises 7 and 8, determine which measures of center and variation best represent the data. Explain your reasoning. 7.

Frequency

1

5

12

20

15

7

Summer Camp Frequency

Number of volunteer 1–2 3–4 5–6 7–8 9–10 11–12 13–14 hours

2

30 24 18 12 6 0

4. DESCRIBING DISTRIBUTIONS The frequency table

8–11

12

12–15

14

16–19

26

20–23

45

24–27

33

Stem 1 2 3 4 5 6

Leaf 1 1 3 4 8 2 3 4 7 8 1 2 4 9 0 3 2 7 9 6

6.

Key: 3 ! 1 = 31

350

Chapter 7

Stem 5 6 7 8 9 10

Leaf 0 0 1 3 6 7 1 4 5 2 4 5 4 6 8 1 3 4

8.

Fundraiser 30 24 18 12 6 0

1–20

21–40 41–60 61–80 81–100

Amount donated (dollars)

9. ANALYZING DATA The table shows the last 24 ATM

withdrawals at a bank. (See Example 2.) a. Display the data in a histogram using seven intervals beginning with 26–50.

In Exercises 5 and 6, describe the shape of the distribution of the data. Explain your reasoning. 5.

9–10 11–12 13–14 15–16 17–18

Age

Frequency

shows the results of a survey that asked people how many hours they spend online per week. Display Hours Frequency the data in a histogram. online Describe the shape of the 0–3 5 distribution. 4–7 7

7–8

9 8 9 5 7 9

Key: 6 ! 3 = 63

Data Analysis and Displays

b. Which measures of center and variation best represent the data? Explain. c. The bank charges a fee for any ATM withdrawal less than $150. How would you interpret the data?

ATM Withdrawals (dollars)

120

100

70

60

40

80

150

80

50

120

60

175

30

50

50

60

200

30

100 70

150 40

110 100

IQ Scores

180

225 190 170 c. The distribution of IQ scores for the human population is symmetric. What happens to the shape of the distribution in part (a) as you include more and more IQ scores from the human population in the data set? ERROR ANALYSIS In Exercises 11 and 12, describe

30

–3

9

and correct the error in the statements about the data displayed in the histogram.

Frequency

Test Scores 20 16 12 8 4 0

12.

✗ ✗

Temperature

16. COMPARING DATA SETS The frequency tables 41–50 51–60 61–70 71–80 81–90 91–100

Percent correct

11.

9

186

–8

170

9

160

–7

230

80

195

9

180

70

170

9

154

18 16 14 12 10 8 6 4 2 0 16 14 12 10 8 6 4 2 0 –6

210

60

180

9

160

Daily High Temperatures

–5

180

50

190

Town A

170

Town B

b. Which measures of center and variation best represent the data? Explain.

shows the distributions of daily high temperatures for two towns over a 50-day period. Compare the distributions using their shapes and appropriate measures of center and variation. (See Example 3.)

Frequency

a. Display the data in a histogram using five intervals beginning with 151–166.

15. COMPARING DATA SETS The double histogram

–4

science. However, IQ scores have been around for years in an attempt to measure human intelligence. The table shows some of the greatest known IQ scores.

40

10. ANALYZING DATA Measuring an IQ is an inexact

Most of the data are on the right. So, the distribution is skewed right.

Because the distribution is skewed, use the standard deviation to describe the variation of the data.

13. USING TOOLS For a large data set, would you use

a stem-and-leaf plot or a histogram to show the distribution of the data? Explain.

show the numbers of entrées in certain price ranges (in dollars) at two different restaurants. Display the data in a double histogram. Compare the distributions using their shapes and appropriate measures of center and variation. Restaurant A

Restaurant B

Price range

Frequency

Price range

Frequency

8–10

5

8–10

0

11–13

9

11–13

2

14–16

12

14–16

5

17–19

4

17–19

7

20–22

3

20–22

8

23–25

0

23–25

6

17. OPEN-ENDED Describe a real-life data set that has a

distribution that is skewed right. 14. REASONING For a symmetric distribution, why is

the mean used to describe the center and the standard deviation used to describe the variation? For a skewed distribution, why is the median used to describe the center and the five-number summary used to describe the variation?

18. OPEN-ENDED Describe a real-life data set that has a

distribution that is skewed left.

Section 7.3

Shapes of Distributions

351

19. COMPARING DATA SETS The table shows the results

22. HOW DO YOU SEE IT? Match the distribution with

of a survey that asked freshmen and sophomores how many songs they have downloaded on their MP3 players. (See Example 4.) Freshmen

Sophomores

Survey size

45

54

Minimum

250

360

Maximum

2150

2400

1st Quartile

800

780

Median

1200

2000

3rd Quartile

1600

2200

Mean

1150

1650

Standard deviation

420

480

the corresponding box-and-whisker plot. a.

b.

c.

A.

a. Make a double box-and-whisker plot that represents the data. Describe the shape of each distribution.

B. C.

b. Compare the number of songs downloaded by freshmen to the number of songs downloaded by sophomores. c. About how many of the freshmen surveyed would you expect to have between 730 and 1570 songs downloaded on their MP3 players?

23. REASONING You record the following waiting times

at a restaurant.

d. If you survey 100 more freshmen, about how many would you expect to have downloaded between 310 and 1990 songs on their MP3 players?

Waiting Times (minutes)

26 40

20. COMPARING DATA SETS You conduct the same

survey as in Exercise 19 but use a different group of freshmen. The results are as follows. Survey size: 60; minimum: 200; maximum: 2400; 1st quartile: 640; median: 1670; 3rd quartile: 2150; mean: 1480; standard deviation: 500

38 35

15 24

8 31

22 42

42 29

25 25

20 0

b. Display the data in a histogram using 10 intervals beginning with 0– 4. What happens when the number of intervals is increased? c. Which histogram best represents the data? Explain your reasoning.

b. Why is the median greater than the mean for this group of freshmen?

24. THOUGHT PROVOKING

The shape of a bimodal distribution is shown. Describe a real-life example of a bimodal distribution.

21. REASONING A data set has a symmetric distribution.

Every value in the data set is doubled. Describe the shape of the new distribution. Are the measures of center and variation affected? Explain.

Maintaining Mathematical Proficiency 25.

352

∣ x − 3 ∣ ≤ 14 Chapter 7

26.

∣ 2x + 4 ∣ > 16

Data Analysis and Displays

18 13

a. Display the data in a histogram using five intervals beginning with 0–9. Describe the shape of the distribution.

a. Compare the number of songs downloaded by this group of freshmen to the number of songs downloaded by sophomores.

Solve the inequality. Graph the solution, if possible.

17 30

Reviewing what you learned in previous grades and lessons

(Section 2.6) 27. 5∣ x + 7 ∣ < 25

28. −2∣ x + 1 ∣ ≥ 18

7.1–7.3

What Did You Learn?

Core Vocabulary measure of center, p. 332 mean, p. 332 median, p. 332 mode, p. 332 outlier, p. 333

measure of variation, p. 333 range, p. 333 standard deviation, p. 334 data transformation, p. 335

box-and-whisker plot, p. 340 quartile, p. 340 five-number summary, p. 340 interquartile range, p. 341

Core Concepts Section 7.1 Measures of Center, p. 332 Measures of Variation, p. 333

Data Transformations Using Addition, p. 335 Data Transformations Using Multiplication, p. 335

Section 7.2 Box-and-Whisker Plot, p. 340

Shapes of Box-and-Whisker Plots, p. 342

Section 7.3 Symmetric and Skewed Distributions, p. 346

Choosing Appropriate Measures, p. 347

Mathematical Practices 1.

Exercises 15 and 16 on page 336 are similar. For each data set, is the outlier much greater than or much less than the rest of the data values? Compare how the outliers affect the means. Explain why this makes sense.

2.

In Exercise 18 on page 351, provide a possible reason for why the distribution is skewed left.

Creating a Positive Study Environment • Set aside an appropriate amount of time for reviewing your notes and the textbook, reworking your notes, and completing homework. • Set up a place for studying at home that is comfortable, but not too comfortable. The place needs to be away from all potential distractions. • Form a study group. Choose students who study well together, help out when someone misses school, and encourage positive attitudes.

353 53 3

7.1–7.3

Quiz

Find the mean, median, and mode of the data set. Which measure of center best represents the data? Explain. (Section 7.1) 1.

2.

Hours Spent on Project

3—12

5

2—12

3

3—12

—2

Waterfall Height (feet)

1

1000

1267

1328

1200

1180

1000

2568

1191

1100

Find the range and standard deviation of each data set. Then compare your results. (Section 7.1) 3. Absent students during a week of school

4. Numbers of points scored

Female: 6, 2, 4, 3, 4 Male: 5, 3, 6, 6, 9

Juniors: 19, 15, 20, 10, 14, 21, 18, 15 Seniors: 22, 19, 29, 32, 15, 26, 30, 19

Make a box-and-whisker plot that represents the data. (Section 7.2) 5. Ages of family members:

6. Minutes of violin practice:

60, 15, 25, 20, 55, 70, 40, 30

20, 50, 60, 40, 40, 30, 60, 40, 50, 20, 20, 35

7. Display the data in a histogram. Describe the shape of the distribution. (Section 7.3) Quiz score

0–2

3–5

6–8

9–11

12–14

Frequency

1

3

6

16

4

8. The table shows the prices of eight mountain bikes in a sporting goods store.

(Section 7.1 and Section 7.2) Price (dollars)

98

119

95

211

130

98

100

125

a. Find the mean, median, mode, range, and standard deviation of the prices. b. Identify the outlier. How does the outlier affect the mean, median, and mode? c. Make a box-and-whisker plot that represents the data. Find and interpret the interquartile range of the data. Identify the shape of the distribution. d. Find the mean, median, mode, range, and standard deviation of the prices when the store offers a 5% discount on all mountain bikes. Time (minutes)

9. The table shows the times of 20 presentations. (Section 7.3)

a. Display the data in a histogram using five intervals beginning with 3–5. b. Which measures of center and variation best represent the data? Explain. c. The presentations are supposed to be 10 minutes long. How would you interpret these results?

354

Chapter 7

Data Analysis and Displays

9

7

10

12

10

11

8

10

10

17

11

5

9

10

4

12

6

14

8

10

7.4

Two-Way Tables Essential Question

How can you read and make a two-way table?

Reading a Two-Way Table Work with a partner. You are the manager of a sports shop. The two-way tables show the numbers of soccer T-shirts in stock at your shop at the beginning and end of the selling season. (a) Complete the totals for the rows and columns in each table. (b) How would you alter the number of T-shirts you order for next season? Explain your reasoning. T-Shirt Size

Color

Beginning of season

S

M

L

XL

XXL

blue/white

5

6

7

6

5

blue/gold

5

6

7

6

5

red/white

5

6

7

6

5

black/white

5

6

7

6

5

black/gold

5

6

7

6

5 145

Total T-Shirt Size

End of season blue/white Color

blue/gold

MODELING WITH MATHEMATICS To be proficient in math, you need to identify important quantities and map their relationships using tools such as graphs and two-way tables.

Total

red/white black/white black/gold

S

M

L

XL

XXL

5 3 4 3 5

4 6 2 4 2

1 5 4 1 3

0 2 1 2 0

2 0 3 1 2

Total

Total

Making a Two-Way Table Work with a partner. The three-dimensional bar graph shows the numbers of hours students work at part-time jobs. a. Make a two-way table showing the data. Use estimation to find the entries in your table. b. Write two observations that summarize the data in your table.

Part-Time Jobs of Students at a High School

160 140 120 100 80 60 40 20 0

Males

Females

0 hours per week 1–7 hours per week 8+ hours per week

Communicate Your Answer 3. How can you read and make a two-way table?

Section 7.4

Two-Way Tables

355

7.4

Lesson

What You Will Learn Find and interpret marginal frequencies.

Core Vocabul Vocabulary larry

Make two-way tables.

two-way table, p. 356 joint frequency, p. 356 marginal frequency, p. 356 joint relative frequency, p. 357 marginal relative frequency, p. 357 conditional relative frequency, p. 358

Use two-way tables to recognize associations in data.

Find relative and conditional relative frequencies.

Finding and Interpreting Marginal Frequencies A two-way table is a frequency table that displays data collected from one source that belong to two different categories. One category of data is represented by rows, and the other is represented by columns. For instance, the two-way table below shows the results of a survey that asked freshmen and sophomores whether they access the Internet using a mobile device, such as a smartphone. The two categories of data are class and mobile access. Class is further divided into freshman and sophomore, and mobile access is further divided into yes and no.

REMEMBER The frequency of an event is the number of times the event occurs.

Mobile Access

Class

categories

Each entry in the table is called a joint frequency. The sums of the rows and columns in a two-way table are called marginal frequencies.

Yes

No

Freshman

55

22

Sophomore

63

12

joint frequency

Finding and Interpreting Marginal Frequencies Find and interpret the marginal frequencies for the two-way table above.

SOLUTION Create a new column and a new row for the marginal frequencies. Then add the entries in each row and column.

The sum of the “total” row should be equal to the sum of the “total” column. Place this sum of the marginal frequencies at the bottom right of your two-way table.

Mobile Access

Class

STUDY TIP

Yes

No

Total

Freshman

55

22

77

77 freshmen responded.

Sophomore

63

12

75

75 sophomores responded.

Total

118

34

152

152 students were surveyed.

118 students access the Internet using a mobile device.

Monitoring Progress

34 students do not access the Internet using a mobile device.

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1. You conduct a technology survey to

356

Chapter 7

Data Analysis and Displays

Tablet Computer

Cell Phone

publish on your school’s website. You survey students in the school cafeteria about the technological devices they own. The results are shown in the two-way table. Find and interpret the marginal frequencies.

Yes

No

Yes

34

124

No

18

67

Making Two-Way Tables Making a Two-Way Table You conduct a survey that asks 286 students in your freshman class whether they play a sport or a musical instrument. One hundred eighteen of the students play a sport, and 64 of those students play an instrument. Ninety-three of the students do not play a sport or an instrument. Organize the results in a two-way table. Include the marginal frequencies.

SOLUTION Step 1 Determine the two categories for the table: sport and instrument.

Yes Sport

Step 2 Use the given joint and marginal frequencies to fill in parts of the table.

Yes

No

64

Total

118 93

No

286

Total

Instrument

Sport

Step 3 Use reasoning to find the missing joint and marginal frequencies. For instance, you can conclude that there are 286 − 118 = 168 students who do not play a sport, and 118 − 64 = 54 students who play a sport but do not play an instrument.

Monitoring Progress

Instrument

Yes

No

Total

Yes

64

54

118

No

75

93

168

Total

139

147

286

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2. You survey students about whether they are getting a summer job. Seventy-five

males respond, with 18 of them responding “no.” Fifty-seven females respond, with 45 of them responding “yes.” Organize the results in a two-way table. Include the marginal frequencies.

Finding Relative and Conditional Relative Frequencies You can display entries of a two-way table as frequency counts (as in Examples 1 and 2) or as relative frequencies.

Core Concept Relative Frequencies A joint relative frequency is the ratio of a frequency that is not in the “total” row or the “total” column to the total number of values or observations. A marginal relative frequency is the sum of the joint relative frequencies in a row or a column. When finding relative frequencies in a two-way table, you can use the corresponding decimals or percents.

Section 7.4

Two-Way Tables

357

Finding Relative Frequencies

Class

Major in Medical Field Yes

No

Junior

124

219

Senior

101

236

The two-way table shows the results of a survey that asked college-bound high school students whether they plan to major in a medical field. Make a two-way table that shows the joint and marginal relative frequencies.

SOLUTION There are 124 + 219 + 101 + 236 = 680 students in the survey. To find the joint relative frequencies, divide each frequency by 680. Then find the sum of each row and each column to find the marginal relative frequencies.

STUDY TIP

Major in Medical Field Yes Junior

Class

The sum of the marginal relative frequencies in the “total” row and the “total” column should each be equal to 1.

Senior

124 — 680 101 — 680

Total

≈ 0.18 ≈ 0.15 0.33

219 — 680 236 — 680

No

Total

≈ 0.32

0.50

≈ 0.35

0.50

0.67

1

About 50% of the students are juniors. About 35% of the students are seniors and are not planning to major in a medical field.

Core Concept Conditional Relative Frequencies A conditional relative frequency is the ratio of a joint relative frequency to the marginal relative frequency. You can find a conditional relative frequency using a row total or a column total of a two-way table.

Finding Conditional Relative Frequencies Use the survey results in Example 3 to make a two-way table that shows the conditional relative frequencies based on the column totals.

SOLUTION Use the marginal relative frequency of each column to calculate the conditional relative frequencies.

When you use column totals, the sum of the conditional relative frequencies for each column should be equal to 1.

Major in Medical Field Yes Class

STUDY TIP

Junior Senior

0.18 0.33 0.15 — ≈ 0.45 0.33 — ≈ 0.55

Monitoring Progress

No

0.32 0.67 0.35 — ≈ 0.52 0.67 — ≈ 0.48

Given that a student is not planning to major in a medical field, the conditional relative frequency that he or she is a junior is about 48%.

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3. Use the survey results in Monitoring Progress Question 2 to make a two-way table

that shows the joint and marginal relative frequencies. What percent of students are not getting a summer job? 4. Use the survey results in Example 3 to make a two-way table that shows the

conditional relative frequencies based on the row totals. Given that a student is a senior, what is the conditional relative frequency that he or she is planning to major in a medical field? 358

Chapter 7

Data Analysis and Displays

Recognizing Associations in Data Recognizing Associations in Data You survey students and find that 40% exercise regularly, 35% eat fruits and vegetables each day, and 52% do not exercise and do not eat fruits and vegetables each day. Is there an association between exercising regularly and eating fruits and vegetables each day?

SOLUTION Use the given information to make a two-way table. Use reasoning to find the missing joint and marginal relative frequencies.

Exercise Regularly Total

Yes

27%

8%

35%

No

13%

52%

65%

Total

40%

60%

100%

Eats Fruit/ Vegetables

No

Exercise Regularly

Vegetables

Yes Eats Fruit/

Use conditional relative frequencies based on the column totals to determine whether there is an association. Of the students who exercise regularly, 67.5% eat fruits and vegetables each day. Of the students who do not exercise regularly, only about 13% eat fruits and vegetables each day. It appears that students who exercise regularly are more likely to eat more fruits and vegetables than students who do not exercise regularly.

Yes

No

0.27 0.4 0.13 No — = 0.325 0.4

0.08 0.6 0.52 — ≈ 0.867 0.6

Yes — = 0.675 — ≈ 0.133

So, there is an association between exercising regularly and eating fruits and vegetables each day. You can also find the conditional relative frequencies by dividing each joint frequency by its corresponding column total or row total.

Recognizing Associations in Data Age

Yes

40

47

42

22

SOLUTION

No

10

25

36

34

Use conditional relative frequencies based on column totals to determine whether there is an association. Based on this sample, 80% of students ages 12–13 share a computer and only about 39% of students ages 18–19 share a computer. The table shows that as age increases, students are less likely to share a computer with other family members. So, there is an association.

Share a Computer

Share a Computer

12–13 14–15 16–17 18–19

The two-way table shows the results of a survey that asked students whether they share a computer at home with other family members. Is there an association between age and sharing a computer?

Monitoring Progress

Age 12–13 Yes No

40 — 50 10 — 50

= 0.8 ≈ 0.2

14–15 47 — 72 25 — 72

≈ 0.65 ≈ 0.35

16–17 42 — 78 36 — 78

≈ 0.54 ≈ 0.46

18–19 22 — 56 34 — 56

≈ 0.39 ≈ 0.61

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5. Using the results of the survey in Monitoring Progress Question 1, is there an

association between owning a tablet computer and owning a cell phone? Explain your reasoning. Section 7.4

Two-Way Tables

359

Exercises

7.4

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE Each entry in a two-way table is called a(n) __________. 2. WRITING When is it appropriate to use a two-way table to organize data? 3. VOCABULARY Explain the relationship between joint relative frequencies, marginal relative

frequencies, and conditional relative frequencies. 4. WRITING Describe two ways you can find conditional relative frequencies.

Monitoring Progress and Modeling with Mathematics You conduct a survey that asks 346 students whether they buy lunch at school. In Exercises 5–8, use the results of the survey shown in the two-way table.

11. USING TWO-WAY TABLES You conduct a survey

that asks students whether they plan to participate in school spirit week. The results are shown in the two-way table. Find and interpret the marginal frequencies.

Yes

No

Freshman

92

86

Sophomore

116

52

Participate in Spirit Week

Class

Class

Buy Lunch at School

5. How many freshmen were surveyed? 6. How many sophomores were surveyed?

Yes

No

Undecided

Freshman

112

56

54

Sophomore

92

68

32

12. USING TWO-WAY TABLES You conduct a survey that

asks college-bound high school seniors about the type of degree they plan to receive. The results are shown in the two-way table. Find and interpret the marginal frequencies.

7. How many students buy lunch at school? 8. How many students do not buy lunch at school?

In Exercises 9 and 10, find and interpret the marginal frequencies. (See Example 1.)

Gender

Set Academic Goals Yes

No

Male

64

168

Female

54

142

Associate’s

Bachelor’s

Master’s

Male

58

126

42

Female

62

118

48

USING STRUCTURE In Exercises 13 and 14, complete the

two-way table.

Dog

Cat

13.

Yes

No

Yes

104

208

No

186

98

Chapter 7

Data Analysis and Displays

Traveled on an Airplane Yes Class

10.

360

Gender

9.

Type of Degree

No

Total

62

Freshman Sophomore

184

Total

274

352

19. USING TWO-WAY TABLES Refer to Exercise 17.

Plan to Attend School Dance Gender

Yes Male

No

Total

24

112

38

Female

What percent of students prefer aerobic exercise? What percent of students are males who prefer anaerobic exercise? 20. USING TWO-WAY TABLES Refer to Exercise 18. What

percent of the sandwiches are on wheat bread? What percent of the sandwiches are turkey on white bread?

196

Total

15. MAKING TWO-WAY TABLES You conduct a survey

that asks 245 students in your school whether they have taken a Spanish or a French class. One hundred nine of the students have taken a Spanish class, and 45 of those students have taken a French class. Eighty-two of the students have not taken a Spanish or a French class. Organize the results in a two-way table. Include the marginal frequencies. (See Example 2.) 16. MAKING TWO-WAY TABLES A car dealership has

98 cars on its lot. Fifty-five of the cars are new. Of the new cars, 36 are domestic cars. There are 15 used foreign cars on the lot. Organize this information in a two-way table. Include the marginal frequencies.

ERROR ANALYSIS In Exercises 21 and 22, describe and

correct the error in using the two-way table. Participate in Fundraiser

Class

14.

Yes

No

Freshman

187

85

Sophomore

123

93

21.

✗

One hundred eighty-seven freshmen responded to the survey.

22.

✗

The two-way table shows the joint relative frequencies. Participate in Fundraiser

Class

Yes

In Exercises 17 and 18, make a two-way table that shows the joint and marginal relative frequencies. (See Example 3.) 17.

Gender

Exercise Preference Aerobic

Anaerobic

Male

88

104

Female

96

62

18.

Freshman

187 — 272

Sophomore

123 — 216

≈ 0.69

≈ 0.31

≈ 0.57

93 — 216

≈ 0.43

23. USING TWO-WAY TABLES A company is hosting an

event for its employees to celebrate the end of the year. It asks the employees whether they prefer a lunch event or a dinner event. It also asks whether they prefer a catered event or a potluck. The results are shown in the two-way table. Make a two-way table that shows the conditional relative frequencies based on the row totals. Given that an employee prefers a lunch event, what is the conditional relative frequency that he or she prefers a catered event? (See Example 4.) Menu

Turkey

Ham

White

452

146

Wheat

328

422

Meal

Meat

Bread

No 85 — 272

Potluck

Catered

Lunch

36

48

Dinner

44

72

Section 7.4

Two-Way Tables

361

24. USING TWO-WAY TABLES The two-way table shows

28. ANALYZING TWO-WAY TABLES Refer to Exercise 12.

the results of a survey that asked students about their preference for a new school mascot. Make a two-way table that shows the conditional relative frequencies based on the column totals. Given that a student prefers a hawk as a mascot, what is the conditional relative frequency that he or she prefers a cartoon mascot?

Is there an association between gender and type of degree? Explain. 29. WRITING Compare Venn diagrams and two-way tables. 30. HOW DO YOU SEE IT? The graph shows the results

of a survey that asked students about their favorite movie genre.

Type Hawk

Favorite Movie Genre

Dragon

Realistic

67

74

51

Cartoon

58

18

24

Number of students

Style

Tiger

25. ANALYZING TWO-WAY TABLES You survey

college-bound seniors and find that 85% plan to live on campus, 35% plan to have a car while at college, and 5% plan to live off campus and not have a car. Is there an association between living on campus and having a car at college? Explain. (See Example 5.)

Male Female

Comedy

Action

a. Display the given information in a two-way table.

and find that 70% watch sports on TV, 48% participate in a sport, and 16% do not watch sports on TV or participate in a sport. Is there an association between participating in a sport and watching sports on TV? Explain.

b. Which of the data displays do you prefer? Explain. 31. PROBLEM SOLVING A box office sells 1809 tickets to

a play, 800 of which are for the main floor. The tickets consist of 2x + y adult tickets on the main floor, x − 40 child tickets on the main floor, x + 2y adult tickets in the balcony, and 3x − y − 80 child tickets in the balcony.

27. ANALYZING TWO-WAY TABLES The two-way table

shows the results of a survey that asked adults whether they participate in recreational skiing. Is there an association between age and recreational skiing? (See Example 6.)

a. Organize this information in a two-way table. b. Find the values of x and y.

Age

c. What percent of tickets are adult tickets?

21–30 31–40 41–50 51–60 61–70 Yes

87

93

68

37

20

No

165

195

148

117

125

d. What percent of child tickets are balcony tickets? 32. THOUGHT PROVOKING Compare “one-way tables”

and “two-way tables.” Is it possible to have a “threeway table?” If so, give an example of a three-way table.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Tell whether the table of values represents a linear or an exponential function. (Section 6.3) 33.

362

x

0

1

2

3

4

y

144

24

4

—3

2

—9

Chapter 7

Horror

Genre

26. ANALYZING TWO-WAY TABLES You survey students

Ski

90 80 70 60 50 40 30 20 10 0

1

Data Analysis and Displays

34.

x

−1

0

1

2

3

y

3

0

−3

−6

−9

7.5

Choosing a Data Display Essential Question

How can you display data in a way that helps

you make decisions?

Displaying Data Work with a partner. Analyze the data and then create a display that best represents the data. Explain your choice of data display. a. A group of schools in New England participated in a 2-month study and reported 3962 animals found dead along roads. birds: 307 reptiles: 75

mammals: 2746 unknown: 689

amphibians: 145

b. The data below show the numbers of black bears killed on a state’s roads from 1993 to 2012. 1993: 30 1994: 37 1995: 46 1996: 33 1997: 43 1998: 35 1999: 43

2000: 47 2001: 49 2002: 61 2003: 74 2004: 88 2005: 82 2006: 109

2007: 99 2008: 129 2009: 111 2010: 127 2011: 141 2012: 135

c. A 1-week study along a 4-mile section of road found the following weights (in pounds) of raccoons that had been killed by vehicles. 13.4 15.2 14.5 20.4

14.8 18.7 9.5 13.6

17.0 18.6 25.4 17.5

12.9 17.2 21.5 18.5

21.3 18.5 17.3 21.5

21.5 9.4 19.1 14.0

16.8 19.4 11.0 13.9

14.8 15.7 12.4 19.0

d. A yearlong study by volunteers in California reported the following numbers of animals killed by motor vehicles.

USING TOOLS STRATEGICALLY To be proficient in math, you need to identify relevant external mathematical resources.

raccoons: 1693 skunks: 1372 ground squirrels: 845 opossum: 763 deer: 761

gray squirrels: 715 cottontail rabbits: 629 barn owls: 486 jackrabbits: 466 gopher snakes: 363

Communicate Your Answer 2. How can you display data in a way that helps you make decisions? 3. Use the Internet or some other reference to find examples of the following types

of data displays. bar graph

circle graph

scatter plot

stem-and-leaf plot

pictograph

line graph

box-and-whisker plot

histogram

dot plot

Section 7.5

Choosing a Data Display

363

7.5

Lesson

What You Will Learn Classify data as quantitative or qualitative. Choose and create appropriate data displays.

Core Vocabul Vocabulary larry

Analyze misleading graphs.

qualitative (categorical) data, p. 364 quantitative data, p. 364 misleading graph, p. 366

Classifying Data Data sets can consist of two types of data: qualitative or quantitative.

Core Concept Types of Data Qualitative data, or categorical data, consist of labels or nonnumerical entries that can be separated into different categories. When using qualitative data, operations such as adding or finding a mean do not make sense. Quantitative data consist of numbers that represent counts or measurements.

STUDY TIP Just because a frequency count can be shown for a data set does not make it quantitative. A frequency count can be shown for both qualitative and quantitative data.

Classifying Data Tell whether the data are qualitative or quantitative. a. prices of used cars at a dealership

b. jersey numbers on a basketball team

c. lengths of songs played at a concert

d. zodiac signs of students in your class

SOLUTION a. Prices are numerical entries. So, the data are quantitative. b. Jersey numbers are numerical, but they are labels. It does not make sense to compare them, and you cannot measure them. So, the data are qualitative. c. Song lengths are numerical measurements. So, the data are quantitative. d. Zodiac signs are nonnumerical entries that can be separated into different categories. So, the data are qualitative.

Monitoring Progress

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Tell whether the data are qualitative or quantitative. Explain your reasoning. 1. telephone numbers in a directory

2. ages of patients at a hospital

3. lengths of videos on a website

4. types of flowers at a florist

Qualitative and quantitative data can be collected from the same data source, as shown below. You can use these types of data together to obtain a more accurate description of a population. Data Source

a student a house

364

Chapter 7

Data Analysis and Displays

Quantitative Data

Qualitative Data

How much do you earn per hour at your job? $10.50 How many square feet of living space is in the house? 2500 ft2

What is your occupation? painter In what city is the house located? Chicago

Choosing and Creating Appropriate Data Displays As shown on page 330, you have learned a variety of ways to display data sets graphically. Choosing an appropriate data display can depend on whether the data are qualitative or quantitative.

Choosing and Creating a Data Display Analyze the data and then create a display that best represents the data. Explain your reasoning. a.

b.

Eye Color Survey Color

Number of students

brown

63

blue

37

hazel

25

green

10

gray

3

amber

2

Speeds of Vehicles (mi/h) Interstate A

65 68 72 68 65 75 68 62 75 77

SOLUTION

Interstate B

67 71 70 65 68 82 59 68 80 75

67 70 65 71 84 77 69 66 73 84

72 78 71 80 81 79 70 69 75 79

a. A circle graph is one appropriate way to display this qualitative data. It shows data as parts of a whole. Step 1 Find the angle measure for each section of the circle graph by multiplying the fraction of students who have each eye color by 360°. Notice that there are 63 + 37 + 25 + 10 + 3 + 2 = 140 students in the survey.

⋅

63 Brown: — 360° ≈ 162° 140

⋅

10 Green: — 360° ≈ 26° 140

37 Blue: — 360° ≈ 95° 140

⋅

25 Hazel: — 360° ≈ 64° 140

⋅

⋅

2 Amber: — 360° ≈ 5° 140

⋅

3 Gray: — 360° ≈ 8° 140

Eye Color Survey

Step 2 Use a protractor to draw the angle measures found in Step 1 on a circle. Then label each section and title the circle graph, as shown.

Gray: 3 Amber: 2

Green: 10

Hazel: 25

b. A double box-and-whisker plot is one appropriate way to display this quantitative data. Use the five-number summary of each data set to create a double box-and-whisker plot.

Brown: 63 Blue: 37

Interstate A 59

66 68

75

82

Interstate B 65 55

60

65

69.5 72.5 70

Section 7.5

79 75

80

84 85

Speed (mi/h)

Choosing a Data Display

365

Monitoring Progress

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5. Display the data in Example 2(a) in another way. 6. Display the data in Example 2(b) in another way.

Analyzing Misleading Graphs Just as there are several ways to display data accurately using graphs, there are several ways to display data that are misleading. A misleading graph is a statistical graph that is not drawn appropriately. This may occur when the creator of a graph wants to give viewers the impression that results are better than they actually are. Below are some questions you can ask yourself when analyzing a statistical graph that will help you recognize when a graph is trying to deceive or mislead. • Does the graph have a title?

• Does the graph need a key?

• Are the numbers of the scale evenly spaced?

• Are all the axes or sections of the graph labeled?

• Does the scale begin at zero? If not, is there a break?

• Are all the components of the graph, such as the bars, the same size?

Analyzing Misleading Graphs Describe how each graph is misleading. Then explain how someone might misinterpret the graph. a.

b. Tuition, Room, and Board at All Colleges and Universities

Mean Hourly Wage for Employees at a Fast-Food Restaurant

18,000 17,500 17,000 16,500 16,000 11 20

10

0–

20 20 1

09

9– 20 0

20 8–

20 0

20

07

–2

00

8

15,500

Academic year

Wage (dollars per hour)

Average cost (dollars)

18,500

9.06 9.05 9.04 9.03 9.02 9.01 9.00 8.98 8.96 8.94 8.92 8.90 0

2008 2009 2010 2011 2012 2013

Year

SOLUTION a. The scale on the vertical axis of the graph starts at $15,500 and does not have a break. This makes it appear that the average cost increased rapidly for the years given. Someone might believe that the average cost more than doubled from 2007 to 2011, when actually, it increased by only about $1500. b. The scale on the vertical axis has very small increments that are not equal. Someone might believe that the greatest increase in the mean hourly wage occurred from 2011 to 2012, when the greatest increase actually occurred from 2009 to 2010.

Monitoring Progress

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7. Redraw the graphs in Example 3 so they are not misleading.

366

Chapter 7

Data Analysis and Displays

Exercises

7.5

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. OPEN-ENDED Describe two ways that a line graph can be misleading. 2. WHICH ONE DOESN’T BELONG? Which data set does not belong with the other three? Explain

your reasoning. ages of people attending a concert

heights of skyscrapers in a city

populations of counties in a state

breeds of dogs at a pet store

Monitoring Progress and Modeling with Mathematics In Exercises 3–8, tell whether the data are qualitative or quantitative. Explain your reasoning. (See Example 1.)

14.

3. brands of cars in a parking lot

Average Precipitation (inches)

January

1.1

July

4.0

February

1.5

August

4.4

4. weights of bears at a zoo

March

2.2

September

4.2

5. budgets of feature films

April

3.7

October

3.5

May

5.1

November

2.1

June

5.5

December

1.8

6. file formats of documents on a computer 7. shoe sizes of students in your class 15.

8. street addresses in a phone book

Grades (out of 100) on a Test

96 84 89 87

In Exercises 9–12, choose an appropriate data display for the situation. Explain your reasoning. 9. the number of students in a marching band each year

74 88 81 95

97 53 52 59

80 77 85 83

62 75 63 100

10. a comparison of students’ grades (out of 100) in two

different classes

16.

11. the favorite sports of students in your class 12. the distribution of teachers by age

In Exercises 13–16, analyze the data and then create a display that best represents the data. Explain your reasoning. (See Example 2.) 13.

Ages of World Cup Winners 2010 Men’s World Cup Winner (Spain)

29 28 22 25

24 30 28 21

23 26 24 24

30 23 21 24

32 26 32 28 27 22 27

2011 Women’s World Cup Winner (Japan)

36 29 22 20

27 26 25 18

24 20 27 23 25 32 27 27 24 23 24 28 24

Colors of Cars that Drive by Your House

white

25

green

3

red

12

silver/gray

27

yellow

1

blue

6

black

21

brown/biege

5

17. DISPLAYING DATA Display the data in Exercise 13 in

another way. 18. DISPLAYING DATA Display the data in Exercise 14 in

another way. 19. DISPLAYING DATA Display the data in Exercise 15 in

another way. Section 7.5

Choosing a Data Display

367

20. DISPLAYING DATA Display the data in Exercise 16 in

another way. In Exercises 21–24, describe how the graph is misleading. Then explain how someone might misinterpret the graph. (See Example 3.) 21.

Annual Sales

voters for a city election. Classmate A says the data should be displayed in a bar graph, while Classmate B says the data would be better displayed in a histogram. Who is correct? Explain. 28. HOW DO YOU SEE IT? The manager of a company

sees the graph shown and concludes that the company is experiencing a decline. What is missing from the graph? Explain why the manager may be mistaken.

104.5

Sales (millions of dollars)

27. MAKING AN ARGUMENT A data set gives the ages of

103.5 102.5 101.5 100.5 99.5

Company Growth

0

2010

2011

2012

2013

0.5

Year

0.4 0.3 0.2

22.

Bicycling

0.1 0

20

Frequency

18

January February March

April

May

16 14 12

29. REASONING A survey asked 100 students about

10

0–29

30–59

the sports they play. The results are shown in the circle graph.

60–119

Minutes

Sports Played

23.

−20

24.

−10

0

40

Golf: 12

Temperature (°F)

80

Decaying Chemical Compound

Football: 18

10,000

Pounds

Basketball: 37

Hockey: 20

1,000

Soccer: 33

100 10 1

0

1

2

3

4

5

6

7

8

a. Explain why the graph is misleading.

9 10

Days

b. What type of data display would be more appropriate for the data? Explain.

25. DISPLAYING DATA Redraw the graph in Exercise 21

30. THOUGHT PROVOKING Use a spreadsheet program

so it is not misleading.

to create a type of data display that is not used in this section.

26. DISPLAYING DATA Redraw the graph in Exercise 22

so it is not misleading.

31. REASONING What type of data display shows the

mode of a data set?

Maintaining Mathematical Proficiency Determine whether the relation is a function. Explain. 32. (−5, −1), (−6, 0), (−5, 1), (−2, 2), (3 , 3)

368

Chapter 7

Data Analysis and Displays

Reviewing what you learned in previous grades and lessons

(Section 3.1)

33. (0, 1), (4, 0), (8, 1), (12, 2), (16, 3)

7.4–7.5

What Did You Learn?

Core Vocabulary two-way table, p. 356 joint frequency, p. 356 marginal frequency, p. 356 joint relative frequency, p. 357 marginal relative frequency, p. 357

conditional relative frequency, p. 358 qualitative (categorical) data, p. 364 quantitative data, p. 364 misleading graph, p. 366

Core Concepts Section 7.4 Joint and Marginal Frequencies, p. 356 Making Two-Way Tables, p. 357 Relative Frequencies, p. 357

Conditional Relative Frequencies, p. 358 Recognizing Associations in Data, p. 359

Section 7.5 Types of Data, p. 364 Choosing and Creating Appropriate Data Displays, p. 365 Analyzing Misleading Graphs, p. 366

Mathematical Practices 1.

Consider the data given in the two-way table for Exercises 5–8 on page 360. Your sophomore friend responded to the survey. Is your friend more likely to have responded “yes” or “no” to buying a lunch? Explain.

2.

Use your answer to Exercise 28 on page 368 to explain why it is important for a company manager to see accurate graphs.

Performance Task:

Retail Therapy Retailers and consumers make decisions based on data, including information from surveys. What items do you enjoy shopping for? Are there differences between what men and women own? How can you use data to answer these questions? To explore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.

369

7

Chapter Review 7.1

Dynamic Solutions available at BigIdeasMath.com

Measures of Center and Variation

(pp. 331–338)

The table shows the number of miles you ran each day for 10 days. Find the mean, median, and mode of the distances.

Miles Run

3.5

4.1

3.5 + 4.0 + 4.4 + 3.9 + 4.3 + 4.1 + 4.3 + 4.5 + 2.0 + 5.0 Mean — x = ————— = 4 10

4.0

4.3

4.4

4.5

Median 2.0, 3.5, 3.9, 4.0, 4.1, 4.3, 4.3, 4.4, 4.5, 5.0

3.9

2.0

4.3

5.0

Order the data.

8.4 2

— = 4.2

Mean of two middle values

Mode 2.0, 3.5, 3.9, 4.0, 4.1, 4.3, 4.3, 4.4, 4.5, 5.0

4.3 occurs most often.

The mean is 4 miles, the median is 4.2 miles, and the mode is 4.3 miles. 1. Use the data in the example above. You run 4.0 miles on Day 11. How does this additional value

affect the mean, median, and mode? Explain. 2. Use the data in the example above. You run 10.0 miles on Day 11. How does this additional value affect the mean, median, and mode? Explain. Find the mean, median, and mode of the data. 3.

4.

Ski Resort Temperatures (°F)

11

3

3

0

−9

−2

10

10

10

0

1

2

3

4

5

Goals per game

Find the range and standard deviation of each data set. Then compare your results. 5.

6.

Bowling Scores Player A

Player B

Tablet Prices Store A

Store B

205

190

228

205

$140

$180

$225

$310

185

200

172

181

$200

$250

$260

$190

210

219

154

240

$150

$190

$190

$285

174

203

235

235

$250

$160

$160

$240

194

230

168

192

Find the values of the measures shown after the given transformation. Mean: 109

Median: 104

Mode: 96

Range: 45

7. Each value in the data set increases by 25. 8. Each value in the data set is multiplied by 0.6.

370

Chapter 7

Data Analysis and Displays

Standard deviation: 3.6

7.2

Box-and-Whisker Plots (pp. 339–344)

Make a box-and-whisker plot that represents the weights (in pounds) of pumpkins sold at a market. 16, 20, 11, 15, 10, 8, 8, 19, 11, 9, 16, 9 Step 1

Order the data. Find the median and the quartiles. lower half least value

upper half

8 8 9 9 10 11 11 15 16 16 19 20

first quartile, 9 median, 11

greatest value

third quartile, 16

Step 2

Draw a number line that includes the least and greatest values. Graph points above the number line for the five-number summary.

Step 3

Draw a box using Q1 and Q3. Draw a line through the median. Draw whiskers from the box to the least and greatest values. median

first quartile

third quartile

least value

greatest value 8

9

10

11

12

13

14

15

16

17

18

19

Weight (pounds)

20

Make a box-and-whisker plot that represents the data. Identify the shape of the distribution. 9. Ages of volunteers at a hospital:

10. Masses (in kilograms) of lions:

14, 17, 20, 16, 17, 14, 21, 18, 22

Shapes of Distributions (pp. 345–352)

Amount

Frequency

0–0.99

9

1–1.99

10

2–2.99

9

3–3.99

7

4– 4.99

4

5–5.99

1

99 5.

99

5–

4.

99

4–

99

3– 3.

99

2– 2.

1– 1.

9

12 10 8 6 4 2 0 0–

The distribution is skewed left. So, use the median to describe the center and the five-number summary to describe the variation.

Amounts of Money in Pocket

0. 9

The histogram shows the amounts of money a group of adults have in their pockets. Describe the shape of the distribution. Which measures of center and variation best represent the data?

Frequency

7.3

120, 230, 180, 210, 200, 200, 230, 160

Amount of money (dollars)

11.

The frequency table shows the amounts (in dollars) of money the students in a class have in their pockets. a. Display the data in a histogram. Describe the shape of the distribution. b. Which measures of center and variation best represent the data? c. Compare this distribution with the distribution shown above using their shapes and appropriate measures of center and variation. Chapter 7

Chapter Review

371

7.4

Two-Way Tables (pp. 355–362)

You conduct a survey that asks 130 students about whether they have an after-school job. Sixty males respond, 38 of which have a job. Twenty-six females do not have a job. Organize the results in a two-way table. Find and interpret the marginal frequencies.

Gender

After-School Job Yes

No

Total

Males

38

22

60

60 males responded.

Females

44

26

70

70 females responded.

Total

82

48

130

130 students were surveyed.

82 students have a job.

48 students do not have a job.

12. The two-way table shows the results of a survey that

Food Court

7.5

Shoppers

asked shoppers at a mall about whether they like the new food court. a. Make a two-way table that shows the joint and marginal relative frequencies. b. Make a two-way table that shows the conditional relative frequencies based on the column totals.

Like

Dislike

Adults

21

79

Teenagers

96

4

Choosing a Data Display (pp. 363–368)

Analyze the data and then create a display that best represents the data. Ages of U.S. Presidents at Inauguration

57

61

57

57

58

57

61

54

68

51

49

64

50

48

65

52

56

46

54

49

51

47

55

55

54

42

51

56

55

51

54

51

60

62

43

55

56

61

52

69

64

46

54

47

A stem-and-leaf plot is one appropriate way to display this quantitative data. It orders numerical data and shows how they are distributed. 4 5 6

Ages of U.S. Presidents at Inauguration 236677899 0111112244444555566677778 0111244589 Key: 5 | 0 = 50

13. Analyze the data in the table at the right and then create a

display that best represents the data. Explain your reasoning. Tell whether the data are qualitative or quantitative. Explain. 14. heights of the members of a basketball team 15. grade level of students in an elementary school

372

Chapter 7

Data Analysis and Displays

Perfect Attendance Class

Number of students

freshman

84

sophomore

42

junior

67

senior

31

7

Chapter Test

Describe the shape of the data distribution. Then determine which measures of center and variation best represent the data. 1.

2.

3.

4. Determine whether each statement is always, sometimes, or never true. Explain your

reasoning. a. The sum of the marginal relative frequencies in the “total” row and the “total” column of a two-way table should each be equal to 1. b. In a box-and-whisker plot, the length of the box to the left of the median and the length of the box to the right of the median are equal. c. Qualitative data are numerical. 5. Find the mean, median, mode, range,

Prices of Shirts at a Clothing Store

and standard deviation of the prices.

$15.50

$18.90

$10.60

$12.25

$7.80

$23.50

$9.75

$21.70

6. Repeat Exercise 5 when all the shirts in the clothing store are 20% off. 7. Which data display best represents the data, a histogram or a stem-and-leaf plot? Explain.

15, 21, 18, 10, 12, 11, 17, 18, 16, 12, 20, 12, 17, 16 8. The tables show the battery lives (in hours) of two brands of laptops.

Brand A

Brand B

a. Make a double box-and-whisker plot that represents the data.

20.75

18.5

10.5

12.5

b. Identify the shape of each distribution.

13.5

16.25

9.5

10.25

c. Which brand’s battery lives are more spread out? Explain.

8.5

13.5

9.0

9.75

d. Compare the distributions using their shapes and appropriate measures of center and variation.

14.5

15.5

8.5

8.5

11.5

16.75

9.0

7.0

Preferred method of exercise

Number of students

walking

20

jogging

28

biking

17

swimming

11

lifting weights

10

dancing

14

9. The table shows the results of a survey that asked students their preferred

method of exercise. Analyze the data and then create a display that best represents the data. Explain your reasoning. 10. You conduct a survey that asks 271 students in your class whether they

are attending the class field trip. One hundred twenty-one males respond, 92 of which are attending the field trip. Thirty-one females are not attending the field trip. a. Organize the results in a two-way table. Find and interpret the marginal frequencies. b. What percent of females are attending the class field trip?

Chapter 7

Chapter Test

373

7

Cumulative Assessment

1. You ask all the students in your grade whether they have a

Cell Phones

Gender

cell phone. The results are shown in the two-way table. Your friend claims that a greater percent of males in your grade have cell phones than females. Do you support your friend’s claim? Justify your answer.

Yes

No

Male

27

12

Female

31

17

2. The function f (x) = a(1.08)x represents the total amount of money (in dollars) in

Account A after x years. The function g(x) = 600(b)x represents the total amount of money (in dollars) in Account B after x years. Fill in values for a and b so that each statement is true. a. When a = ____ and b = ____, Account B has a greater initial amount and increases at a faster rate than Account A. b. When a = ____ and b = ____, Account B has a lesser initial amount than Account A but increases at a faster rate than Account A. c. When a = ____ and b = ____, Account B and Account A have the same initial amount, and Account B increases at a slower rate than Account A.

3. Classify the shape of each distribution as symmetric, skewed left, or skewed right.

a.

b.

c.

d.

4. Complete the equation so that the solution of the system of equations is (4, −5).

y−3=

x

1 y = —x − 7 2 5. Your friend claims the line of best fit for the data shown in the scatter plot has a

correlation coefficient close to 1. Is your friend correct? Explain your reasoning.

y 4 2 0

374

Chapter 7

Data Analysis and Displays

0

2

4

x

6. The box-and-whisker plot represents the lengths (in minutes) of project presentations

at a science fair. Find the interquartile range of the data. What does this represent in the context of the situation?

1

2

3

4

5

6

7

8

9

Presentation length (minutes)

10

A 7; The middle half of the presentation lengths vary by no more than 7 minutes. ○ B 3; The presentation lengths vary by no more than 3 minutes. ○ C 3; The middle half of the presentation lengths vary by no more than 3 minutes. ○ D 7; The presentation lengths vary by no more than 7 minutes. ○ 7. Scores in a video game can be between 0 and 100. Use the data set shown to fill in

a value for x so that each statement is true. Video Game Scores

a. When x = ____, the mean of the scores is 45.5. b. When x = ____, the median of the scores is 47. c. When x = ____, the mode of the scores is 63. d. When x = ____, the range of the scores is 71.

36

28

48

x

42

57

63

52

8. The table shows the altitudes of a hang glider that descends at a constant rate. How long

will it take for the hang glider to descend to an altitude of 100 feet? Justify your answer.

A 25 seconds ○

Time (seconds), t

Altitude (feet), y

0

450

10

350

20

250

C 45 seconds ○

30

150

D 55 seconds ○

B 35 seconds ○

9. A traveler walks and takes a shuttle bus to get to a terminal of an airport. The function

y = D(x) represents the traveler’s distance (in feet) after x minutes. a. Estimate and interpret D(2).

y

b. Use the graph to find the solution of the equation D(x) = 3500. Explain the meaning of the solution.

3000

c. How long does the traveler wait for the shuttle bus?

2000

d. How far does the traveler ride on the shuttle bus? e. What is the total distance that the traveler walks before and after riding the shuttle bus? Chapter 7

(15, 3500) (13, 3000) D (4, 1000)

1000 0

(12, 1000) 0

4

8

12

16 x

Cumulative Assessment

375

8 8.1 8.2 8.3 8.4 8.5 8.6

Basics of Geometry Points, Lines, and Planes Measuring and Constructing Segments Using Midpoint and Distance Formulas Perimeter and Area in the Coordinate Plane Measuring and Constructing Angles Describing Pairs of Angles SEE the Big Idea

Alamillo Alamil Al illlo Bridge Brid idge (p. (p. 429) 429)

( 425)) Soccer (p.

Shed ((p. 409))

Skateboard (p. (p 396)

Sulfur Hexafluoride S ulf lfur H exafl fluorid ide ((p. p. 38 383) 3)

Maintaining Mathematical Proficiency Finding Absolute Value Example 1

Simplify ∣ −7 − 1 ∣.

∣ −7 − 1 ∣ = ∣ −7 + (−1) ∣

Add the opposite of 1.

= ∣ −8 ∣

Add.

=8

Find the absolute value.

∣ −7 − 1 ∣ = 8

Simplify the expression. 1.

∣ 8 − 12 ∣

2.

∣ −6 − 5 ∣

3.

∣ 4 + (–9) ∣

4.

∣ 13 + (−4) ∣

5.

∣ 6 − (−2) ∣

6.

∣ 5 − (−1) ∣

7.

∣ −8 − (−7) ∣

8.

∣ 8 − 13 ∣

9.

∣ −14 − 3 ∣

Finding the Area of a Triangle Example 2

Find the area of the triangle. 5 cm 18 cm

A = —12 bh

Write the formula for area of a triangle.

1

= —2 (18)(5)

Substitute 18 for b and 5 for h.

1

= —2(90)

Multiply 18 and 5.

= 45

Multiply —12 and 90.

The area of the triangle is 45 square centimeters.

Find the area of the triangle. 10.

11.

12.

7 yd

22 m

24 yd

16 in.

25 in. 14 m

13. ABSTRACT REASONING Describe the possible values for x and y when ∣ x − y ∣ > 0. What does it

mean when ∣ x − y ∣ = 0? Can ∣ x − y ∣ < 0? Explain your reasoning.

Dynamic Solutions available at BigIdeasMath.com

377

Mathematical Practices

Mathematically proficient students carefully specify units of measure.

Specifying Units of Measure

Core Concept Customary Units of Length

Metric Units of Length

1 foot = 12 inches 1 yard = 3 feet 1 mile = 5280 feet = 1760 yards

1 centimeter = 10 millimeters 1 meter = 1000 millimeters 1 kilometer = 1000 meters

1

in.

2

3

1 in. = 2.54 cm cm

1

2

3

4

5

6

7

8

9

Converting Units of Measure Find the area of the rectangle in square centimeters. Round your answer to the nearest hundredth.

2 in.

SOLUTION

6 in.

Use the formula for the area of a rectangle. Convert the units of length from customary units to metric units. Area = (Length)(Width)

Formula for area of a rectangle

= (6 in.)(2 in.)

[ (

2.54 cm = (6 in.) — 1 in.

Substitute given length and width.

) ] [ (2 in.)( 2.541 in.cm ) ] —

Multiply each dimension by the conversion factor.

= (15.24 cm)(5.08 cm)

Multiply.

≈ 77.42 cm2

Multiply and round to the nearest hundredth.

The area of the rectangle is about 77.42 square centimeters.

Monitoring Progress Find the area of the polygon using the specified units. Round your answer to the nearest hundredth. 1. triangle (square inches)

2 cm

2 cm

2. parallelogram (square centimeters)

2 in.

2.5 in.

3. The distance between two cities is 120 miles. What is the distance in kilometers? Round your answer

to the nearest whole number. 378

Chapter 8

Basics of Geometry

8.1

Points, Lines, and Planes Essential Question

How can you use dynamic geometry software to visualize geometric concepts? Using Dynamic Geometry Software Work with a partner. Use dynamic geometry software to draw several points. Also, draw some lines, line segments, and rays. What is the difference between a line, a line segment, and a ray? Sample B A

G F C

E

D

Intersections of Lines and Planes

UNDERSTANDING MATHEMATICAL TERMS To be proficient in math, you need to understand definitions and previously established results. An appropriate tool, such as a software package, can sometimes help.

Work with a partner. a. Describe and sketch the ways in which two lines can intersect or not intersect. Give examples of each using the lines formed by the walls, floor, and ceiling in your classroom. b. Describe and sketch the ways in which a line and a plane can intersect or not intersect. Give examples of each using the walls, floor, and ceiling in your classroom. c. Describe and sketch the ways in which two planes can intersect or not intersect. Give examples of each using the walls, floor, and ceiling in your classroom.

Q

P

B A

Exploring Dynamic Geometry Software Work with a partner. Use dynamic geometry software to explore geometry. Use the software to find a term or concept that is unfamiliar to you. Then use the capabilities of the software to determine the meaning of the term or concept.

Communicate Your Answer 4. How can you use dynamic geometry software to visualize geometric concepts?

Section 8.1

Points, Lines, and Planes

379

8.1

Lesson

What You Will Learn Name points, lines, and planes.

Core Vocabul Vocabulary larry

Name segments and rays.

undefined terms, p. 380 point, p. 380 line, p. 380 plane, p. 380 collinear points, p. 380 coplanar points, p. 380 defined terms, p. 381 line segment, or segment, p. 381 endpoints, p. 381 ray, p. 381 opposite rays, p. 381 intersection, p. 382

Solve real-life problems involving lines and planes.

Sketch intersections of lines and planes.

Using Undefined Terms In geometry, the words point, line, and plane are undefined terms. These words do not have formal definitions, but there is agreement about what they mean.

Core Concept Undefined Terms: Point, Line, and Plane Point

A

A point has no dimension. A dot represents a point.

point A

A line has one dimension. It is represented by a line with two arrowheads, but it extends without end.

Line

Through any two points, there is exactly one line. You can use any two points on a line to name it. A plane has two dimensions. It is represented by a shape that looks like a floor or a wall, but it extends without end.

A

line , line AB (AB), or line BA (BA)

Plane

Through any three points not on the same line, there is exactly one plane. You can use three points that are not all on the same line to name a plane.

B

A

M C

B

plane M, or plane ABC

Collinear points are points that lie on the same line. Coplanar points are points that lie in the same plane.

Naming Points, Lines, and Planes a. Give two other names for ⃖""⃗ PQ and plane R. b. Name three points that are collinear. Name four points that are coplanar.

SOLUTION

Q

T

V S

n

P

m

R

a. Other names for ⃖""⃗ PQ are ⃖""⃗ QP and line n. Other names for plane R are plane SVT and plane PTV. b. Points S, P, and T lie on the same line, so they are collinear. Points S, P, T, and V lie in the same plane, so they are coplanar.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1. Use the diagram in Example 1. Give two other names for ⃖""⃗ ST . Name a point

that is not coplanar with points Q, S, and T.

380

Chapter 8

Basics of Geometry

Using Defined Terms In geometry, terms that can be described using known words such as point or line are called defined terms.

Core Concept Defined Terms: Segment and Ray

line

The definitions below use line AB (written as ⃖!!⃗ AB) and points A and B.

A

Segment The line segment AB, or segment AB, — ) consists of the endpoints A and B (written as AB and all points on ⃖!!⃗ AB that are between A and B. — can also be named BA —. Note that AB

B segment

endpoint

endpoint

A

B

Ray The ray AB (written as !!!⃗ AB ) consists of the

ray

endpoint A and all points on ⃖!!⃗ AB that lie on the

endpoint

same side of A as B.

A

Note that !!!⃗ AB and !!!⃗ BA are different rays.

B endpoint

Opposite Rays If point C lies on ⃖!!⃗ AB between

A and B, then !!!⃗ CA and !!!⃗ CB are opposite rays.

A

B

A C

B

Segments and rays are collinear when they lie on the same line. So, opposite rays are collinear. Lines, segments, and rays are coplanar when they lie in the same plane.

Naming Segments, Rays, and Opposite Rays

—. a. Give another name for GH

COMMON ERROR

In Example 2, !!!⃗ JG and !!!⃗ JF have a common endpoint, but they are not collinear. So, they are not opposite rays.

E

b. Name all rays with endpoint J. Which of these rays are opposite rays?

G J

F

H

SOLUTION

— is HG —. a. Another name for GH b. The rays with endpoint J are !!!⃗ JE , !!!⃗ JG , !!!⃗ JF , and !!!⃗ JH . The pairs of opposite rays

with endpoint J are !!!⃗ JE and !!!⃗ JF , and !!!⃗ JG and !!!⃗ JH .

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use the diagram. M

K P

L

N

—. 2. Give another name for KL 3. Are !!!⃗ KP and !!!⃗ PK the same ray? Are !!!⃗ NP and !!!!⃗ NM the same ray? Explain.

Section 8.1

Points, Lines, and Planes

381

Sketching Intersections Two or more geometric figures intersect when they have one or more points in common. The intersection of the figures is the set of points the figures have in common. Some examples of intersections are shown below. m

A

q

n The intersection of two different lines is a point.

The intersection of two different planes is a line.

Sketching Intersections of Lines and Planes a. Sketch a plane and a line that is in the plane. b. Sketch a plane and a line that does not intersect the plane. c. Sketch a plane and a line that intersects the plane at a point.

SOLUTION a.

b.

c.

Sketching Intersections of Planes Sketch two planes that intersect in a line.

SOLUTION Step 1 Draw a vertical plane. Shade the plane. Step 2 Draw a second plane that is horizontal. Shade this plane a different color. Use dashed lines to show where one plane is hidden. Step 3 Draw the line of intersection.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

4. Sketch two different lines that intersect a plane

B

at the same point. Use the diagram. 5. Name the intersection of ⃖""⃗ PQ and line k. 6. Name the intersection of plane A and plane B. 7. Name the intersection of line k and plane A.

382

Chapter 8

Basics of Geometry

P

k M A

Q

Solving Real-Life Problems Modeling with Mathematics The diagram shows a molecule of sulfur hexafluoride, the most potent greenhouse gas in the world. Name two different planes that contain line r. p A q D

B

E

G

r

F C

SOLUTION Electric utilities use sulfur hexafluoride as an insulator. Leaks in electrical equipment contribute to the release of sulfur hexafluoride into the atmosphere.

1. Understand the Problem In the diagram, you are given three lines, p, q, and r, that intersect at point B. You need to name two different planes that contain line r. 2. Make a Plan The planes should contain two points on line r and one point not on line r. 3. Solve the Problem Points D and F are on line r. Point E does not lie on line r. So, plane DEF contains line r. Another point that does not lie on line r is C. So, plane CDF contains line r. Note that you cannot form a plane through points D, B, and F. By definition, three points that do not lie on the same line form a plane. Points D, B, and F are collinear, so they do not form a plane. 4. Look Back The question asks for two different planes. You need to check whether plane DEF and plane CDF are two unique planes or the same plane named differently. Because point C does not lie on plane DEF, plane DEF and plane CDF are different planes.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use the diagram that shows a molecule of phosphorus pentachloride. s G J

K

H

L

I

8. Name two different planes that contain line s. 9. Name three different planes that contain point K. 10. Name two different planes that contain !!!⃗ HJ .

Section 8.1

Points, Lines, and Planes

383

Exercises

8.1

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Compare collinear points and coplanar points. 2. WHICH ONE DOESN’T BELONG? Which term does not belong with the other three?

Explain your reasoning.

— AB

⃖""⃗ FG

plane CDE

"""⃗ HI

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, use the diagram. B

In Exercises 11–16, use the diagram. (See Example 2.) C

t

S B

A D

T

s

E

A

3. Name four points.

C

D

4. Name two lines. 5. Name the plane that contains points A, B, and C. 6. Name the plane that contains points A, D, and E.

In Exercises 7–10, use the diagram. (See Example 1.) g

—? 11. What is another name for BD —? 12. What is another name for AC 13. What is another name for ray """⃗ AE? 14. Name all rays with endpoint E. 15. Name two pairs of opposite rays.

W Q V

E

R

S

f

T

16. Name one pair of rays that are not opposite rays.

In Exercises 17–24, sketch the figure described. (See Examples 3 and 4.) 17. plane P and line ℓ intersecting at one point

7. Give two other names for ⃖"""⃗ WQ.

18. plane K and line m intersecting at all points on line m 19. """⃗ AB and ⃖""⃗ AC

8. Give another name for plane V.

20. """"⃗ MN and """⃗ NX

9. Name three points that are collinear. Then name

21. plane M and """⃗ NB intersecting at B

a fourth point that is not collinear with these three points.

22. plane M and """⃗ NB intersecting at A

10. Name a point that is not coplanar with R, S, and T.

384

Chapter 8

Basics of Geometry

23. plane A and plane B not intersecting 24. plane C and plane D intersecting at ⃖""⃗ XY

ERROR ANALYSIS In Exercises 25 and 26, describe

and correct the error in naming opposite rays in the diagram.

In Exercises 35–38, name the geometric term modeled by the object. 35.

C A

B

X Y

D

E

25.

26.



36.

!!!⃗ AD and !!!⃗ AC are opposite rays.



37.

38.

— and YE — are opposite rays. YC

In Exercises 27–34, use the diagram. B

I

A

C

In Exercises 39–44, use the diagram to name all the points that are not coplanar with the given points.

D F

G

39. N, K, and L

K

L

40. P, Q, and N E

J

H

41. P, Q, and R

27. Name a point that is collinear with points E and H.

42. R, K, and N

28. Name a point that is collinear with points B and I.

43. P, S, and K

29. Name a point that is not collinear with points E

44. Q, K, and L

and H. 30. Name a point that is not collinear with points B and I.

N

M

R

Q

S

P

45. CRITICAL THINKING Given two points on a line and

31. Name a point that is coplanar with points D, A, and B.

a third point not on the line, is it possible to draw a plane that includes the line and the third point? Explain your reasoning.

32. Name a point that is coplanar with points C, G, and F.

46. CRITICAL THINKING Is it possible for one point to be

33. Name the intersection of plane AEH and plane FBE.

in two different planes? Explain your reasoning.

34. Name the intersection of plane BGF and plane HDG.

Section 8.1

Points, Lines, and Planes

385

47. REASONING Explain why a four-legged chair may

53. x ≥ 5 or x ≤ −2

rock from side to side even if the floor is level. Would a three-legged chair on the same level floor rock from side to side? Why or why not?

54.

∣x∣ ≤ 0

55. MODELING WITH MATHEMATICS Use the diagram. J

48. THOUGHT PROVOKING You are designing the living

K

room of an apartment. Counting the floor, walls, and ceiling, you want the design to contain at least eight different planes. Draw a diagram of your design. Label each plane in your design.

L

P N

Q

M

49. LOOKING FOR STRUCTURE Two coplanar intersecting

lines will always intersect at one point. What is the greatest number of intersection points that exist if you draw four coplanar lines? Explain. 50. HOW DO YOU SEE IT? You and your friend walk

a. Name two points that are collinear with P.

in opposite directions, forming opposite rays. You were originally on the corner of Apple Avenue and Cherry Court.

b. Name two planes that contain J. c. Name all the points that are in more than one plane.

N

Apple Ave

E

CRITICAL THINKING In Exercises 56–63, complete the statement with always, sometimes, or never. Explain your reasoning.

.

Rd.

S

Rose

Cherry Ct.

W

56. A line ____________ has endpoints.

Daisy Dr.

57. A line and a point ____________ intersect. 58. A plane and a point ____________ intersect. 59. Two planes ____________ intersect in a line.

a. Name two possibilities of the road and direction you and your friend may have traveled.

60. Two points ____________ determine a line.

b. Your friend claims he went north on Cherry Court, and you went east on Apple Avenue. Make an argument as to why you know this could not have happened.

61. Any three points ____________ determine a plane. 62. Any three points not on the same line ____________

determine a plane.

MATHEMATICAL CONNECTIONS In Exercises 51–54,

63. Two lines that are not parallel ___________ intersect.

graph the inequality on a number line. Tell whether the graph is a segment, a ray or rays, a point, or a line.

64. ABSTRACT REASONING Is it possible for three planes

51. x ≤ 3

to never intersect? intersect in one line? intersect in one point? Sketch the possible situations.

52. −7 ≤ x ≤ 4

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Find the absolute value. (Skills Review Handbook) 65.

∣6 + 2∣

66.

∣3 − 9∣

67.

∣ −8 − 2 ∣

68.

∣ 7 − 11 ∣

Solve the equation. (Section 1.1) 69. 18 + x = 43

386

Chapter 8

70. 36 + x = 20

Basics of Geometry

71. x − 15 = 7

72. x − 23 = 19

8.2

Measuring and Constructing Segments Essential Question

How can you measure and construct a

line segment?

Measuring Line Segments Using Nonstandard Units 12

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

H INC

5

3 2 1

4

6 5 4

3

9 8 7

2

29

30

To be proficient in math, you need to explain to yourself the meaning of a problem and look for entry points to its solution.

a. Draw a line segment that has a length of 6 inches. b. Use a standard-sized paper clip to measure the length of the line segment. Explain how you measured the line segment in “paper clips.”

1

MAKING SENSE OF PROBLEMS

11 10

c. Write conversion factors from paper clips to inches and vice versa. 1 paper clip = 1 in. =

in.

paper clip

d. A straightedge is a tool that you can use to draw a straight line. An example of a straightedge is a ruler. Use only a pencil, straightedge, paper clip, and paper to draw another line segment that is 6 inches long. Explain your process.

Measuring Line Segments Using Nonstandard Units Work with a partner.

CONNECTIONS TO GEOMETRY You will learn more about the Pythagorean Theorem and its useful applications in a future course.

a. Fold a 3-inch by 5-inch index card fo ld on one of its diagonals. 3 in. b. Use the Pythagorean Theorem to algebraically determine the length of the diagonal in inches. 5 in. Use a ruler to check your answer. c. Measure the length and width of the index card in paper clips. d. Use the Pythagorean Theorem to algebraically determine the length of the diagonal in paper clips. Then check your answer by measuring the length of the diagonal in paper clips. Does the Pythagorean Theorem work for any unit of measure? Justify your answer.

Measuring Heights Using Nonstandard Units Work with a partner. Consider a unit of length that is equal to the length of the diagonal you found in Exploration 2. Call this length “1 diag.” How tall are you in diags? Explain how you obtained your answer.

Communicate Your Answer 4. How can you measure and construct a line segment?

Section 8.2

Measuring and Constructing Segments

387

CM

Work with a partner.

8.2 Lesson

What You Will Learn Use the Ruler Postulate. Copy segments and compare segments for congruence.

Core Vocabul Vocabulary larry

Use the Segment Addition Postulate.

postulate, p. 388 axiom, p. 388 coordinate, p. 388 distance, p. 388 construction, p. 389 congruent segments, p. 389 between, p. 390

Using the Ruler Postulate In geometry, a rule that is accepted without proof is called a postulate or an axiom. A rule that can be proved is called a theorem, as you will see later. The Ruler Postulate shows how to find the distance between two points on a line.

Postulate

names of points

Ruler Postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point.

A

B

x1

x2

coordinates of points

The distance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B.

A

B

AB

x2

x1

AB = "x2 − x1"

Using the Ruler Postulate —

Measure the length of ST to the nearest tenth of a centimeter. S

T

SOLUTION Align one mark of a metric ruler with S. Then estimate the coordinate of T. For example, when you align S with 2, T appears to align with 5.4. S cm

1

T

2

3

4

ST = ∣ 5.4 – 2 ∣ = 3.4

5

6

Ruler Postulate

— is about 3.4 centimeters. So, the length of ST

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use a ruler to measure the length of the segment to the nearest —18 inch. 1.

3.

388

Chapter 8

M

U

Basics of Geometry

N

V

2.

4.

P

W

Q

X

Constructing and Comparing Congruent Segments A construction is a geometric drawing that uses a limited set of tools, usually a compass and straightedge.

Copying a Segment Use a compass and straightedge to construct a line segment —. that has the same length as AB

A

B

SOLUTION Step 1

Step 2

A

B

Step 3

A

C

B

C

Draw a segment Use a straightedge —. to draw a segment longer than AB Label point C on the new segment.

Measure length Set your —. compass at the length of AB

A

B

C

D

Copy length Place the compass at C. Mark point D on the new segment. — has the same length as AB —. So, CD

Core Concept Congruent Segments

READING In the diagram, the red tick marks indicate — AB ≅ — CD . When there is more than one pair of congruent segments, use multiple tick marks.

Line segments that have the same length are called congruent segments. You — is equal to the length of CD —,” or you can say “AB — is can say “the length of AB — congruent to CD .” The symbol ≅ means “is congruent to.” A

B

Lengths are equal. AB = CD

Segments are congruent. — ≅ CD — AB

C

D

“is equal to”

“is congruent to”

Comparing Segments for Congruence Plot J(−3, 4), K(2, 4), L(1, 3), and M(1, −2) in a coordinate plane. Then determine — and LM — are congruent. whether JK

SOLUTION y

J(−3, 4)

K(2, 4) L(1, 3) 2

−4

−2

JK = ∣ 2 − (−3) ∣ = 5

Ruler Postulate

To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints. 2

−2

Plot the points, as shown. To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints.

4 x

M(1, −2)

LM = ∣ −2 − 3 ∣ = 5

Ruler Postulate

— — have the same length. So, JK — ≅ LM —. JK and LM

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

5. Plot A(−2, 4), B(3, 4), C(0, 2), and D(0, −2) in a coordinate plane. Then

— and CD — are congruent. determine whether AB Section 8.2

Measuring and Constructing Segments

389

Using the Segment Addition Postulate When three points are collinear, you can say that one point is between the other two. A

D B

E F

C

Point B is between points A and C.

Point E is not between points D and F.

Postulate Segment Addition Postulate If B is between A and C, then AB + BC = AC.

AC

If AB + BC = AC, then B is between A and C.

A

AB

B

BC

C

Using the Segment Addition Postulate a. Find DF.

23

D

b. Find GH.

35

E

F

36 F

21

G

H

SOLUTION a. Use the Segment Addition Postulate to write an equation. Then solve the equation to find DF.

CONNECTIONS TO ALGEBRA In this step, you are applying the Substitution Property of Equality that you learned about in Section 1.1.

DF = DE + EF

Segment Addition Postulate

DF = 23 + 35

Substitute 23 for DE and 35 for EF.

DF = 58

Add.

b. Use the Segment Addition Postulate to write an equation. Then solve the equation to find GH. FH = FG + GH

Segment Addition Postulate

36 = 21 + GH

Substitute 36 for FH and 21 for FG.

15 = GH

Subtract 21 from each side.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use the diagram at the right. 6. Use the Segment Addition Postulate to find XZ. 7. In the diagram, WY = 30. Can you use the

Segment Addition Postulate to find the distance between points W and Z ? Explain your reasoning.

144 J 37 K

390

Chapter 8

L

8. Use the diagram at the left to find KL.

Basics of Geometry

23 X

50 Y

W

Z

Using the Segment Addition Postulate The cities shown on the map lie approximately in a straight line. Find the distance from Tulsa, Oklahoma, to St. Louis, Missouri.

S St. Louis 738 mi T Tulsa 377 mi L Lubbock

SOLUTION 1. Understand the Problem You are given the distance from Lubbock to St. Louis and the distance from Lubbock to Tulsa. You need to find the distance from Tulsa to St. Louis. 2. Make a Plan Use the Segment Addition Postulate to find the distance from Tulsa to St. Louis. 3. Solve the Problem Use the Segment Addition Postulate to write an equation. Then solve the equation to find TS. LS = LT + TS 738 = 377 + TS 361 = TS

Segment Addition Postulate Substitute 738 for LS and 377 for LT. Subtract 377 from each side.

So, the distance from Tulsa to St. Louis is about 361 miles.

4. Look Back Does the answer make sense in the context of the problem? The distance from Lubbock to St. Louis is 738 miles. By the Segment Addition Postulate, the distance from Lubbock to Tulsa plus the distance from Tulsa to St. Louis should equal 738 miles. 377 + 361 = 738



Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

9. The cities shown on the map lie approximately in a straight line. Find the distance

from Albuquerque, New Mexico, to Provo, Utah.

Provo P

680 mi

Albuquerque A

231 mi C

Section 8.2

Carlsbad

Measuring and Constructing Segments

391

8.2

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check —

1. WRITING Explain how XY and XY are different. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. 7 3 A

B

C

Find AC + CB.

Find BC − AC.

Find AB.

Find CA + BC.

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, use a ruler to measure the length of the segment to the nearest tenth of a centimeter. (See Example 1.)

In Exercises 15–22, find FH. (See Example 3.) 15.

8

F

14

G

3.

H

4. 16.

5.

7 19

6. F

CONSTRUCTION In Exercises 7 and 8, use a compass and

straightedge to construct a copy of the segment.

17.

12

7. Copy the segment in Exercise 3.

11

H

G

F

8. Copy the segment in Exercise 4.

In Exercises 9–14, plot the points in a coordinate plane. Then determine whether — AB and — CD are congruent. (See Example 2.)

H

G

18. F

4 G

15

9. A(−4, 5), B(−4, 8), C(2, −3), D(2, 0) H

10. A(6, −1), B(1, −1), C(2, −3), D(4, −3) 11. A(8, 3), B(−1, 3), C(5, 10), D(5, 3)

19. 37 H

12. A(6, −8), B(6, 1), C(7, −2), D(−2, −2) 13. A(−5, 6), B(−5, −1), C(−4, 3), D(3, 3) 14. A(10, −4), B(3, −4), C(−1, 2), D(−1, 5)

392

Chapter 8

Basics of Geometry

F

13

G

20.

26. MODELING WITH MATHEMATICS In 2003, a remote-

22 F

15

H

controlled model airplane became the first ever to fly nonstop across the Atlantic Ocean. The map shows the airplane’s position at three different points during its flight. Point A represents Cape Spear, Newfoundland, point B represents the approximate position after 1 day, and point C represents Mannin Bay, Ireland. The airplane left from Cape Spear and landed in Mannin Bay. (See Example 4.)

G

21. F 42 H 22 G

22.

North America

G 53

40

B A

Europe

C 601 mi

1282 mi

Atlantic Ocean

H

a. Find the total distance the model airplane flew.

F

ERROR ANALYSIS In Exercises 23 and 24, describe and

correct the error in finding the length of — AB . A

B

b. The model airplane’s flight lasted nearly 38 hours. Estimate the airplane’s average speed in miles per hour. 27. USING STRUCTURE Determine whether the statements

are true or false. Explain your reasoning. cm

1

2

3

4

5

6

C

23.

B



AB = 1 − 4.5 = −3.5

A

D

E

H F

24.



a. B is between A and C.

AB = ∣ 1 + 4.5 ∣ = 5.5

b. C is between B and E. c. D is between A and H.

25. ATTENDING TO PRECISION The diagram shows an

insect called a walking stick. Use the ruler to estimate the length of the abdomen and the length of the thorax to the nearest —14 inch. How much longer is the walking stick’s abdomen than its thorax? How many times longer is its abdomen than its thorax? abdomen

d. E is between C and F. 28. MATHEMATICAL CONNECTIONS Write an expression

for the length of the segment.

— a. AC

A x+2 B

thorax

7x − 3

C

— b. QR 13y + 25 INCH

1

2

3

4

5

6

7

P

8y + 5

Q

R

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Section 8.2

Measuring and Constructing Segments

393

29. MATHEMATICAL CONNECTIONS Point S is between

33. REASONING You travel from City X to City Y. You

—. Use the information to write an points R and T on RT equation in terms of x. Then solve the equation and find RS, ST, and RT. a. RS = 2x + 10 ST = x − 4 RT = 21

b. RS = 3x − 16 ST = 4x − 8 RT = 60

c. RS = 2x − 8 ST = 11 RT = x + 10

d. RS = 4x − 9 ST = 19 RT = 8x − 14

know that the round-trip distance is 647 miles. City Z, a city you pass on the way, is 27 miles from City X. Find the distance from City Z to City Y. Justify your answer. 34. HOW DO YOU SEE IT? The bar graph shows the

win-loss record for a lacrosse team over a period of three years. Explain how you can apply the Ruler Postulate and the Segment Addition Postulate when interpreting a stacked bar graph like the one shown. Win-Loss Record Wins

30. THOUGHT PROVOKING Is it possible to design a table

where no two legs have the same length? Assume that the endpoints of the legs must all lie in the same plane. Include a diagram as part of your answer.

Losses

Year 1 Year 2 Year 3 0

31. MODELING WITH MATHEMATICS You have to walk

2

from Room 103 to Room 117. 103

107

113

117

109

111

115

8

10

12

form a segment, and the points (d, e) and (d, f ) form a segment. Create an equation assuming the segments are congruent. Are there any letters not used in the equation? Explain.

22 ft 105

6

35. ABSTRACT REASONING The points (a, b) and (c, b)

86 ft 101

4

Number of games

119

121

36. MATHEMATICAL CONNECTIONS In the diagram,

— ≅ BC —, AC — ≅ CD —, and AD = 12. Find the AB lengths of all segments in the diagram. Suppose you choose one of the segments at random. What is the probability that the measure of the segment is greater than 3? Explain your reasoning.

a. How many feet do you travel from Room 103 to Room 117? b. You can walk 4.4 feet per second. How many minutes will it take you to get to Room 117? c. Why might it take you longer than the time in part (b)?

D B

A

32. MAKING AN ARGUMENT Your friend and your cousin

discuss measuring with a ruler. Your friend says that you must always line up objects at the zero on a ruler. Your cousin says it does not matter. Decide who is correct and explain your reasoning.

C

37. CRITICAL THINKING Is it possible to use the Segment

Addition Postulate to show FB > CB or that  AC > DB? Explain your reasoning. A

Maintaining Mathematical Proficiency

D

F

C

Reviewing what you learned in previous grades and lessons

Simplify. (Skills Review Handbook) −4 + 6 2

—

39. √ 20 + 5

38. —

—

40. √ 25 + 9

7+6 2

41. —

Solve the equation. (Section 1.2 and Section 1.3) 42. 5x + 7 = 9x − 17

394

Chapter 8

3+y 2

43. — = 6

Basics of Geometry

−5 + x 2

44. — = −9

45. −6x − 13 = −x − 23

B

8.3

Using Midpoint and Distance Formulas Essential Question

How can you find the midpoint and length of a line segment in a coordinate plane? Finding the Midpoint of a Line Segment Work with a partner. Use centimeter graph paper.

—, where the points A a. Graph AB and B are as shown. —, b. Explain how to bisect AB — that is, to divide AB into two congruent line segments. Then — and use the result bisect AB —. to find the midpoint M of AB c. What are the coordinates of the midpoint M? d. Compare the x-coordinates of A, B, and M. Compare the y-coordinates of A, B, and M. How are the coordinates of the midpoint M related to the coordinates of A and B?

MAKING SENSE OF PROBLEMS To be proficient in math, you need to check your answers and continually ask yourself, “Does this make sense?”

4

A(3, 4)

2

−4

−2

2

4

−2

B(−5, −2)

−4

Finding the Length of a Line Segment Work with a partner. Use centimeter graph paper.

a. Add point C to your graph as shown. b. Use the Pythagorean Theorem —. to find the length of AB c. Use a centimeter ruler to verify the length you found in part (b). d. Use the Pythagorean Theorem and point M from Exploration 1 to find the lengths of — and MB —. What can AM you conclude?

15 14 13 12 11

4

10

A(3, 3 4)

9 8 7

2

6 5 4 34 −4

−2 2

2

4

2 1 cm

B(−5, B(− −5, − −2) 2)

−2 2

C(3, −2) 2

−4

Communicate Your Answer 3. How can you find the midpoint and length of a line segment in a coordinate

plane? 4. Find the coordinates of the midpoint M and the length of the line segment whose

endpoints are given. a. D(−10, −4), E(14, 6) Section 8.3

b. F(−4, 8), G(9, 0) Using Midpoint and Distance Formulas

395

8.3 Lesson

What You Will Learn Find segment lengths using midpoints and segment bisectors. Use the Midpoint Formula.

Core Vocabul Vocabulary larry

Use the Distance Formula.

midpoint, p. 396 segment bisector, p. 396

Midpoints and Segment Bisectors

Core Concept Midpoints and Segment Bisectors The midpoint of a segment is the point that divides the segment into two congruent segments.

READING

A

The word bisect means “to cut into two equal parts.”

M

B

—

M is the midpoint of AB .

— —

So, AM ≅ MB and AM = MB.

A segment bisector is a point, ray, line, line segment, or plane that intersects the segment at its midpoint. A midpoint or a segment bisector bisects a segment. C M

A

B

D

—. ⃖##⃗ CD is a segment bisector of AB — —

So, AM ≅ MB and AM = MB.

Finding Segment Lengths

— bisects XY — at point T, and XT = 39.9 cm. I the skateboard design, VW In Find F XY.

Y

SOLUTION

—. So, XT = TY = 39.9 cm. Point T is the midpoint of XY P XY = XT + TY

V T

Segment Addition Postulate

= 39.9 + 39.9

Substitute.

= 79.8

Add.

W

— is 79.8 centimeters. So, the length of XY

Monitoring Progress

X

Help in English and Spanish at BigIdeasMath.com

—. Then find PQ. Identify the segment bisector of PQ 1.

2.

7

18 P

M

Q N

396

Chapter 8

Basics of Geometry

2

27 P

M

Q

Using Algebra with Segment Lengths

—. Find the length of VM —. Point M is the midpoint of VW 4x − 1 V

3x + 3 M

W

SOLUTION Step 1 Write and solve an equation. Use the fact that VM = MW. VM = MW

Write the equation.

4x − 1 = 3x + 3

Substitute.

x−1=3 x=4

Check Because VM = MW, the length of — MW should be 15. MW = 3x + 3 = 3(4) + 3 = 15

Subtract 3x from each side.



Add 1 to each side.

Step 2 Evaluate the expression for VM when x = 4. S VM = 4x − 1 = 4(4) − 1 = 15

— is 15. So, the length of VM

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

—. 3. Identify the segment bisector of PQ

— 4. Identify the segment bisector of RS.

Then find MQ.

Then find RS. n

5x − 3 P

11 − 2x M

4x + 3 Q

R

6x − 12 M

S

Bisecting a Segment

— by paper folding. Then find the midpoint M of AB. Construct a segment bisector of AB —

SOLUTION Step 1

Step 2

Step 3

Draw the segment — on a piece Draw AB of paper.

Fold the paper Fold the paper so that B is on top of A.

Label the midpoint Label point M. Compare AM, MB, and AB.

AM = MB = —12 AB

Section 8.3

Using Midpoint and Distance Formulas

397

Using the Midpoint Formula You can use the coordinates of the endpoints of a segment to find the coordinates of the midpoint.

Core Concept The Midpoint Formula The coordinates of the midpoint of a segment are the averages of the x-coordinates and of the y-coordinates of the endpoints. If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the midpoint M — has coordinates of AB x1 + x2 y1 + y2 —, — . 2 2

(

y y2

B(x2, y2)

y1 + y2 2 y1

)

M

(

x1 + x2 y1 + y2 , 2 2

)

A(x1, y1) x1 + x2 2

x1

x2

x

Using the Midpoint Formula

— are R(1, –3) and S(4, 2). Find the coordinates of the a. The endpoints of RS midpoint M.

— is M(2, 1). One endpoint is J(1, 4). Find the coordinates b. The midpoint of JK of endpoint K. SOLUTION

y

a. Use the Midpoint Formula.

) (

(

5 1 1 + 4 −3 + 2 M —, — = M —, − — 2 2 2 2

S(4, 2)

2

) (

1

)

The coordinates of the midpoint M are —52 , −—2 .

2 −2

b. Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula. Step 1 Find x. 1+x —=2 2 1+x=4 x=3

Step 2 Find y. 4+y —=1 2 4+y=2

Monitoring Progress

x

R(1, −3)

4

y

J(1, 4)

2

y = −2

The coordinates of endpoint K are (3, −2).

4

M(?, ?)

M(2, 1) 2

−2

4

x

K(x, y)

Help in English and Spanish at BigIdeasMath.com

— are A(1, 2) and B(7, 8). Find the coordinates of the 5. The endpoints of AB midpoint M.

— are C(−4, 3) and D(−6, 5). Find the coordinates of the 6. The endpoints of CD midpoint M.

— is M(2, 4). One endpoint is T(1, 1). Find the coordinates of 7. The midpoint of TU endpoint U.

— is M(−1, −2). One endpoint is W(4, 4). Find the coordinates 8. The midpoint of VW of endpoint V. 398

Chapter 8

Basics of Geometry

Using the Distance Formula You can use the Distance Formula to find the distance between two points in a coordinate plane.

Core Concept The Distance Formula

READING The red mark at the corner of the triangle that makes a right angle indicates a right triangle.

y

B(x2, y2)

If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is

"y2 − y1" A(x1, y1)

——

AB = √(x2 – x1)2 + (y2 – y1)2 .

"x2 − x1" C(x2, y1) x

The Distance Formula is related to the Pythagorean Theorem, which you will see again when you work with right triangles in a future course. Distance Formula

Pythagorean Theorem

(AB)2 = (x2 − x1)2 + (y2 − y1)2

c 2 = a2 + b 2

B(x2, y2)

y

c

b

"y2 − y1" A(x1, y1)

"x2 − x1"

a

C(x2, y1) x

Using the Distance Formula Your school is 4 miles east and 1 mile south of your apartment. A recycling center, where your class is going on a field trip, is 2 miles east and 3 miles north of your apartment. Estimate the distance between the recycling center and your school.

SOLUTION You can model the situation using a coordinate plane with your apartment at the origin (0, 0). The coordinates of the recycling center and the school are R(2, 3) and S(4, −1), respectively. Use the Distance Formula. Let (x1, y1) = (2, 3) and (x2, y2) = (4, −1). ——

RS = √ (x2 − x1)2 + (y2 − y1)2 ——

= √ (4 −

2)2

+ (−1 −

—

READING The symbol ≈ means “is approximately equal to.”

3)2

Distance Formula Substitute.

= √ 22 + (−4)2

Subtract.

= √ 4 + 16

Evaluate powers.

— —

4

= √ 20

Add.

≈ 4.5

Use a calculator.

N

R(2, 3)

2

2

W −2

E

S(4, −1) S

So, the distance between the recycling center and your school is about 4.5 miles.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

9. In Example 4, a park is 3 miles east and 4 miles south of your apartment. Find the

distance between the park and your school. Section 8.3

Using Midpoint and Distance Formulas

399

8.3

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY If a point, ray, line, line segment, or plane intersects a segment at its midpoint,

then what does it do to the segment?

—, with endpoints A(−7, 5) and B(4, −6), 2. COMPLETE THE SENTENCE To find the length of AB you can use the _____________.

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, identify the segment bisector of — RS . Then find RS. (See Example 1.) 3.

9.

k

17 R

In Exercises 9 and 10, identify the segment bisector of —. Then find XY. (See Example 2.) XY

M

3x + 1

8x − 24

X

S

M

Y

N

4. A R

10.

9

M

S

5.

5x + 8 X

22 R

M

6.

n 9x + 12 M

Y

S

CONSTRUCTION In Exercises 11–14, copy the segment

s

and construct a segment bisector by paper folding. Then label the midpoint M.

12 R

M

S

11.

A

B

12.

In Exercises 7 and 8, identify the segment bisector of — JK . Then find JM. (See Example 2.) 7.

7x + 5 J

C

13.

K

E

14.

8. 3x + 15 J

F

8x M

D

G

8x + 25 M

K

H

400

Chapter 8

Basics of Geometry

In Exercises 15–18, the endpoints of — CD are given. Find the coordinates of the midpoint M. (See Example 3.) 15. C(3, −5) and D(7, 9) 16. C(−4, 7) and D(0, −3)

36. PROBLEM SOLVING In baseball, the strike zone is

18. C(−8, −6) and D(−4, 10)

In Exercises 19–22, the midpoint M and one endpoint of — GH are given. Find the coordinates of the other endpoint. (See Example 3.) 20. H(−3, 7) and M(−2, 5)

21. H(−2, 9) and M(8, 0)

(

the Midpoint Formula. a. Explain how to find the midpoint when given the two endpoints in your own words. b. Explain how to find the other endpoint when given one endpoint and the midpoint in your own words.

17. C(−2, 0) and D(4, 9)

19. G(5, −6) and M(4, 3)

35. WRITING Your friend is having trouble understanding

13

22. G(−4, 1) and M −— 2 , −6

the region a baseball needs to pass through for the umpire to declare it a strike when the batter does not swing. The top of the strike zone is a horizontal plane passing through the midpoint of the top of the batter’s shoulders and the top of the uniform pants when the player is in a batting stance. Find the height of T. (Note: All heights are in inches.)

)

In Exercises 23–30, find the distance between the two points. (See Example 4.)

60

23. A(13, 2) and B(7, 10)

24. C(−6, 5) and D(−3, 1)

T

25. E(3, 7) and F(6, 5)

26. G(−5, 4) and H(2, 6)

42

27. J(−8, 0) and K(1, 4)

28. L(7, −1) and M(−2, 4)

29. R(0, 1) and S(6, 3.5)

30. T(13, 1.6) and V(5.4, 3.7)

22

ERROR ANALYSIS In Exercises 31 and 32, describe and

0

correct the error in finding the distance between A(6, 2) and B(1, −4).



AB = (6 − 1)2 + [ 2 − (−4) ]2 = 25 + 36 = 61

32.



37. MODELING WITH MATHEMATICS The figure shows

the position of three players during part of a water polo match. Player A throws the ball to Player B, who then throws the ball to Player C.

= 52 + 62

y 16

———

AB = √ (6 − 2)2 + [ 1 − (−4) ]2 —

= √ 42 + 52

—

= √ 16 + 25 —

= √ 41

Distance (m)

31.

Player B

8

0

COMPARING SEGMENTS In Exercises 33 and 34, the endpoints of two segments are given. Find each segment length. Tell whether the segments are congruent. If they are not congruent, state which segment length is greater.

— AB : A(0, 2), B(−3, 8) and — CD : C(−2, 2), D(0, −4) —

(24, 14)

12

(18, 7)

Player A

4

≈ 6.4

33.

Player C

(8, 4) 0

4

8

12

16

20

24

28 x

Distance (m)

a. How far did Player A throw the ball? Player B? b. How far would Player A have to throw the ball to throw it directly to Player C?

—

34. EF : E(1, 4), F(5, 1) and GH : G(−3, 1), H(1, 6)

Section 8.3

Using Midpoint and Distance Formulas

401

38. MODELING WITH MATHEMATICS Your school is

41. MATHEMATICAL CONNECTIONS Two points are

20 blocks east and 12 blocks south of your house. The mall is 10 blocks north and 7 blocks west of your house. You plan on going to the mall right after school. Find the distance between your school and the mall assuming there is a road directly connecting the school and the mall. One block is 0.1 mile.

located at (a, c) and (b, c). Find the midpoint and the distance between the two points.

— contains midpoint M and 42. HOW DO YOU SEE IT? AB points C and D, as shown. Compare the lengths. If you cannot draw a conclusion, write impossible to tell. Explain your reasoning.

39. PROBLEM SOLVING A path goes around a triangular

park, as shown.

Distance (yd)

A y

60

D

M

b. AC and MB c. MC and MD

20

d. MB and DB R

Q 0

20

40

60

80

x

43. ABSTRACT REASONING Use the diagram in

— represent locations you Exercise 42. The points on AB pass on your commute to work. You travel from your home at location A to location M before realizing that you left your lunch at home. You could turn around to get your lunch and then continue to work at location B. Or you could go to work and go to location D for lunch today. You want to choose the option that involves the least distance you must travel. Which option should you choose? Explain your reasoning.

Distance (yd)

a. Find the distance around the park to the nearest yard. b. A new path and a bridge are constructed from —. Find QM to point Q to the midpoint M of PR the nearest yard. c. A man jogs from P to Q to M to R to Q and back to P at an average speed of 150 yards per minute. About how many minutes does it take? Explain your reasoning.

44. THOUGHT PROVOKING Describe three ways to

divide a rectangle into two congruent regions. Do the regions have to be triangles? Use a diagram to support your answer.

40. MAKING AN ARGUMENT Your friend claims there

is an easier way to find the length of a segment than the Distance Formula when the x-coordinates of the endpoints are equal. He claims all you have to do is subtract the y-coordinates. Do you agree with his statement? Explain your reasoning.

— is 45. ANALYZING RELATIONSHIPS The length of XY

— is M, and C is on 24 centimeters. The midpoint of XY 2 — — so that XM so that XC is —3 of XM. Point D is on MY —? MD is —34 of MY. What is the length of CD

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Find the perimeter and area of the figure. (Skills Review Handbook) 46.

47.

48.

49. 5m

3m

10 ft

13 yd

12 yd

4m 5 cm

5 yd 5 yd

3 ft

Solve the inequality. Graph the solution. (Section 2.2 and Section 2.3) 50. a + 18 < 7

402

B

a. AM and MB

P

40

0

C

Chapter 8

51. y − 5 ≥ 8

Basics of Geometry

52. −3x > 24

z 4

53. — ≤ 12

8.1–8.3

What Did You Learn?

Core Vocabulary undefined terms, p. 380 point, p. 380 line, p. 380 plane, p. 380 collinear points, p. 380 coplanar points, p. 380 defined terms, p. 381

line segment, or segment, p. 381 endpoints, p. 381 ray, p. 381 opposite rays, p. 381 intersection, p. 382 postulate, p. 388 axiom, p. 388

coordinate, p. 388 distance, p. 388 construction, p. 389 congruent segments, p. 389 between, p. 390 midpoint, p. 396 segment bisector, p. 396

Core Concepts Section 8.1 Undefined Terms: Point, Line, and Plane, p. 380 Defined Terms: Segment and Ray, p. 381

Intersections of Lines and Planes, p. 382

Section 8.2 Ruler Postulate, p. 388 Congruent Segments, p. 389

Segment Addition Postulate, p. 390

Section 8.3 Midpoints and Segment Bisectors, p. 396 The Midpoint Formula, p. 398

The Distance Formula, p. 399

Mathematical Practices 1.

Sketch an example of the situation described in Exercise 49 on page 386 in a coordinate plane. Label your figure.

2.

Explain how you arrived at your answer for Exercise 35 on page 394.

3.

What assumptions did you make when solving Exercise 43 on page 402?

Keeping Your Mind Focused • Keep a notebook just for vocabulary, formulas, and core concepts. • Review this notebook before completing homework and before tests.

403 40 03

8.1–8.3

Quiz

Use the diagram. (Section 8.1) 1. Name four points.

2. Name three collinear points.

3. Name two lines.

4. Name three coplanar points.

5. Name the plane that is

6. Give two names for the plane

shaded green.

H G

that is shaded blue.

7. Name three line segments.

K

A L B

F

C

D

E

8. Name three rays.

Sketch the figure described. (Section 8.1) 9. !!!⃗ QR and ⃖!!⃗ QS

10. plane P intersecting ⃖!!⃗ YZ at Z

— and CD — are congruent. (Section 8.2) Plot the points in a coordinate plane. Then determine whether AB 11. A(−3, 3), B(1, 3), C(3, 2), D(3, −2)

12. A(−8, 7), B(1, 7), C(−3, −6), D(5, −6)

Find AC. (Section 8.2) 13.

A

13

26

B

C

14.

62 A

11

C

B

Find the coordinates of the midpoint M and the distance between the two points. (Section 8.3) 15. J(4, 3) and K(2, −3)

16. L(−4, 5) and N(5, −3)

—. Then find RS. 18. Identify the segment bisector of RS (Section 8.3)

17. P(−6, −1) and Q(1, 2) 6x − 2

3x + 7

R

M

S

— is M(0, 1). One endpoint is J(−6, 3). Find the coordinates of 19. The midpoint of JK endpoint K. (Section 8.3) 20. Your mom asks you to run some errands on your way home from school. She wants

you to stop at the post office and the grocery store, which are both on the same straight road between your school and your house. The distance from your school to the post office is 376 yards, the distance from the post office to your house is 929 yards, and the distance from the grocery store to your house is 513 yards. (Section 8.2) a. Where should you stop first? b. What is the distance from the post office to the grocery store? c. What is the distance from your school to your house? d. You walk at a speed of 75 yards per minute. How long does it take you to walk straight home from school? Explain your answer.

y

8 6

(0, 6)

(6, 6)

21. The figure shows a coordinate plane on a baseball field.

The distance from home plate to first base is 90 feet. The pitching mound is the midpoint between home plate and second base. Find the distance from home plate to second base. Find the distance between home plate and the pitching mound. Explain how you found your answers. (Section 8.3) 404

Chapter 8

Basics of Geometry

4

(3, 3)

2 0

(0, 0) 0

2

(6, 0) 4

6

8 x

8.4

Perimeter and Area in the Coordinate Plane Essential Question

How can you find the perimeter and area of a

polygon in a coordinate plane? Finding the Perimeter and Area of a Quadrilateral Work with a partner. a. On a piece of centimeter graph paper, draw quadrilateral ABCD in a coordinate plane. Label the points A(1, 4), B(−3, 1), C(0, −3), and D(4, 0).

CONNECTIONS TO ALGEBRA In this exploration, you expand your work on perimeter and area into the coordinate plane.

4

2

B(−3, 1)

b. Find the perimeter of quadrilateral ABCD.

A(1, 4)

D(4, 0) −4

c. Are adjacent sides of quadrilateral ABCD perpendicular to each other? How can you tell?

−2

2

4

−2

C(0, −3)

d. What is the definition of a square? Is quadrilateral ABCD a square? Justify your answer. Find the area of quadrilateral ABCD.

−4

Finding the Area of a Polygon Work with a partner.

LOOKING FOR STRUCTURE To be proficient in math, you need to visualize single objects as being composed of more than one object.

a. Partition quadrilateral ABCD into four right triangles and one square, as shown. Find the coordinates of the vertices for the five smaller polygons.

4

2

B(−3, 1)

b. Find the areas of the five smaller polygons.

A(1, 4)

S

P D(4, 0)

−4

Area of Triangle BPA:

−2 2

R

Q 2

4

−2 2

Area of Triangle AQD: Area of Triangle DRC:

C(0, −3) −4

Area of Triangle CSB: Area of Square PQRS:

c. Is the sum of the areas of the five smaller polygons equal to the area of quadrilateral ABCD? Justify your answer.

Communicate Your Answer 3. How can you find the perimeter and area of a polygon in a coordinate plane? 4. Repeat Exploration 1 for quadrilateral EFGH, where the coordinates of the

vertices are E(−3, 6), F(−7, 3), G(−1, −5), and H(3, −2). Section 8.4

Perimeter and Area in the Coordinate Plane

405

8.4 Lesson

What You Will Learn Classify polygons. Inscribe regular polygons.

Core Vocabul Vocabulary larry

Find perimeters and areas of polygons in the coordinate plane.

inscribed polygon, p. 407 circumscribed circle, p. 407 Previous polygon side vertex n-gon convex concave regular polygon

Classifying Polygons

Core Concept Polygons In geometry, a figure that lies in a plane is called a plane figure. Recall that a polygon is a closed plane figure formed by three or more line segments called sides. Each side intersects exactly two sides, one at each vertex, so that no two sides with a common vertex are collinear. You can name a polygon by listing the vertices in consecutive order.

C side BC

vertex D D

B

A E polygon ABCDE

The number of sides determines the name of a polygon, as shown in the table. You can also name a polygon using the term n-gon, where n is the number of sides. For instance, a 14-gon is a polygon with 14 sides. Number of sides

Type of polygon

3

Triangle

4

Quadrilateral

5

Pentagon

6

Hexagon

7

Heptagon

8

Octagon

9

Nonagon

10

Decagon

12

Dodecagon

n

n-gon

interior

convex polygon

A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon that is not convex is concave. When all the sides of a polygon have the same length and all the angles have the same measure, it is a regular polygon.

interior

concave polygon

Classifying Polygons Classify each polygon by the number of sides. Tell whether it is convex or concave. a.

b.

SOLUTION a. The polygon has four sides. So, it is a quadrilateral. The polygon is concave. b. The polygon has six sides. So, it is a hexagon. The polygon is convex.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Classify the polygon by the number of sides. Tell whether it is convex or concave. 1.

406

Chapter 8

Basics of Geometry

2.

Inscribing Regular Polygons

Core Concept Inscribed Polygon A polygon is an inscribed polygon when all its vertices lie on a circle. The circle that contains the vertices is a circumscribed circle.

circumscribed circle inscribed polygon

Inscribing Regular Polygons Use a compass to construct each polygon inscribed in a circle, where the radius is equal to the length of the segment shown. a. regular hexagon

b. equilateral triangle

SOLUTION The constructions of the regular hexagon and equilateral triangle are the same except for the last step. Step 1

Step 2

C

Construct a circle Draw a point and label it C. Set the compass to the length of the given segment. Place the compass at C and draw a full circle.

Step 3

Step 4

C

C

Mark a point Mark a point anywhere on the circle.

Mark another point Keep the same compass setting. Place the compass at the point you marked and mark another point on the circle.

Step 5(a)

C

Repeat Step 3 Repeat Step 3 until you have marked six points.

Step 5(b)

C

C

Connect the points Use a straightedge to connect the marked points, as shown. The resulting figure is a regular hexagon. Section 8.4

Connect the points Use a straightedge to connect the marked points, as shown. The resulting figure is an equilateral triangle. Perimeter and Area in the Coordinate Plane

407

Finding Perimeter and Area in the Coordinate Plane Finding Perimeter in the Coordinate Plane Find the perimeter of △ABC with vertices A(−2, 3), B(3, −3), and C(−2, −3).

READING You can read the notation △ABC as “triangle A B C.”

SOLUTION Step 1 Draw the triangle in a coordinate plane. Then find the length of each side. Side — AB : AB = √ (x2 − x1)2 + (y2 − y1)2

——

A(−2, 3)

4

y

———

2

−4

2

4 x

−2

C(−2, −3)

B(3, −3)

−4

Distance Formula

= √ [3 − (−2)]2 + (−3 − 3)2

Substitute.

≈ 7.81

Use a calculator.

Side — BC : BC = ∣ −2 − 3 ∣ = 5

Ruler Postulate

Side — CA : CA = ∣ 3 − (−3) ∣ = 6

Ruler Postulate

Step 2 Find the sum of the side lengths. AB + BC + CA ≈ 7.81 + 5 + 6 = 18.81 So, the perimeter of △ABC is about 18.81 units.

Finding Area in the Coordinate Plane 4

y

Find the area of △DEF with vertices D(1, 3), E(4, −3), and F(−4, −3).

D(1, 3)

SOLUTION

2

−4

2

4 x

−2

F(−4, −3)

−4

Step 1 Draw the triangle in a coordinate plane by plotting the vertices and connecting them. Step 2 Find the lengths of the base and height. The length of the base is

E(4, −3)

FE = ∣ 4 − (−4) ∣ = ∣ 8 ∣ = 8 units.

—. By counting The height is the distance from point D to line segment FE grid lines, you can determine that the height is 6 units.

REMEMBER Perimeter has linear units, such as feet or meters. Area has square units, such as square feet or square meters.

Step 3 Substitute the values for the base and height into the formula for the area of a triangle. A = —12 bh = —12 (8)(6) = 24 So, the area of △DEF is 24 square units.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Find the perimeter of the polygon with the given vertices. 3. D(−3, 2), E(4, 2), F(4, −3)

4. K(−1, 1), L(4, 1), M(2, −2), N(−3, −2)

Find the area of the polygon with the given vertices. 5. G(2, 2), H(3, −1), J(−2, −1) 6. N(−1, 1), P(2, 1), Q(2, −2), R(−1, −2)

408

Chapter 8

Basics of Geometry

Modeling with Mathematics You are building a shed in your backyard. The diagram shows the four vertices of the shed. Each unit in the coordinate plane represents 1 foot. Find the area of the floor of the shed.

8

y

G(2, 7)

H(8, 7)

K(2, 2)

J(8, 2)

6 4 2

1 ft 2

4

6

8 x

1 ft

SOLUTION 1. Understand the Problem You are given the coordinates of a shed. You need to find the area of the floor of the shed. 2. Make a Plan The shed is rectangular, so use the coordinates to find the length and width of the shed. Then use a formula to find the area. 3. Solve the Problem Step 1 Find the length and width. Length GH = ∣ 8 − 2 ∣ = 6

Ruler Postulate

Width GK = ∣ 7 − 2 ∣ = 5

Ruler Postulate

The shed has a length of 6 feet and a width of 5 feet. Step 2 Substitute the values for the length and width into the formula for the area of a rectangle. A = ℓw

Write the formula for area of a rectangle.

= (6)(5)

Substitute.

= 30

Multiply.

So, the area of the floor of the shed is 30 square feet.

4. Look Back Make sure your answer makes sense in the context of the problem. Because you are finding an area, your answer should be in square units. An 2

y

M(2, 2) N(6, 2)



x

í

Monitoring Progress



Help in English and Spanish at BigIdeasMath.com

7. You are building a patio in your school’s courtyard. In the diagram at the left, the

í

1 ft

answer of 30 square feet makes sense in the context of the problem.

R(2, -3)

P(6, -3)

coordinates represent the four vertices of the patio. Each unit in the coordinate plane represents 1 foot. Find the area of the patio.

1 ft

Section 8.4

Perimeter and Area in the Coordinate Plane

409

Exercises

8.4

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE The perimeter of a square with side length s is P = ____. 2. WRITING What formulas can you use to find the area of a triangle in a coordinate plane?

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, classify the polygon by the number of sides. Tell whether it is convex or concave. (See Example 1.) 3.

4.

14.

y

F(−2, 4) E(−2, 2) −4

A(0, 4) 2

B(2, 0)

−2

4x

D(0, −2)

5.

6.

C(2, −2)

In Exercises 15–18, find the area of the polygon with the given vertices. (See Example 3.) 15. E(3, 1), F(3, −2), G(−2, −2) CONSTRUCTION In Exercises 7 and 8, use a compass

to construct a circle, where the radius is equal to the length of the given segment. Then inscribe a hexagon and an equilateral triangle in the circle.

16. J(−3, 4), K(4, 4), L(3, −3) 17. W(0, 0), X(0, 3), Y(−3, 3), Z(−3, 0)

7.

18. N(−2, 1), P(3, 1), Q(3, −1), R(−2, −1)

8.

In Exercises 19–22, use the diagram.

In Exercises 9–14, find the perimeter of the polygon with the given vertices. (See Example 2.)

y

A(−5, 4)

9. G(2, 4), H(2, −3), J(−2, −3), K(−2, 4) 10. Q(−3, 2), R(1, 2), S(1, −2), T(−3, −2) 11. U(−2, 4), V(3, 4), W(3, −4) 12. X(−1, 3), Y(3, 0), Z(−1, −2) 13.

4 2

F(−2, 1) −4

−2

2

−4

2 −4

−2

P(−1, −2)

2

D(4, −5)

19. Find the perimeter of △CDE.

N(2, 0) 4

x

M(4, 0)

20. Find the perimeter of rectangle BCEF. 21. Find the area of △ABF. 22. Find the area of quadrilateral ABCD.

410

Chapter 8

Basics of Geometry

6 x

E(2, −3)

−6

L(1, 4)

4

C(4, −1)

−2

y 4

B(0, 3)

ERROR ANALYSIS In Exercises 23 and 24, describe and

26. Determine which points are the remaining vertices of

correct the error in finding the perimeter or area of the polygon.

a rectangle with a perimeter of 14 units.

23.

B C(−2, −2) and D(−2, 2) ○



y

4

A A(2, −2) and B(2, −1) ○ C E(−2, −2) and F(2, −2) ○

2

D G(2, 0) and H(−2, 0) ○

−2

2

4

x

27. USING STRUCTURE Use the diagram.

P = 2ℓ+ 2w = 2(4) + 2(3) = 14

y 4

J(0, 4)

G(2, 2)

F(0, 2)

The perimeter is 14 units.

K(4, 4)

L(4, 0)

E(0, 0)

24.

−2



y 4

A(4, 3)

a. Find the areas of square EFGH and square EJKL. What happens to the area when the perimeter of square EFGH is doubled?

2 2

−2

C(1, 1)

4

6x

b. Is this true for every square? Explain.

b = ∣ 5 −1 ∣ = 4 h=

28. MODELING WITH MATHEMATICS You are

growing zucchini plants in your garden. In the figure, the entire garden is rectangle QRST. Each unit in the coordinate plane represents 1 foot. (See Example 4.)

——

√(5 − 4)2 + (1 − 3)2 —

= √5

≈ 2.2 A = —12 bh ≈ —12(4)(2.2) = 4.4

14

The area is about 4.4 square units.

12

R(7, 13)

6

Q(2, 1)

4 −2

Q(1, 13)

8

y 2

y

10

In Exercises 25 and 26, use the diagram.

P(−2, 1)

x

H(2, 0)

−2

B(5, 1)

4

2

2

U(1, 4) V(4, 4)

x 2

−2

T(1, 1) W(4, 1) S(7, 1) 2

25. Determine which point is the remaining vertex of a

4

6

8

10 x

triangle with an area of 4 square units.

a. Find the area of the garden.

A R(2, 0) ○

b. Zucchini plants require 9 square feet around each plant. How many zucchini plants can you plant?

B S(−2, −1) ○

c. You decide to use square TUVW to grow lettuce. You can plant four heads of lettuce per square foot. How many of each vegetable can you plant? Explain.

C T(−1, 0) ○ D U(2, −2) ○ Section 8.4

Perimeter and Area in the Coordinate Plane

411

29. MODELING WITH MATHEMATICS You are going for

32. THOUGHT PROVOKING Your bedroom has an area of

a hike in the woods. You hike to a waterfall that is 4 miles east of where you left your car. You then hike to a lookout point that is 2 miles north of your car. From the lookout point, you return to your car. a. Map out your route in a coordinate plane with your car at the origin. Let each unit in the coordinate plane represent 1 mile. Assume you travel along straight paths.

350 square feet. You are remodeling to include an attached bathroom that has an area of 150 square feet. Draw a diagram of the remodeled bedroom and bathroom in a coordinate plane. 33. PROBLEM SOLVING Use the diagram. L(−2, 2)

b. How far do you travel during the entire hike?

−3

M(2, 2)

−1

1

P(−2, −2)

3 x

N(2, −2)

a. Find the perimeter and area of the square.

30. HOW DO YOU SEE IT? Without performing any

b. Connect the midpoints of the sides of the given square to make a quadrilateral. Is this quadrilateral a square? Explain your reasoning.

calculations, determine whether the triangle or the rectangle has a greater area. Which one has a greater perimeter? Explain your reasoning.

−4

y

1

c. When you leave the waterfall, you decide to hike to an old wishing well before going to the lookout point. The wishing well is 3 miles north and 2 miles west of the lookout point. How far do you travel during the entire hike?

4

3

c. Find the perimeter and area of the quadrilateral you made in part (b). Compare this area to the area you found in part (a).

y

2

34. MAKING AN ARGUMENT Your friend claims that a

x

rectangle with the same perimeter as △QRS will have the same area as the triangle. Is your friend correct? Explain your reasoning.

−2 −4

y

Q(−2, 1)

31. MATHEMATICAL CONNECTIONS The lines

2

1

y1 = 2x − 6, y2 = −3x + 4, and y3 = −—2 x + 4 are the sides of a right triangle.

2

a. Use slopes to determine which sides are perpendicular.

R(−2, −2)

x

S(2, −2)

b. Find the vertices of the triangle. c. Find the perimeter and area of the triangle.

Maintaining Mathematical Proficiency

35. REASONING Triangle ABC has a perimeter of

12 units. The vertices of the triangle are A(x, 2), B(2, −2), and C(−1, 2). Find the value of x. Reviewing what you learned in previous grades and lessons

Solve the equation. (Section 1.2 and Section 1.3) 36. 3x − 7 = 2

37. 5x + 9 = 4

38. x + 4 = x − 12

39. 4x − 9 = 3x + 5

40. 11 − 2x = 5x − 3

41. — = 4x − 3

x+1 2

42. Use a compass and straightedge to construct a copy of the line segment. X

412

Chapter 8

Basics of Geometry

Y

(Section 8.2)

8.5

Measuring and Constructing Angles Essential Question

How can you measure and classify an angle?

Measuring and Classifying Angles Work with a partner. Find the degree measure of each of the following angles. Classify each angle as acute, right, or obtuse. D

C

0 10 180 170 1 20 3 60 15 0 4 01 0 40

E

a. ∠AOB e. ∠COE

B

170 180 60 0 1 20 10 0 15 0 30 14 0 4

80 90 10 0 70 10 0 90 80 110 1 70 2 0 60 0 11 60 0 13 2 0 1 5 0 50 0 13

O

b. ∠AOC f. ∠COD

c. ∠BOC g. ∠BOD

A

d. ∠BOE h. ∠AOE

Drawing a Regular Polygon Work with a partner. a. Use a ruler and protractor to draw the triangular pattern shown at the right. b. Cut out the pattern and use it to draw three regular hexagons, as shown below.

ATTENDING TO PRECISION To be proficient in math, you need to calculate and measure accurately and efficiently.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in.

2 in. 120°

2 in. 2 in. 2 in. 2 in.

c. The sum of the angle measures of a polygon with n sides is equal to 180(n − 2)°. Do the angle measures of your hexagons agree with this rule? Explain. d. Partition your hexagons into smaller polygons, as shown below. For each hexagon, find the sum of the angle measures of the smaller polygons. Does each sum equal the sum of the angle measures of a hexagon? Explain.

Communicate Your Answer 3. How can you measure and classify an angle?

Section 8.5

Measuring and Constructing Angles

413

8.5 Lesson

What You Will Learn Name angles.

Core Vocabul Vocabulary larry

Measure and classify angles.

angle, p. 414 vertex, p. 414 sides of an angle, p. 414 interior of an angle, p. 414 exterior of an angle, p. 414 measure of an angle, p. 415 acute angle, p. 415 right angle, p. 415 obtuse angle, p. 415 straight angle, p. 415 congruent angles, p. 416 angle bisector, p. 418

Use the Angle Addition Postulate to find angle measures.

Identify congruent angles. Bisect angles.

Naming Angles C

An angle is a set of points consisting of two different rays that have the same endpoint, called the vertex. The rays are the sides of the angle.

vertex 1

You can name an angle in several different ways.

A

• Use its vertex, such as ∠A. • Use a point on each ray and the vertex, such as ∠BAC or ∠CAB. • Use a number, such as ∠1.

Previous protractor degrees

sides

The region that contains all the points between the sides of the angle is the interior of the angle. The region that contains all the points outside the angle is the exterior of the angle.

B

exterior interior

Naming Angles

COMMON ERROR When a point is the vertex of more than one angle, you cannot use the vertex alone to name the angle.

A lighthouse keeper measures the angles formed by the lighthouse at point M and three boats. Name three angles shown in the diagram.

J lighthouse ligh htho o ouse

SOLUTION ∠JMK or ∠KMJ

K

M

∠KML or ∠LMK L

∠JML or ∠LMJ

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Write three names for the angle. 1.

2.

Q

414

Chapter 8

Basics of Geometry

E 2

R

P

3.

X 1 Z

Y F

D

Measuring and Classifying Angles A protractor helps you approximate the measure of an angle. The measure is usually given in degrees.

Protractor Postulate Consider ⃖!!⃗ OB and a point A on one side of ⃖!!⃗ OB. The rays of the form !!!⃗ OA can be matched one to one with the real numbers from 0 to 180. The measure of ∠AOB, which can be written as m∠AOB, is equal to the absolute value of the difference between the real numbers matched with !!!⃗ OA and !!!⃗ OB on a protractor.

80 90 10 0 70 10 0 90 80 110 1 70 2 60 0 110 60 0 1 2 3 50 0 1 50 0 3 1

A

O

B

170 180 60 0 1 20 10 0 15 0 30 14 0 4

You can classify angles according to their measures.

Core Concept Types of Angles

A

A

A

acute angle Measures greater than 0° and less than 90°

A

right angle

obtuse angle

straight angle

Measures 90°

Measures greater than 90° and less than 180°

Measures 180°

Measuring and Classifying Angles Find the measure of each angle. Then classify each angle. b. ∠JHL

c. ∠LHK

SOLUTION a. !!!⃗ HG lines up with 0° on the outer scale of the protractor. !!!⃗ HK passes through 125° on the outer scale. So, m∠GHK = 125°. It is an obtuse angle.

G

L

M

K

H

J

170 180 60 0 1 20 10 0 15 0 30 14 0 4

a. ∠GHK

80 90 10 0 70 10 0 90 80 110 1 70 2 60 0 110 60 0 1 2 3 50 0 1 50 0 13

0 10 180 170 1 20 3 60 15 0 4 01 0 40

Most protractors have an inner and an outer scale. When measuring, make sure you are using the correct scale.

Postulate

0 10 180 170 1 20 3 60 15 0 4 01 0 40

COMMON ERROR

b. !!!⃗ HJ lines up with 0° on the inner scale of the protractor. !!!⃗ HL passes through 90°. So, m∠JHL = 90°. It is a right angle. c. !!!⃗ HL passes through 90°. !!!⃗ HK passes through 55° on the inner scale. So, m∠LHK = ∣ 90 − 55 ∣ = 35°. It is an acute angle.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use the diagram in Example 2 to find the angle measure. Then classify the angle. 4. ∠JHM

5. ∠MHK

Section 8.5

6. ∠MHL

Measuring and Constructing Angles

415

Identifying Congruent Angles You can use a compass and straightedge to construct an angle that has the same measure as a given angle.

Copying an Angle Use a compass and straightedge to construct an angle that has the same measure as ∠A. In this construction, the center of an arc is the point where the compass point rests. The radius of an arc is the distance from the center of the arc to a point on the arc drawn by the compass.

SOLUTION Step 1

Step 2

Step 3 C

A

A

Step 4 C

A

B

C A

B

B

F D

E

D

Draw a segment Draw an angle such as ∠A, as shown. Then draw a segment. Label a point D on the segment.

E

D

Draw arcs Draw an arc with center A. Using the same radius, draw an arc with center D.

F E

D

Draw an arc Label B, C, and E. Draw an arc with radius BC and center E. Label the intersection F.

DF. Draw a ray Draw """⃗ ∠EDF ≅ ∠BAC.

Two angles are congruent angles when they have the same measure. In the construction above, ∠A and ∠D are congruent angles. So, m∠A = m∠D

The measure of angle A is equal to the measure of angle D.

∠A ≅ ∠D.

Angle A is congruent to angle D.

and

Identifying Congruent Angles a. Identify the congruent angles labeled in the quilt design. b. m∠ADC = 140°. What is m∠EFG?

G H E

SOLUTION

F

a. There are two pairs of congruent angles:

READING

∠ABC ≅ ∠FGH

In diagrams, matching arcs indicate congruent angles. When there is more than one pair of congruent angles, use multiple arcs.

and ∠ADC ≅ ∠EFG.

b. Because ∠ADC ≅ ∠EFG, m∠ADC = m∠EFG.

C D

So, m∠EFG = 140°. A

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

7. Without measuring, is ∠DAB ≅ ∠FEH in Example 3? Explain your reasoning.

Use a protractor to verify your answer. 416

Chapter 8

Basics of Geometry

B

Using the Angle Addition Postulate

Postulate Angle Addition Postulate R

Words

If P is in the interior of ∠RST, then the measure of ∠RST is equal to the sum of the measures of ∠RSP and ∠PST.

m∠RST m∠RSP

S

m∠PST

Symbols If P is in the interior of

P

∠RST, then

T

m∠RST = m∠RSP + m∠PST.

Finding Angle Measures Given that m∠LKN = 145°, find m∠LKM and m∠MKN.

M (2x + 10)° L (4x − 3)° K

SOLUTION

N

Step 1 Write and solve an equation to find the value of x. m∠LKN = m∠LKM + m∠MKN

Angle Addition Postulate

145° = (2x + 10)° + (4x − 3)°

Substitute angle measures.

145 = 6x + 7

Combine like terms.

138 = 6x

Subtract 7 from each side.

23 = x

Divide each side by 6.

Step 2 Evaluate the given expressions when x = 23.

⋅ ⋅

m∠LKM = (2x + 10)° = (2 23 + 10)° = 56° m∠MKN = (4x − 3)° = (4 23 − 3)° = 89° So, m∠LKM = 56°, and m∠MKN = 89°.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Find the indicated angle measures. 8. Given that ∠KLM is a straight angle,

find m∠KLN and m∠NLM.

9. Given that ∠EFG is a right angle,

find m∠EFH and m∠HFG.

N

E (2x + 2)°

(10x − 5)° (4x + 3)° K

L

(x + 1)°

M F

Section 8.5

H

G

Measuring and Constructing Angles

417

Bisecting Angles

X

An angle bisector is a ray that divides an angle into two angles that are congruent. In the figure, !!!⃗ YW bisects ∠XYZ, so ∠XYW ≅ ∠ZYW.

W

Y

You can use a compass and straightedge to bisect an angle.

Z

Bisecting an Angle Construct an angle bisector of ∠A with a compass and straightedge.

SOLUTION Step 2 C

Step 3 C

C

G

15 14 13 12

2

9 8 7

1

11 10

in.

Step 1

6 5 2 cm

1

B

5

6

Draw an arc Draw an angle such as ∠A, as shown. Place the compass at A. Draw an arc that intersects both sides of the angle. Label the intersections B and C.

A

B

4

A

B

3

A

4 3

Draw arcs Place the compass at C. Draw an arc. Then place the compass point at B. Using the same radius, draw another arc.

Draw a ray Label the intersection G. Use a straightedge to draw a ray through A and G.

!!!⃗ AG bisects ∠A.

Using a Bisector to Find Angle Measures

!!!⃗ QS bisects ∠PQR, and m∠PQS = 24°. Find m∠PQR. SOLUTION

P

Step 1 Draw a diagram. Q

Step 2 Because !!!⃗ QS bisects ∠PQR, m∠PQS = m∠RQS. So, m∠RQS = 24°. Use the Angle Addition Postulate to find m∠PQR. m∠PQR = m∠PQS + m∠RQS

S

24°

R

Angle Addition Postulate

= 24° + 24°

Substitute angle measures.

= 48°

Add.

So, m∠PQR = 48°.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

!!!⃗. 10. Angle MNP is a straight angle, and !!!⃗ NQ bisects ∠MNP. Draw ∠MNP and NQ Use arcs to mark the congruent angles in your diagram. Find the angle measures of these congruent angles. 418

Chapter 8

Basics of Geometry

Exercises

8.5

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE Two angles are ___________ angles when they have the same measure. 2. WHICH ONE DOESN’T BELONG? Which angle name does not belong with the other three? Explain

your reasoning. B 3

A

∠BCA

1

2

∠BAC

∠1

C

∠CAB

Monitoring Progress and Modeling with Mathematics In Exercises 3−6, write three names for the angle. (See Example 1.) 4.

A

80 90 10 0 70 10 0 90 80 110 1 70 20 60 0 110 60 13 2 50 0 1 50 0 13

C

0 10 180 170 1 20 60

G H

5.

6. Q

J

S 8

1 K

R

L

O

B

9. m∠AOC

10. m∠BOD

11. m∠COD

12. m∠EOD

correct the error in finding the angle measure. Use the diagram from Exercises 9−12. 13.

8.

M

E

ERROR ANALYSIS In Exercises 13 and 14, describe and

In Exercises 7 and 8, name three different angles in the diagram. (See Example 1.) 7.

A

D

C

170 180 60 0 1 20 10 0 15 0 30 14 0 4

B

F

3 15 0 4 01 0 40

3.

In Exercises 9−12, find the angle measure. Then classify the angle. (See Example 2.)



m∠BOC = 30°



m∠DOE = 65°

G C

N

H K

14. J F

Section 8.5

Measuring and Constructing Angles

419

CONSTRUCTION In Exercises 15 and 16, use a compass

and straightedge to copy the angle. 15.

In Exercises 25−30, find the indicated angle measures. Dynamic Solutions at BigIdeasMath.com (See Exampleavailable 4.)

16.

25. m∠ABC = 95°. Find m∠ABD and m∠DBC. (2x + 23)°

A

D (9x − 5)°

In Exercises 17–20, m∠AED = 34° and m∠EAD = 112°. (See Example 3.)

B

C

26. m∠XYZ = 117°. Find m∠XYW and m∠WYZ. A

B

Y

Z (−10x + 65)°

(6x + 44)° E

D

C

X W

17. Identify the angles congruent to ∠AED.

27. ∠LMN is a straight angle. Find m∠LMP and m∠NMP.

18. Identify the angles congruent to ∠EAD.

P

19. Find m∠BDC. 20. Find m∠ADB.

(−16x + 13)° (−20x + 23)° L

In Exercises 21−24, find the indicated angle measure. 21. Find m∠ABC.

22. Find m∠LMN.

A

B

28. ∠ABC is a straight angle. Find m∠ABX and m∠CBX. X

85°

21°

23°

N (14x + 70)° (20x + 8)°

M

C

23. m∠RST = 114°. Find m∠RSV.

A

B

29. Find m∠RSQ and m∠TSQ. (15x − 43)°

V

R Q

R 72° S

T

24. ∠GHK is a straight angle. Find m∠LHK. L

(8x + 18)° S

T

30. Find m∠DEH and m∠FEH. (10x + 21)° F H

79° G

H

K

13x° D

420

Chapter 8

N

P

L D

37°

M

Basics of Geometry

E

C

CONSTRUCTION In Exercises 31 and 32, copy the angle.

42. ANALYZING RELATIONSHIPS The map shows the

Then construct the angle bisector with a compass and straightedge. 31.

intersections of three roads. Malcom Way intersects Sydney Street at an angle of 162°. Park Road intersects Sydney Street at an angle of 87°. Find the angle at which Malcom Way intersects Park Road.

32.

Malcom Way Park

ey

Sydn

et Stre

Road

In Exercises 33–36, !!!⃗ QS bisects ∠PQR. Use the diagram and the given angle measure to find the indicated angle measures. (See Example 5.)

43. ANALYZING RELATIONSHIPS In the sculpture shown P

in the photograph, the measure of ∠LMN is 76° and the measure of ∠PMN is 36°. What is the measure of ∠LMP?

S

Q

R L

33. m∠PQS = 63°. Find m∠RQS and m∠PQR.

P

34. m∠RQS = 71°. Find m∠PQS and m∠PQR. 35. m∠PQR = 124°. Find m∠PQS and m∠RQS.

N

M

36. m∠PQR = 119°. Find m∠PQS and m∠RQS.

In Exercises 37– 40, !!!⃗ BD bisects ∠ABC. Find m∠ABD, m∠CBD, and m∠ABC. 37.

38.

A (6x + 14)°

A

USING STRUCTURE In Exercises 44–46, use the diagram D

D

B

B (7x − 18)° C

(3x + 29)° B

of the roof truss.

(3x + 6)°

C

E

39.

40.

(−4x + 33)° A

D

A

A (8x + 35)° D

G D

(2x + 81)° B

C

(11x + 23)° B

C

F

!!!⃗ bisects ∠ABC and ∠DEF, 44. In the roof truss, BG

C

m∠ABC = 112°, and ∠ABC ≅ ∠DEF. Find the measure of each angle.

41. WRITING Explain how to find m∠ABD when you are

given m∠ABC and m∠CBD. D

a. m∠DEF

b. m∠ABG

c. m∠CBG

d. m∠DEG

45. In the roof truss, ∠DGF is a straight angle and !!!⃗ GB

A

bisects ∠DGF. Find m∠DGE and m∠FGE. B

C

46. Name an example of each of the four types of angles

according to their measures in the diagram. Section 8.5

Measuring and Constructing Angles

421

47. MATHEMATICAL CONNECTIONS In ∠ABC, !!!⃗ BX is in

53. CRITICAL THINKING Two acute angles are added

the interior of the angle, m∠ABX is 12 more than 4 times m∠CBX, and m∠ABC = 92°. a. Draw a diagram to represent the situation.

together. What type(s) of angle(s) do they form? Explain your reasoning. 54. HOW DO YOU SEE IT? Use the diagram.

b. Write and solve an equation to find m∠ABX and m∠CBX.

W

48. THOUGHT PROVOKING The angle between the

V

minute hand and the hour hand of a clock is 90°. What time is it? Justify your answer. 46° X

49. ABSTRACT REASONING Classify the angles that

result from bisecting each type of angle. a. acute angle

b. right angle

c. obtuse angle

d. straight angle

Y

Z

a. Is it possible for ∠XYZ to be a straight angle? Explain your reasoning. b. What can you change in the diagram so that ∠XYZ is a straight angle?

50. ABSTRACT REASONING Classify the angles that

result from drawing a ray in the interior of each type of angle. Include all possibilities and explain your reasoning.

55. WRITING Explain the process of bisecting an angle in

a. acute angle

b. right angle

56. ANALYZING RELATIONSHIPS !!!⃗ SQ bisects ∠RST, !!!⃗ SP

c. obtuse angle

d. straight angle

your own words. Compare it to bisecting a segment. bisects ∠RSQ, and !!!⃗ SV bisects ∠RSP. The measure of ∠VSP is 17°. Find m∠TSQ. Explain.

51. CRITICAL THINKING The ray from the origin through

(4, 0) forms one side of an angle. Use the numbers below as x- and y-coordinates to create each type of angle in a coordinate plane. −2

−1

0

1

a. acute angle

b. right angle

c. obtuse angle

d. straight angle

57. ABSTRACT REASONING A bubble level is a tool used

to determine whether a surface is horizontal, like the top of a picture frame. If the bubble is not exactly in the middle when the level is placed on the surface, then the surface is not horizontal. What is the most realistic type of angle formed by the level and a horizontal line when the bubble is not in the middle? Explain your reasoning.

2

52. MAKING AN ARGUMENT Your friend claims it is

possible for a straight angle to consist of two obtuse angles. Is your friend correct? Explain your reasoning.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Solve the equation. (Section 1.1 and Section 1.2) 58. x + 67 = 180

59. x + 58 = 90

60. 16 + x = 90

61. 109 + x = 180

62. (6x + 7) + (13x + 21) = 180

63. (3x + 15) + (4x − 9) = 90

64. (11x − 25) + (24x + 10) = 90

65. (14x − 18) + (5x + 8) = 180

422

Chapter 8

Basics of Geometry

8.6

Describing Pairs of Angles Essential Question

How can you describe angle pair relationships and use these descriptions to find angle measures? Finding Angle Measures Work with a partner. The five-pointed star has a regular pentagon at its center. a. What do you notice about the following angle pairs? x° and y°

v°

y° and z°

y° z° w° x° 108°

x° and z° b. Find the values of the indicated variables. Do not use a protractor to measure the angles. x= y= z= w= v= Explain how you obtained each answer.

Finding Angle Measures Work with a partner. A square is divided by its diagonals into four triangles. a. What do you notice about the following angle pairs? a° and b° c° and d °

e°

c° and e°

d°

c°

b. Find the values of the indicated variables. Do not use a protractor to measure the angles.

a° b°

c= d= e= Explain how you obtained each answer.

ATTENDING TO PRECISION To be proficient in math, you need to communicate precisely with others.

Communicate Your Answer 3. How can you describe angle pair relationships and use these descriptions

to find angle measures? 4. What do you notice about the angle measures of complementary angles,

supplementary angles, and vertical angles? Section 8.6

Describing Pairs of Angles

423

8.6 Lesson

What You Will Learn Identify complementary and supplementary angles. Identify linear pairs and vertical angles.

Core Vocabul Vocabulary larry complementary angles, p. 424 supplementary angles, p. 424 adjacent angles, p. 424 linear pair, p. 426 vertical angles, p. 426 Previous vertex sides of an angle interior of an angle opposite rays

Using Complementary and Supplementary Angles Pairs of angles can have special relationships. The measurements of the angles or the positions of the angles in the pair determine the relationship.

Core Concept Complementary and Supplementary Angles A

C 115°

1

3 4

20°

2

70°

65°

B

∠1 and ∠2

∠A and ∠B

D

∠3 and ∠4

complementary angles Two positive angles whose measures have a sum of 90°. Each angle is the complement of the other.

∠C and ∠D

supplementary angles Two positive angles whose measures have a sum of 180°. Each angle is the supplement of the other.

Adjacent Angles Complementary angles and supplementary angles can be adjacent angles or nonadjacent angles. Adjacent angles are two angles that share a common vertex and side, but have no common interior points. common side 5

7

6

8

common vertex

∠5 and ∠6 are adjacent angles.

∠7 and ∠8 are nonadjacent angles.

Identifying Pairs of Angles

COMMON ERROR In Example 1, ∠DAC and ∠DAB share a common vertex and a common side. But they also share common interior points. So, they are not adjacent angles.

In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

SOLUTION

R

D C

127° A 37°

53° S

T

B

Because 37° + 53° = 90°, ∠BAC and ∠RST are complementary angles. Because 127° + 53° = 180°, ∠CAD and ∠RST are supplementary angles. Because ∠BAC and ∠CAD share a common vertex and side, they are adjacent angles.

424

Chapter 8

Basics of Geometry

Finding Angle Measures a. ∠1 is a complement of ∠2, and m∠1 = 62°. Find m∠2.

COMMON ERROR Do not confuse angle names with angle measures.

b. ∠3 is a supplement of ∠4, and m∠4 = 47°. Find m∠3.

SOLUTION a. Draw a diagram with complementary adjacent angles to illustrate the relationship.

1

62°

m∠2 = 90° − m∠1 = 90° − 62° = 28° b. Draw a diagram with supplementary adjacent angles to illustrate the relationship.

47° 4

m∠3 = 180° − m∠4 = 180° − 47° = 133°

Monitoring Progress

2

3

Help in English and Spanish at BigIdeasMath.com

In Exercises 1 and 2, use the figure.

F

1. Name a pair of complementary angles, a pair of

G 41°

H

131°

supplementary angles, and a pair of adjacent angles. 2. Are ∠KGH and ∠LKG adjacent angles? Are

K

∠FGK and ∠FGH adjacent angles? Explain.

49° L

3. ∠1 is a complement of ∠2, and m∠2 = 5°. Find m∠1. 4. ∠3 is a supplement of ∠4, and m∠3 = 148°. Find m∠4.

Real-Life Application W When viewed from the side, the frame of a bball-return net forms a pair of supplementary aangles with the ground. Find m∠BCE aand m∠ECD.

E (x + 2)° (4x + 8)° C

D

B

SOLUTION S

Step 1 Use the fact that the sum of the measures of supplementary angles is 180°. S m∠BCE + m∠ECD = 180°

Write an equation.

(4x + 8)° + (x + 2)° = 180°

Substitute angle measures.

5x + 10 = 180 x = 34

Combine like terms. Solve for x.

Step 2 Evaluate the given expressions when x = 34. S m∠BCE = (4x + 8)° = (4 ∙ 34 + 8)° = 144° m∠ECD = (x + 2)° = (34 + 2)° = 36° So, m∠BCE = 144° and m∠ECD = 36°.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

5. ∠LMN and ∠PQR are complementary angles. Find the measures of the angles

when m∠LMN = (4x − 2)° and m∠PQR = (9x + 1)°. Section 8.6

Describing Pairs of Angles

425

Using Other Angle Pairs

Core Concept Linear Pairs and Vertical Angles Two adjacent angles are a linear pair when their noncommon sides are opposite rays. The angles in a linear pair are supplementary angles.

Two angles are vertical angles when their sides form two pairs of opposite rays.

common side 3 1 2 noncommon side noncommon side

∠1 and ∠2 are a linear pair.

4 5

6

∠3 and ∠6 are vertical angles. ∠4 and ∠5 are vertical angles.

Identifying Angle Pairs

COMMON ERROR In Example 4, one side of ∠1 and one side of ∠3 are opposite rays. But the angles are not a linear pair because they are nonadjacent.

Identify all the linear pairs and all the vertical angles in the figure.

SOLUTION 1

To find vertical angles, look for angles formed by intersecting lines.

2 4

3 5

∠1 and ∠5 are vertical angles. To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays. ∠1 and ∠4 are a linear pair. ∠4 and ∠5 are also a linear pair.

Finding Angle Measures in a Linear Pair Two angles form a linear pair. The measure of one angle is five times the measure of the other angle. Find the measure of each angle.

SOLUTION Step 1 Draw a diagram. Let x° be the measure of one angle. The measure of the other angle is 5x°. 5x°

x°

Step 2 Use the fact that the angles of a linear pair are supplementary to write an equation. x° + 5x° = 180° 6x = 180 x = 30

Write an equation. Combine like terms. Divide each side by 6.

The measures of the angles are 30° and 5(30°) = 150°. 426

Chapter 8

Basics of Geometry

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

6. Do any of the numbered angles

in the figure form a linear pair? Which angles are vertical angles? Explain your reasoning. 1 6

7. The measure of an angle is twice

the measure of its complement. Find the measure of each angle.

2 5

3 4

8. Two angles form a linear pair.

The measure of one angle is 1—12 times the measure of the other angle. Find the measure of each angle.

Concept Summary Interpreting a Diagram There are some things you can conclude from a diagram, and some you cannot. For example, here are some things that you can conclude from the diagram below. D

A

E

B

C

YOU CAN CONCLUDE • All points shown are coplanar. • Points A, B, and C are collinear, and B is between A and C. • ⃖""⃗ AC, """⃗ BD, and """⃗ BE intersect at point B. • ∠DBE and ∠EBC are adjacent angles, and ∠ABC is a straight angle. • Point E lies in the interior of ∠DBC. Here are some things you cannot conclude from the diagram above. YOU CANNOT CONCLUDE

— ≅ BC —. • AB

• ∠DBE ≅ ∠EBC. • ∠ABD is a right angle. To make such conclusions, the following information must be given. D

A

Section 8.6

E

B

C

Describing Pairs of Angles

427

Exercises

8.6

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Explain what is different between adjacent angles and vertical angles. 2. WHICH ONE DOESN’T BELONG? Which one does not belong with the other three?

Explain your reasoning.

A

1 2

50°

40°

3

T

4

65°

B

S

25°

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, use the figure. (See Example 1.) M F

124° 41° G

H

56° 34° N 49° J P

L

E

K

3. Name a pair of adjacent complementary angles. 4. Name a pair of adjacent supplementary angles. 5. Name a pair of nonadjacent complementary angles. 6. Name a pair of nonadjacent supplementary angles.

In Exercises 7–10, find the angle measure. (See Example 2.) 7. ∠1 is a complement of ∠2, and m∠1 = 23°.

Find m∠2. 8. ∠3 is a complement of ∠4, and m∠3 = 46°.

Find m∠4. 9. ∠5 is a supplement of ∠6, and m∠5 = 78°.

Find m∠6.

12.

B (15x − 2)° C (7x + 4)° A

D

13. ∠UVW and ∠XYZ are complementary angles,

m∠UVW = (x − 10)°, and m∠XYZ = (4x − 10)°.

14. ∠EFG and ∠LMN are supplementary angles,

(

)

m∠EFG = (3x + 17)°, and m∠LMN = —12x − 5 °. In Exercises 15–18, use the figure. (See Example 4.) 15. Identify the linear pair(s) that

include ∠1.

16. Identify the linear pair(s) that

include ∠7.

17. Are ∠6 and ∠8 vertical angles?

Explain your reasoning.

1 2 3 5 4 6 7

9 8

18. Are ∠2 and ∠5 vertical angles?

Explain your reasoning.

10. ∠7 is a supplement of ∠8, and m∠7 = 109°.

Find m∠8.

In Exercises 19–22, find the measure of each angle. (See Example 5.)

In Exercises 11–14, find the measure of each angle. (See Example 3.)

19. Two angles form a linear pair. The measure of one

11.

20. Two angles form a linear pair. The measure of one

T

(3x + 5)° Q

428

angle is twice the measure of the other angle. angle is —13 the measure of the other angle.

(10x − 7)° R

Chapter 8

21. The measure of an angle is nine times the measure of S

Basics of Geometry

its complement.

1

22. The measure of an angle is —4 the measure of its

complement.

ERROR ANALYSIS In Exercises 23 and 24, describe

and correct the error in identifying pairs of angles in the figure.

28. REASONING The foul lines of a baseball field

intersect at home plate to form a right angle. A batter hits a fair ball such that the path of the baseball forms an angle of 27° with the third base foul line. What is the measure of the angle between the first base foul line and the path of the baseball? 29. CONSTRUCTION Construct a linear pair where one

angle measure is 115°.

3 1

2

4

30. CONSTRUCTION Construct a pair of adjacent angles

that have angle measures of 45° and 97°. 23.

24.

 

31. PROBLEM SOLVING m∠U = 2x°, and m∠V = 4m∠U.

∠2 and ∠4 are adjacent.

Which value of x makes ∠U and ∠V complements of each other? A. 25

∠1 and ∠3 form a linear pair.

B. 9

C. 36

D. 18

MATHEMATICAL CONNECTIONS In Exercises 32–35,

write and solve an algebraic equation to find the measure of each angle based on the given description.

In Exercises 25 and 26, the picture shows the Alamillo Bridge in Seville, Spain. In the picture, m∠1 = 58° and m∠2 = 24°.

32. The measure of an angle is 6° less than the measure of

its complement. 33. The measure of an angle is 12° more than twice the

measure of its complement. 1

34. The measure of one angle is 3° more than —2 the

measure of its supplement.

1

35. Two angles form a linear pair. The measure of

2

one angle is 15° less than —23 the measure of the other angle.

25. Find the measure of the supplement of ∠1. 26. Find the measure of the supplement of ∠2. 27. PROBLEM SOLVING The arm of a crossing gate

moves 42° from a vertical position. How many more degrees does the arm have to move so that it is horizontal?

CRITICAL THINKING In Exercises 36–41, tell whether the statement is always, sometimes, or never true. Explain your reasoning. 36. Complementary angles are adjacent. 37. Angles in a linear pair are supplements of each other. 38. Vertical angles are adjacent. 39. Vertical angles are supplements of each other. 40. If an angle is acute, then its complement is greater

than its supplement. 42°

41. If two complementary angles are congruent, then the

measure of each angle is 45°. 42. WRITING Explain why the supplement of an acute

A. 42°

B. 138°

C. 48°

D. 90°

angle must be obtuse. 43. WRITING Explain why an obtuse angle does not have

a complement. Section 8.6

Describing Pairs of Angles

429

44. THOUGHT PROVOKING Sketch an intersection of

48. MAKING AN ARGUMENT Light from a flashlight

roads. Identify any supplementary, complementary, or vertical angles.

strikes a mirror and is reflected so that the angle of reflection is congruent to the angle of incidence. Your classmate claims that ∠QPR is congruent to ∠TPU regardless of the measure of ∠RPS. Is your classmate correct? Explain your reasoning.

45. ATTENDING TO PRECISION Use the figure. V U

S angle of angle of incidence reflection

130° W

X

T

R

T

a. Find m∠UWV, m∠TWU, and m∠TWX. b. You write the measures of ∠TWU, ∠TWX, ∠UWV, and ∠VWX on separate pieces of paper and place the pieces of paper in a box. Then you pick two pieces of paper out of the box at random. What is the probability that the angle measures you choose are supplementary? Explain your reasoning.

Q

C

B

E

b. What do you notice about the measures of vertical angles? Explain your reasoning.

c. ∠CAD ≅ ∠EAF. d.

D

a. Write expressions for the measures of ∠BAE, ∠DAE, and ∠CAB.

D

C

b. Points C, A, and F are collinear.

y° A

conclude that each statement is true based on the figure. Explain your reasoning.

U

49. DRAWING CONCLUSIONS Use the figure.

46. HOW DO YOU SEE IT? Tell whether you can

— ≅ AF —. a. CA

P

B

— ≅ AE —. BA

A

E

50. MATHEMATICAL CONNECTIONS Let m∠1 = x°,

m∠2 = y1°, and m∠3 = y2°. ∠2 is the complement of ∠1, and ∠3 is the supplement of ∠1.

F

e. ⃖$$⃗ CF, ⃖$$⃗ BE, and ⃖$$⃗ AD intersect at point A.

a. Write equations for y1 as a function of x and for y2 as a function of x. What is the domain of each function? Explain.

f. ∠BAC and ∠CAD are complementary angles. g. ∠DAE is a right angle.

b. Graph each function and describe its range. 47. REASONING ∠KJL and ∠LJM

are complements, and ∠MJN and ∠LJM are complements. Can you show that ∠KJL ≅ ∠MJN? Explain your reasoning.

K

51. MATHEMATICAL CONNECTIONS The sum of the

L x°

M

J N

Maintaining Mathematical Proficiency

measures of two complementary angles is 74° greater than the difference of their measures. Find the measure of each angle. Explain how you found the angle measures.

Reviewing what you learned in previous grades and lessons

Determine whether the statement is always, sometimes, or never true. Explain your reasoning. (Skills Review Handbook) 52. An integer is a whole number.

53. An integer is an irrational number.

54. An irrational number is a real number.

55. A whole number is negative.

56. A rational number is an integer.

57. A natural number is an integer.

58. A whole number is a rational number.

59. An irrational number is negative.

430

Chapter 8

Basics of Geometry

8.4–8.6

What Did You Learn?

Core Vocabulary inscribed polygon, p. 407 circumscribed circle, p. 407 angle, p. 414 vertex, p. 414 sides of an angle, p. 414 interior of an angle, p. 414 exterior of an angle, p. 414

measure of an angle, p. 415 acute angle, p. 415 right angle, p. 415 obtuse angle, p. 415 straight angle, p. 415 congruent angles, p. 416 angle bisector, p. 418

complementary angles, p. 424 supplementary angles, p. 424 adjacent angles, p. 424 linear pair, p. 426 vertical angles, p. 426

Core Concepts Section 8.4 Classifying Polygons, p. 406 Inscribed Polygon, p. 407 Finding Perimeter and Area in the Coordinate Plane, p. 408

Section 8.5 Protractor Postulate, p. 415 Types of Angles, p. 415

Angle Addition Postulate, p. 417 Bisecting Angles, p. 418

Section 8.6 Complementary and Supplementary Angles, p. 424 Adjacent Angles, p. 424

Linear Pairs and Vertical Angles, p. 426 Interpreting a Diagram, p. 427

Mathematical Practices 1.

How could you explain your answers to Exercise 31 on page 412 to a friend who is unable to hear?

2.

What tool(s) could you use to verify your answers to Exercises 25–30 on page 420?

3.

Your friend says that the angles in Exercise 28 on page 429 are supplementary angles. Explain why you agree or disagree.

Performance Task:

Building Bridges Beam, arch, truss, and suspension are all types of bridges. The type chosen for a specific span depends on many factors, including the length of the bridge and how it handles tension and compression. What mathematical concepts are involved in the complexities of building bridges? To explore the answer to this question and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.

431

8

Chapter Review 8.1

Dynamic Solutions available at BigIdeasMath.com

Points, Lines, and Planes (pp. 379–386)

Use the diagram at the right. Give another name for plane P. Then name a line in the plane, a ray, a line intersecting the plane, and three collinear points.

m D

You can find another name for plane P by using any three points in the plane that are not on the same line. So, another name for plane P is plane FAB.

F

P

C

A

B

A line in the plane is ⃖""⃗ AB, a ray is """⃗ CB, a line intersecting the plane is ⃖""⃗ CD, and three collinear points are A, C, and B. Use the diagram. 1. Give another name for plane M.

h P

2. Name a line in the plane. 3. Name a line intersecting the plane.

Z

X

4. Name two rays.

Y

N

M

5. Name a pair of opposite rays. 6. Name a point not in plane M.

8.2

Measuring and Constructing Segments (pp. 387–394)

a. Find AC.

A

AC = AB + BC

12

25

B

C

Segment Addition Postulate

= 12 + 25

Substitute 12 for AB and 25 for BC.

= 37

Add.

So, AC = 37. 57

b. Find EF. D

DF = DE + EF

39

E

F

Segment Addition Postulate

57 = 39 + EF

Substitute 57 for DF and 39 for DE.

18 = EF

Subtract 39 from each side.

So, EF = 18. Find XZ. 7.

X

17

Y

24

Z

8.

38 A

27

X

9. Plot A(8, −4), B(3, −4), C(7, 1), and D(7, −3) in a coordinate plane.

— and CD — are congruent. Then determine whether AB

432

Chapter 8

Basics of Geometry

Z

g

8.3

Using Midpoint and Distance Formulas (pp. 395–402)

— are A(6, −1) and B(3, 5). Find the coordinates of the midpoint M. The endpoints of AB Then find the distance between points A and B. Use the Midpoint Formula.

(

6

) ( )

6 + 3 –1 + 5 9 M —, — = M — , 2 2 2 2

4 2

Use the Distance Formula. ——

√

AB = (x2 – x1)2 + (y2 – y1)2

——

= √(3 −

6)2

+ [5 −

—

=√

(−3)2

+

y

B(3, 5)

(–1)]2

62

Distance Formula

2 −2

Substitute.

4

A(6, −1)

x

Subtract.

—

= √9 + 36

Evaluate powers.

= √45

Add.

≈ 6.7 units

Use a calculator.

—

( )

So, the midpoint is M —92 , 2 , and the distance is about 6.7 units. Find the coordinates of the midpoint M. Then find the distance between points S and T. 10. S(−2, 4) and T(3, 9)

11. S(6, −3) and T(7, −2)

— is M(6, 3). One endpoint is J(14, 9). Find the coordinates of endpoint K. 12. The midpoint of JK — where AM = 3x + 8 and MB = 6x − 4. Find AB. 13. Point M is the midpoint of AB 8.4

Perimeter and Area in the Coordinate Plane

(pp. 405–412)

Find the perimeter and area of rectangle ABCD with vertices A(−3, 4), B(6, 4), C(6, −1), and D(−3, −1). Draw the rectangle in a coordinate plane. Then find the length and width using the Ruler Postulate. Length Width

y

A

B

AB = ∣ −3 − 6 ∣ = 9 BC = ∣ 4 − (−1) ∣ = 5

2

Substitute the values for the length and width into the formulas for the perimeter and area of a rectangle. P = 2ℓ + 2w = 2(9) + 2(5) = 18 + 10 = 28

D

−2

2

4

−2

x

C

A = ℓw = (9)(5) = 45

So, the perimeter is 28 units, and the area is 45 square units. Find the perimeter and area of the polygon with the given vertices. 14. W(5, −1), X(5, 6), Y(2, −1), Z(2, 6)

15. E(6, −2), F(6, 5), G(−1, 5)

Chapter 8

Chapter Review

433

8.5

Measuring and Constructing Angles (pp. 413–422)

Given that m∠DEF = 87°, find m∠DEG and m∠GEF. Step 1

m∠DEF = m∠DEG + m∠GEF 87° = (6x + 13)° + (2x + 10)° 87 = 8x + 23 64 = 8x 8=x Step 2

D

Write and solve an equation to find the value of x.

(6x + 13)° G

Angle Addition Postulate Substitute angle measures.

(2x + 10)°

E

Combine like terms.

F

Subtract 23 from each side. Divide each side by 8.

Evaluate the given expressions when x = 8. m∠DEG = (6x + 13)° = (6 ∙ 8 + 13)° = 61° m∠GEF = (2x + 10)° = (2 ∙ 8 + 10)° = 26°

So, m∠DEG = 61°, and m∠GEF = 26°. Find m∠ABD and m∠CBD. 16. m∠ABC = 77°

17. m∠ABC = 111°

A

A

(3x + 22)° D B

(−10x + 58)° D

(5x − 17)°

(6x + 41)° B

C

C

18. Find the measure of the angle using a protractor.

8.6

Describing Pairs of Angles (pp. 423–430)

a. ∠1 is a complement of ∠2, and m∠1 = 54°. Find m∠2. Draw a diagram with complementary adjacent angles to illustrate the relationship. m∠2 = 90° − m∠1 = 90° − 54° = 36°

54° 1 2

b. ∠3 is a supplement of ∠4, and m∠4 = 68°. Find m∠3. Draw a diagram with supplementary adjacent angles to illustrate the relationship. m∠3 = 180° − m∠4 = 180° − 68° = 112° ∠1 and ∠2 are complementary angles. Given m∠1, find m∠2. 19. m∠1 = 12°

20. m∠1 = 83°

∠3 and ∠4 are supplementary angles. Given m∠3, find m∠4. 21. m∠3 = 116°

434

Chapter 8

Basics of Geometry

22. m∠3 = 56°

68°

4 3

8

Chapter Test

—. Explain how you found your answer. Find the length of QS 1.

Q

12

R

19

S

2.

59 Q

47

S

R

Find the coordinates of the midpoint M. Then find the distance between the two points. 3. A(−4, −8) and B(−1, 4)

4. C(−1, 7) and D(−8, −3)

— is M(1, −1). One endpoint is E(−3, 2). Find the coordinates of 5. The midpoint of EF endpoint F. Use the diagram to decide whether the statement is true or false. 6. Points A, R, and B are collinear.

E

⃖##⃗ and ⃖##⃗ 7. BW AT are lines. 8. ###⃗ BR and ###⃗ RT are opposite rays.

D

B

RT W

A

9. Plane D could also be named plane ART.

Find the perimeter and area of the polygon with the given vertices. Explain how you found your answer. 10. P(−3, 4), Q(1, 4), R(−3, −2), S(3, −2)

11. J(−1, 3), K (5, 3), L(2, −2) C

12. In the diagram, ∠AFE is a straight angle and ∠CFE is a right angle.

D

B

Identify all supplementary and complementary angles. Explain. Then find m∠DFE, m∠BFC, and m∠BFE.

27° 39°

13. Use the clock at the left.

A

F

E

a. What is the measure of the acute angle created when the clock is at 10:00? b. What is the measure of the obtuse angle created when the clock is at 5:00? c. Find a time where the hour and minute hands create a straight angle. 14. Sketch a figure that contains a plane and two lines that intersect the plane at one point. 15. Your parents decide they would like to install a rectangular swimming pool in the

backyard. There is a 15-foot by 20-foot rectangular area available. Your parents request a 3-foot edge around each side of the pool. Draw a diagram of this situation in a coordinate plane. What is the perimeter and area of the largest swimming pool that will fit? 16. The picture shows the arrangement of balls in a game of boccie. The object of the

game is to throw your ball closest to the small, white ball, which is called the pallino. The green ball is the midpoint between the red ball and the pallino. The distance between the green ball and the red ball is 10 inches. The distance between the yellow ball and the pallino is 8 inches. Which ball is closer to the pallino, the green ball or the yellow ball? Explain.

Chapter 8

Chapter Test

435

8

Cumulative Assessment

1. The picture shows an aerial view of a city. Use the streets highlighted in red to identify all

congruent angles. Assume all streets are straight angles. B

A

37° 55°

E

F

G

C

H J

P

K M

N

D

55°

47°

I

L

47° O

Q

R

2. The box-and-whisker plot represents the lengths (in minutes) of songs on an album.

Which statement about the data displayed in the box-and-whisker plot is true?

2

3

4

5

6

7

8

Song length (minutes)

A The range of the song lengths is 2 minutes. ○ B The distribution of the song lengths is skewed left. ○ C 75% of the song lengths are less than 6 minutes. ○ D 50% of the song lengths are greater than 4 minutes. ○ 3. Order the terms so that each consecutive term builds off the previous term.

plane

segment

line

point

ray

4. The frequency table shows the numbers of magazine subscriptions sold by students in

your grade. Display the data in a histogram. Then classify the shape of the distribution as symmetric, skewed left, or skewed right. Number of subscriptions sold Frequency

436

Chapter 8

0–4

5–9

10–14

15–19

20–24

25–29

30–34

3

8

10

12

20

24

15

Basics of Geometry

5. Use the steps in the construction to explain how you know that !!!⃗ AG is the angle bisector

Step 2

Step 3

C

C

C

G

15 14 13 12

2

8 7

10

1

11 9

in.

of ∠CAB. Step 1

6 5

cm

1 5

A

B

4

A

6

B

3

A

4 3 2

B

6. The domain of the function shown is all integers in the interval −3 <  x  ≤  3. Find all the

ordered pairs that are solutions of the equation y = f (x). f (x) = 4x − 5

7. Three roads come to an intersection point that the people in your town call Five Corners,

as shown in the figure. M

N

W ay

rH

an

ill

W

N Buffalo Road

Sa

K

E S

lts

rte

m

Ca

L

J P

a. Identify all vertical angles. b. Identify all linear pairs. c. You are traveling east on Buffalo Road and decide to turn left onto Carter Hill. Name the angle of the turn you made. ∠KJL

∠KJM

∠KJN

∠KJP

∠LJM

∠LJN

∠LJP

∠MJN

∠MJP

∠NJP

1

8. Consider the equation y = −—3 x + 2.

a. Graph the equation in a coordinate plane. b. Does the equation represent a linear or nonlinear function? c. Is the domain discrete or continuous?

Chapter 8

Cumulative Assessment

437

9 9.1 9.2 9.3 9.4 9.5

Reasoning and Proofs Conditional Statements Inductive and Deductive Reasoning Postulates and Diagrams Proving Statements about Segments and Angles Proving Geometric Relationships

Airport Runway (p. 480)

Sculpture S l t ((p. 476)

SEE the Big Idea

Cit St ( 469) City Streett (p.

Tiger (p. (p 457)

Guitar (p. G i ( 443) 3)

Maintaining Mathematical Proficiency Finding the nth Term of an Arithmetic Sequence Example 1 Write an equation for the nth term of the arithmetic sequence 2, 5, 8, 11, . . ..  Then find a20. The first term is 2, and the common difference is 3. an = a1 + (n − 1)d

Equation for an arithmetic sequence

an = 2 + (n − 1)3

Substitute 2 for a1 and 3 for d.

an = 3n − 1

Simplify.

Use the equation to find the 20th term. an = 3n − 1

Write the equation.

a20 = 3(20) − 1

Substitute 20 for n.

= 59

Simplify.

The 20th term of the arithmetic sequence is 59.

Write an equation for the nth term of the arithmetic sequence. Then find a50. 1. 3, 9, 15, 21, . . .

2. −29, −12, 5, 22, . . .

3. 2.8, 3.4, 4.0, 4.6, . . .

1 1 2 5 4. —, —, —, —, . . . 3 2 3 6

5. 26, 22, 18, 14, . . .

6. 8, 2, −4, −10, . . .

Solving Multi-Step Equations Example 2

Solve 5x + 6 = 21. 5x + 6 = −6

21 −6

Write the equation. Subtract 6 from each side.

5x =

15

Simplify.

5x 5

—

15 5

Divide each side by 5.

—=

x=3

Simplify.

The solution is x = 3.

Solve the equation. 7. 2x + 2 = 10 10. 3q − 8 = 11

8. 15 = 5y + 20 11. 4r − 5 = 24

9. 4z − 7 = 1 12. 29 = 7s + 8

13. ABSTRACT REASONING Can you use the equation for an arithmetic sequence to write an

equation for the sequence 3, 9, 27, 81, . . . ? Explain your reasoning.

Dynamic Solutions available at BigIdeasMath.com



Mathematical Practices

Mathematically proficient students distinguish correct reasoning from flawed reasoning.

Using Correct Reasoning

Core Concept Deductive Reasoning When you use deductive reasoning, you start with two or more true statements and deduce or infer the truth of another statement. Here is an example. 1. Premise:

If a polygon is a triangle, then the sum of its angle measures is 180°.

2. Premise:

Polygon ABC is a triangle.

3. Conclusion: The sum of the angle measures of polygon ABC is 180°.

This pattern for deductive reasoning is called a syllogism.

Recognizing Flawed Reasoning The syllogisms below represent common types of flawed reasoning. Explain why each conclusion is not valid. a. When it rains, the ground gets wet. The ground is wet. Therefore, it must have rained.

b. If △ABC is equilateral, then it is isosceles. △ABC is not equilateral. Therefore, it must not be isosceles.

c. All squares are polygons. All trapezoids are quadrilaterals. Therefore, all squares are quadrilaterals.

d. No triangles are quadrilaterals. Some quadrilaterals are not squares. Therefore, some squares are not triangles.

SOLUTION a. The ground may be wet for another reason. b. A triangle can be isosceles but not equilateral. c. All squares are quadrilaterals, but not because all trapezoids are quadrilaterals. d. No squares are triangles.

Monitoring Progress Decide whether the syllogism represents correct or flawed reasoning. If flawed, explain why the conclusion is not valid. 1. All triangles are polygons.

Figure ABC is a triangle. Therefore, figure ABC is a polygon. 3. If polygon ABCD is a square, then it is a rectangle.

Polygon ABCD is a rectangle. Therefore, polygon ABCD is a square.




Chapter 9

Reasoning and Proofs

2. No trapezoids are rectangles.

Some rectangles are not squares. Therefore, some squares are not trapezoids. 4. If polygon ABCD is a square, then it is a rectangle.

Polygon ABCD is not a square. Therefore, polygon ABCD is not a rectangle.



Conditional Statements Essential Question

When is a conditional statement true or false?

A conditional statement, symbolized by p → q, can be written as an “if-then statement” in which p is the hypothesis and q is the conclusion. Here is an example. If a polygon is a triangle, then the sum of its angle measures is 180 °. hypothesis, p

conclusion, q

Determining Whether a Statement Is True or False Work with a partner. A hypothesis can either be true or false. The same is true of a conclusion. For a conditional statement to be true, the hypothesis and conclusion do not necessarily both have to be true. Determine whether each conditional statement is true or false. Justify your answer. a. If yesterday was Wednesday, then today is Thursday. b. If an angle is acute, then it has a measure of 30°. c. If a month has 30 days, then it is June. d. If an even number is not divisible by 2, then 9 is a perfect cube.

Determining Whether a Statement Is True or False

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to distinguish correct logic or reasoning from that which is flawed.

Work with a partner. Use the points in the coordinate plane to determine whether each statement is true or false. Justify your answer. a. △ABC is a right triangle. b. △BDC is an equilateral triangle. c. △BDC is an isosceles triangle. d. Quadrilateral ABCD is a trapezoid. e. Quadrilateral ABCD is a parallelogram.

6

A

y

D

4 2

C

B −6

−4

−2

2

4

6x

−2 −4 −6

Determining Whether a Statement Is True or False Work with a partner. Determine whether each conditional statement is true or false. Justify your answer. a. If △ADC is a right triangle, then the Pythagorean Theorem is valid for △ADC. b. If ∠ A and ∠ B are complementary, then the sum of their measures is 180°. c. If figure ABCD is a quadrilateral, then the sum of its angle measures is 180°. d. If points A, B, and C are collinear, then they lie on the same line. e. If ⃖##⃗ AB and ⃖##⃗ BD intersect at a point, then they form two pairs of vertical angles.

Communicate Your Answer 4. When is a conditional statement true or false? 5. Write one true conditional statement and one false conditional statement that are

different from those given in Exploration 3. Justify your answer. Section 9.1

Conditional Statements





Lesson

What You Will Learn Write conditional statements.

Core Vocabul Vocabulary larry

Use definitions written as conditional statements.

conditional statement, p. 442 if-then form, p. 442 hypothesis, p. 442 conclusion, p. 442 negation, p. 442 converse, p. 443 inverse, p. 443 contrapositive, p. 443 equivalent statements, p. 443 biconditional statement, p. 445 truth value, p. 446 truth table, p. 446

Make truth tables.

Write biconditional statements.

Writing Conditional Statements

Core Concept Conditional Statement A conditional statement is a logical statement that has two parts, a hypothesis p and a conclusion q. When a conditional statement is written in if-then form, the “if” part contains the hypothesis and the “then” part contains the conclusion. Words

If p, then q.

Previous perpendicular lines

p → q (read as “p implies q”)

Symbols

Rewriting a Statement in If-Then Form Use red to identify the hypothesis and blue to identify the conclusion. Then rewrite the conditional statement in if-then form. a. All birds have feathers.

b. You are in Texas if you are in Houston.

SOLUTION a. All birds have feathers.

b. You are in Texas if you are in Houston.

If an animal is a bird, then it has feathers.

If you are in Houston, then you are in Texas.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use red to identify the hypothesis and blue to identify the conclusion. Then rewrite the conditional statement in if-then form. 2. 2x + 7 = 1, because x = −3.

1. All 30° angles are acute angles.

Core Concept Negation The negation of a statement is the opposite of the original statement. To write the negation of a statement p, you write the symbol for negation (∼) before the letter. So, “not p” is written ∼p. Words

Symbols

not p

∼p

Writing a Negation Write the negation of each statement. a. The ball is red.

b. The cat is not black.

SOLUTION a. The ball is not red. 

Chapter 9

Reasoning and Proofs

b. The cat is black.

Core Concept Related Conditionals Consider the conditional statement below.

COMMON ERROR Just because a conditional statement and its contrapositive are both true does not mean that its converse and inverse are both false. The converse and inverse could also both be true.

Symbols

p→q

Words

If p, then q.

Converse

To write the converse of a conditional statement, exchange the hypothesis and the conclusion.

Words

If q, then p.

Symbols

q→p

Inverse To write the inverse of a conditional statement, negate both the

hypothesis and the conclusion. Words

Symbols

If not p, then not q.

∼p → ∼q

Contrapositive To write the contrapositive of a conditional statement, first

write the converse. Then negate both the hypothesis and the conclusion. Words

If not q, then not p.

Symbols

∼q → ∼p

A conditional statement and its contrapositive are either both true or both false. Similarly, the converse and inverse of a conditional statement are either both true or both false. In general, when two statements are both true or both false, they are called equivalent statements.

Writing Related Conditional Statements L p be “you are a guitar player” and let q be “you are a musician.” Write each Let sstatement in words. Then decide whether it is true or false. aa. the conditional statement p → q b. the converse q → p b cc. the inverse ∼p → ∼q d. the contrapositive ∼q → ∼p d

SOLUTION S aa. Conditional: If you are a guitar player, then you are a musician. true; Guitar players are musicians. b. Converse: If you are a musician, then you are a guitar player. false; Not all musicians play the guitar. c. Inverse: If you are not a guitar player, then you are not a musician. false; Even if you do not play a guitar, you can still be a musician. d. Contrapositive: If you are not a musician, then you are not a guitar player. true; A person who is not a musician cannot be a guitar player.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

In Exercises 3 and 4, write the negation of the statement. 3. The shirt is green.

4. The shoes are not red.

5. Repeat Example 3. Let p be “the stars are visible” and let q be “it is night.”

Section 9.1

Conditional Statements



Using Definitions You can write a definition as a conditional statement in if-then form or as its converse. Both the conditional statement and its converse are true for definitions. For example, recall the definition of perpendicular lines. If two lines intersect to form a right angle, then they are perpendicular lines. You can also write the definition using the converse: If two lines are perpendicular lines, then they intersect to form a right angle.

m

You can write “lineℓ is perpendicular to line m” asℓ⊥ m. ⊥m

Using Definitions Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned.

B

a. ⃖##⃗ AC ⊥ ⃖##⃗ BD b. ∠AEB and ∠CEB are a linear pair.

A

c. ###⃗ EA and ###⃗ EB are opposite rays.

E

C

D

SOLUTION a. This statement is true. The right angle symbol in the diagram indicates that the lines intersect to form a right angle. So, you can say the lines are perpendicular. b. This statement is true. By definition, if the noncommon sides of adjacent angles are opposite rays, then the angles are a linear pair. Because ###⃗ EA and ###⃗ EC are opposite rays, ∠AEB and ∠CEB are a linear pair. c. This statement is false. Point E does not lie on the same line as A and B, so the rays are not opposite rays.

Monitoring Progress

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Use the diagram. Decide whether the statement is true. Explain your answer using the definitions you have learned. F

G M J

6. ∠JMF and ∠FMG are supplementary.

—. 7. Point M is the midpoint of FH

8. ∠JMF and ∠HMG are vertical angles.

⃖##⃗ ⊥ ⃖##⃗ 9. FH JG



Chapter 9

Reasoning and Proofs

H

Writing Biconditional Statements

Core Concept Biconditional Statement When a conditional statement and its converse are both true, you can write them as a single biconditional statement. A biconditional statement is a statement that contains the phrase “if and only if.” Words

p if and only if q

Symbols

p↔q

Any definition can be written as a biconditional statement.

Writing a Biconditional Statement Rewrite the definition of perpendicular lines as a single biconditional statement. Definition If two lines intersect to form a right angle, then they are perpendicular lines.

SOLUTION Let p be “two lines intersect to form a right angle” and let q be “they are perpendicular lines.” Use red to identify p and blue to identify q. Write the definition p → q.

s

t

Definition If two lines intersect to form a right angle, then they are perpendicular lines. Write the converse q → p.

s⊥t

Converse If two lines are perpendicular lines, then they intersect to form a right angle. Use the definition and its converse to write the biconditional statement p ↔ q. Biconditional Two lines intersect to form a right angle if and only if they are perpendicular lines.

Monitoring Progress

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10. Rewrite the definition of a right angle as a single biconditional statement.

Definition If an angle is a right angle, then its measure is 90°. 11. Rewrite the definition of congruent segments as a single biconditional statement.

Definition If two line segments have the same length, then they are congruent segments. 12. Rewrite the statements as a single biconditional statement.

If Mary is in theater class, then she will be in the fall play. If Mary is in the fall play, then she must be taking theater class. 13. Rewrite the statements as a single biconditional statement.

If you can run for President, then you are at least 35 years old. If you are at least 35 years old, then you can run for President.

Section 9.1

Conditional Statements



Making Truth Tables

Dynamic Solutions available at BigIdeasMath.com The truth value of a statement is either true (T) or false (F). You can determine the conditions under which a conditional statement is true by using a truth table. The truth table below shows the truth values for hypothesis p and conclusion q. Conditional

p

q

p→q

T

T

T

T

F

F

F

T

T

F

F

T

The conditional statement p → q is only false when a true hypothesis produces a false conclusion. Two statements are logically equivalent when they have the same truth table.

Making a Truth Table Use the truth table above to make truth tables for the converse, inverse, and contrapositive of a conditional statement p → q.

SOLUTION The truth tables for the converse and the inverse are shown below. Notice that the converse and the inverse are logically equivalent because they have the same truth table. Converse

Inverse

p

q

q→p

p

q

∼p

∼q

∼p → ∼q

T

T

T

T

T

F

F

T

T

F

T

T

F

F

T

T

F

T

F

F

T

T

F

F

F

F

T

F

F

T

T

T

The truth table for the contrapositive is shown below. Notice that a conditional statement and its contrapositive are logically equivalent because they have the same truth table. Contrapositive

p

q

∼q

∼p

∼q → ∼p

T

T

F

F

T

T

F

T

F

F

F

T

F

T

T

F

F

T

T

T

Monitoring Progress

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14. Make a truth table for the conditional statement p → ∼q. 15. Make a truth table for the conditional statement ∼( p → q).



Chapter 9

Reasoning and Proofs



Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY What type of statements are either both true or both false? 2. WHICH ONE DOESN’T BELONG? Which statement does not belong with the other three? Explain your reasoning.

If today is Tuesday, then tomorrow is Wednesday.

If it is Independence Day, then it is July.

If an angle is acute, then its measure is less than 90°.

If you are an athlete, then you play soccer.

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, copy the conditional statement. Underline the hypothesis and circle the conclusion. 3. If a polygon is a pentagon, then it has five sides. 4. If two lines form vertical angles, then they intersect. 5. If you run, then you are fast. 6. If you like math, then you like science.

In Exercises 7–12, rewrite the conditional statement in if-then form. (See Example 1.) 7. 9x + 5 = 23, because x = 2. 8. Today is Friday, and tomorrow is the weekend. 9. You are in a band, and you play the drums. 10. Two right angles are supplementary angles.

18. Let p be “you are in math class” and let q be “you are

in Geometry.” 19. Let p be “you do your math homework” and let q be

“you will do well on the test.” 20. Let p be “you are not an only child” and let q be “you

have a sibling.” 21. Let p be “it does not snow” and let q be “I will run

outside.” 22. Let p be “the Sun is out” and let q be “it is daytime.” 23. Let p be “3x − 7 = 20” and let q be “x = 9.” 24. Let p be “it is Valentine’s Day” and let q be “it is

February.”

11. Only people who are registered are allowed to vote.

In Exercises 25–28, decide whether the statement about the diagram is true. Explain your answer using the definitions you have learned. (See Example 4.)

12. The measures of complementary angles sum to 90°.

25. m∠ABC = 90°

In Exercises 13–16, write the negation of the statement. (See Example 2.) 13. The sky is blue.

14. The lake is cold.

15. The ball is not pink.

16. The dog is not a Lab.

In Exercises 17–24, write the conditional statement p → q, the converse q → p, the inverse ∼p → ∼q, and the contrapositive ∼q → ∼p in words. Then decide whether each statement is true or false. (See Example 3.) 17. Let p be “two angles are supplementary” and let q be

⃖%%⃗ ⊥ ⃖%%⃗ 26. PQ ST P

A S C

B

Q

—. 28. M is the midpoint of AB

27. m∠2 + m∠3 = 180° Q

A 2

M

T

3 N

M

B

P

“the measures of the angles sum to 180°.” Section 9.1

Conditional Statements



In Exercises 29–32, rewrite the definition of the term as In Exercises 39–44, create a truth table for the logical Dynamic Solutions(See available at 6.) BigIdeasMath.com a biconditional statement. (See Example 5.) statement. Example 29. The midpoint of a segment is the point that divides the

segment into two congruent segments. 30. Two angles are vertical angles when their sides form

two pairs of opposite rays. 31. Adjacent angles are two angles that share a common

vertex and side but have no common interior points. 32. Two angles are supplementary angles when the sum

of their measures is 180°. In Exercises 33–36, rewrite the statements as a single biconditional statement. (See Example 5.) 33. If a polygon has three sides, then it is a triangle.

If a polygon is a triangle, then it has three sides. 34. If a polygon has four sides, then it is a quadrilateral.

39. ∼p → q 40. ∼q → p 41. ∼(∼p → ∼q) 42. ∼( p → ∼q) 43. q → ∼p 44. ∼(q → p) 45. USING STRUCTURE The statements below describe

three ways that rocks are formed. Igneous rock is formed from the cooling of molten rock.

If a polygon is a quadrilateral, then it has four sides. 35. If an angle is a right angle, then it measures 90°.

If an angle measures 90°, then it is a right angle.

Sedimentary rock is formed from pieces of other rocks.

36. If an angle is obtuse, then it has a measure between

90° and 180°. If an angle has a measure between 90° and 180°, then it is obtuse. 37. ERROR ANALYSIS Describe and correct the error in

rewriting the conditional statement in if-then form.



Conditional statement All high school students take four English courses. If-then form If a high school student takes four courses, then all four are English courses.

38. ERROR ANALYSIS Describe and correct the error in

writing the converse of the conditional statement.



Conditional statement If it is raining, then I will bring an umbrella. Converse If it is not raining, then I will not bring an umbrella.



Chapter 9

Reasoning and Proofs

Metamorphic rock is formed by changing temperature, pressure, or chemistry.

a. Write each statement in if-then form. b. Write the converse of each of the statements in part (a). Is the converse of each statement true? Explain your reasoning. c. Write a true if-then statement about rocks that is different from the ones in parts (a) and (b). Is the converse of your statement true or false? Explain your reasoning. 46. MAKING AN ARGUMENT Your friend claims the

statement “If I bought a shirt, then I went to the mall” can be written as a true biconditional statement. Your sister says you cannot write it as a biconditional. Who is correct? Explain your reasoning. 47. REASONING You are told that the contrapositive

of a statement is true. Will that help you determine whether the statement can be written as a true biconditional statement? Explain your reasoning.

48. PROBLEM SOLVING Use the conditional statement to

identify the if-then statement as the converse, inverse, or contrapositive of the conditional statement. Then use the symbols to represent both statements.

53. MATHEMATICAL CONNECTIONS Can the statement

“If x2 − 10 = x + 2, then x = 4” be combined with its converse to form a true biconditional statement?

54. CRITICAL THINKING The largest natural arch in

Conditional statement If I rode my bike to school, then I did not walk to school.

the United States is Landscape Arch, located in Thompson, Utah. It spans 290 feet.

If-then statement If I did not ride my bike to school, then I walked to school. p

q

∼

→

↔

USING STRUCTURE In Exercises 49–52, rewrite the

conditional statement in if-then form. Then underline the hypothesis and circle the conclusion. 49.

a. Use the information to write at least two true conditional statements. b. Which type of related conditional statement must also be true? Write the related conditional statements. 50.

c. What are the other two types of related conditional statements? Write the related conditional statements. Then determine their truth values. Explain your reasoning. 55. REASONING Which statement has the same meaning

as the given statement?

51.

Given statement You can watch a movie after you do your homework.

A If you do your homework, then you can watch a ○ movie afterward.

B If you do not do your homework, then you can ○ watch a movie afterward.

C If you cannot watch a movie afterward, then do ○ 52.

your homework.

D If you can watch a movie afterward, then do not ○ do your homework. 56. THOUGHT PROVOKING Write three conditional

statements, where one is always true, one is always false, and one depends on the person interpreting the statement.

Section 9.1

Conditional Statements



57. CRITICAL THINKING One example of a conditional

60. DRAWING CONCLUSIONS You measure the heights of

statement involving dates is “If today is August 31, then tomorrow is September 1.” Write a conditional statement using dates from two different months so that the truth value depends on when the statement is read.

your classmates to get a data set. a. Tell whether this statement is true: If x and y are the least and greatest values in your data set, then the mean of the data is between x and y. b. Write the converse of the statement in part (a). Is the converse true? Explain your reasoning.

58. HOW DO YOU SEE IT? The Venn diagram represents

c. Copy and complete the statement below using mean, median, or mode to make a conditional statement that is true for any data set. Explain your reasoning.

all the musicians at a high school. Write three conditional statements in if-then form describing the relationships between the various groups of musicians.

If a data set has a mean, median, and a mode, then the _______ of the data set will always be a data value.

musicians

chorus

band

jazz band

61. WRITING Write a conditional statement that is true,

but its converse is false. 62. CRITICAL THINKING Write a series of if-then

statements that allow you to find the measure of each angle, given that m∠1 = 90°. Use the definition of linear pairs.

59. MULTIPLE REPRESENTATIONS Create a Venn diagram

representing each conditional statement. Write the converse of each conditional statement. Then determine whether each conditional statement and its converse are true or false. Explain your reasoning.

4 1 3 2

a. If you go to the zoo to see a lion, then you will see a cat.

63. WRITING Advertising slogans such as “Buy these

b. If you play a sport, then you wear a helmet.

shoes! They will make you a better athlete!” often imply conditional statements. Find an advertisement or write your own slogan. Then write it as a conditional statement.

c. If this month has 31 days, then it is not February.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Find the pattern. Then draw the next two figures in the sequence.

(Skills Review Handbook)

64.

65.

Find the pattern. Then write the next two numbers. 66. 1, 3, 5, 7, . . .

67. 12, 23, 34, 45, . . .

4 8 16

68. 2, —3 , —9 , — ,... 27



Chapter 9

(Section 4.6 and Section 6.5)

69. 1, 4, 16, 64, . . .

Reasoning and Proofs



Inductive and Deductive Reasoning Essential Question

How can you use reasoning to solve problems?

Recall that a conjecture is an unproven statement about a general mathematical concept that is based on observations.

Writing a Conjecture Work with a partner. Write a conjecture about the pattern. Then use your conjecture to draw the 10th object in the pattern. 1

2

3

4

5

6

7

a.

b.

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to justify your conclusions and communicate them to others.

c.

Using a Venn Diagram Work with a partner. Use the Venn diagram to determine whether the statement is true or false. Justify your answer. Assume that no region of the Venn diagram is empty. a. If an item has Property B, then it has Property A. b. If an item has Property A, then it has Property B. Property C c. If an item has Property A, then it has Property C. d. Some items that have Property A do not have Property B. e. If an item has Property C, then it does not have Property B. f. Some items have both Properties A and C. g. Some items have both Properties B and C.

Property A Property B

Reasoning and Venn Diagrams Work with a partner. Draw a Venn diagram that shows the relationship between different types of quadrilaterals: squares, rectangles, parallelograms, trapezoids, rhombuses, and kites. Then write several conditional statements that are shown in your diagram, such as “If a quadrilateral is a square, then it is a rectangle.”

Communicate Your Answer 4. How can you use reasoning to solve problems? 5. Give an example of how you used reasoning to solve a real-life problem.

Section 9.2

Inductive and Deductive Reasoning



 Lesson

What You Will Learn Use inductive reasoning. Use deductive reasoning.

Core Vocabul Vocabulary larry inductive reasoning, p. 452 counterexample, p. 453 deductive reasoning, p. 454

Using Inductive Reasoning

Core Concept

Previous conjecture

Inductive Reasoning Recall that a conjecture is an unproven statement about a general mathematical concept that is based on observations. You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case.

Describing a Visual Pattern Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. Figure 1

Figure 2

Figure 3

SOLUTION Each circle is divided into twice as many equal regions as the figure number. Sketch the fourth figure by dividing a circle into eighths. Shade the section just above the horizontal segment at the left. Figure 4

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1. Sketch the fifth figure in the pattern in Example 1.

Sketch the next figure in the pattern. 2.

3.



Chapter 9

Reasoning and Proofs

Making and Testing a Conjecture Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers.

SOLUTION Step 1 Find a pattern using a few groups of small numbers.

⋅

3 + 4 + 5 = 12 = 4 3

⋅

7 + 8 + 9 = 24 = 8 3

⋅

10 + 11 + 12 = 33 = 11 3 Step 2 Make a conjecture.

⋅

16 + 17 + 18 = 51 = 17 3

Conjecture The sum of any three consecutive integers is three times the second number. Step 3 Test your conjecture using other numbers. For example, test that it works with the groups −1, 0, 1 and 100, 101, 102.

⋅ 

−1 + 0 + 1 = 0 = 0 3

⋅ 

100 + 101 + 102 = 303 = 101 3

Core Concept Counterexample To show that a conjecture is true, you must show that it is true for all cases. You can show that a conjecture is false, however, by finding just one counterexample. A counterexample is a specific case for which the conjecture is false.

Finding a Counterexample A student makes the following conjecture about the sum of two numbers. Find a counterexample to disprove the student’s conjecture. Conjecture The sum of two numbers is always more than the greater number.

SOLUTION To find a counterexample, you need to find a sum that is less than the greater number. −2 + (−3) = −5 −5 ≯ −2 Because a counterexample exists, the conjecture is false.

Monitoring Progress

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4. Make and test a conjecture about the sign of the product of any three

negative integers. 5. Make and test a conjecture about the sum of any five consecutive integers.

Find a counterexample to show that the conjecture is false. 6. The value of x2 is always greater than the value of x. 7. The sum of two numbers is always greater than their difference.

Section 9.2

Inductive and Deductive Reasoning



Using Deductive Reasoning

Core Concept Deductive Reasoning Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. This is different from inductive reasoning, which uses specific examples and patterns to form a conjecture.

Laws of Logic Law of Detachment

If the hypothesis of a true conditional statement is true, then the conclusion is also true. Law of Syllogism

If hypothesis p, then conclusion q.

If these statements are true,

If hypothesis q, then conclusion r. If hypothesis p, then conclusion r.

then this statement is true.

Using the Law of Detachment If two segments have the same length, then they are congruent. You know that BC = XY. Using the Law of Detachment, what statement can you make?

SOLUTION Because BC = XY satisfies the hypothesis of a true conditional statement, the conclusion is also true.

— ≅ XY —. So, BC Using the Law of Syllogism If possible, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements. a. If x2 > 25, then x2 > 20. If x > 5, then x2 > 25. b. If a polygon is regular, then all angles in the interior of the polygon are congruent. If a polygon is regular, then all its sides are congruent.

SOLUTION a. Notice that the conclusion of the second statement is the hypothesis of the first statement. The order in which the statements are given does not affect whether you can use the Law of Syllogism. So, you can write the following new statement. If x > 5, then x2 > 20. b. Neither statement’s conclusion is the same as the other statement’s hypothesis. You cannot use the Law of Syllogism to write a new conditional statement.



Chapter 9

Reasoning and Proofs

Using Inductive and Deductive Reasoning What conclusion can you make about the product of an even integer and any other integer?

SOLUTION

MAKING SENSE OF PROBLEMS In geometry, you will frequently use inductive reasoning to make conjectures. You will also use deductive reasoning to show that conjectures are true or false. You will need to know which type of reasoning to use.

Step 1 Look for a pattern in several examples. Use inductive reasoning to make a conjecture. (−2)(2) = −4

(−1)(2) = −2

2(2) = 4

3(2) = 6

(−2)(−4) = 8

(−1)(−4) = 4

2(−4) = −8

3(−4) = −12

Conjecture Even integer • Any integer = Even integer Step 2 Let n and m each be any integer. Use deductive reasoning to show that the conjecture is true. 2n is an even integer because any integer multiplied by 2 is even. 2nm represents the product of an even integer 2n and any integer m. 2nm is the product of 2 and an integer nm. So, 2nm is an even integer. The product of an even integer and any integer is an even integer.

Comparing Inductive and Deductive Reasoning Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. a. Each time Monica kicks a ball up in the air, it returns to the ground. So, the next time Monica kicks a ball up in the air, it will return to the ground. b. All reptiles are cold-blooded. Parrots are not cold-blooded. Sue’s pet parrot is not a reptile.

SOLUTION a. Inductive reasoning, because a pattern is used to reach the conclusion. b. Deductive reasoning, because facts about animals and the laws of logic are used to reach the conclusion.

Monitoring Progress

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8. If 90° < m∠R < 180°, then ∠R is obtuse. The measure of ∠R is 155°. Using the

Law of Detachment, what statement can you make? 9. Use the Law of Syllogism to write a new conditional statement that follows

from the pair of true statements. If you get an A on your math test, then you can go to the movies. If you go to the movies, then you can watch your favorite actor. 10. Use inductive reasoning to make a conjecture about the sum of a number and

itself. Then use deductive reasoning to show that the conjecture is true. 11. Decide whether inductive reasoning or deductive reasoning is used to reach the

conclusion. Explain your reasoning. All multiples of 8 are divisible by 4. 64 is a multiple of 8. So, 64 is divisible by 4. Section 9.2

Inductive and Deductive Reasoning





Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY How does the prefix “counter-” help you understand the term counterexample? 2. WRITING Explain the difference between inductive reasoning and deductive reasoning.

Monitoring Progress and Modeling with Mathematics In Exercises 3–8, describe the pattern. Then write or draw the next two numbers, letters, or figures. (See Example 1.) 3. 1, −2, 3, −4, 5, . . .

4. 0, 2, 6, 12, 20, . . .

5. Z, Y, X, W, V, . . .

6. J, F, M, A, M, . . .

In Exercises 17–20, use the Law of Detachment to determine what you can conclude from the given information, if possible. (See Example 4.) 17. If you pass the final, then you pass the class. You

passed the final. 18. If your parents let you borrow the car, then you will

go to the movies with your friend. You will go to the movies with your friend.

7.

19. If a quadrilateral is a square, then it has four right

angles. Quadrilateral QRST has four right angles.

8.

20. If a point divides a line segment into two congruent

line segments, then the point is a midpoint. Point P — into two congruent line segments. divides LH In Exercises 9–12, make and test a conjecture about the given quantity. (See Example 2.) 9. the product of any two even integers 10. the sum of an even integer and an odd integer

In Exercises 21–24, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements, if possible. (See Example 5.) 21. If x < −2, then ∣ x ∣ > 2. If x > 2, then ∣ x ∣ > 2. 1

11. the quotient of a number and its reciprocal 12. the quotient of two negative integers

In Exercises 13–16, find a counterexample to show that the conjecture is false. (See Example 3.) 13. The product of two positive numbers is always greater

than either number. n+1 n

14. If n is a nonzero integer, then — is always greater

than 1.

15. If two angles are supplements of each other, then one

of the angles must be acute. 16.



— into two line segments. So, the A line s divides MN —. line s is a segment bisector of MN

Chapter 9

Reasoning and Proofs

1

22. If a = 3, then 5a = 15. If —2 a = 1—2 , then a = 3. 23. If a figure is a rhombus, then the figure is a

parallelogram. If a figure is a parallelogram, then the figure has two pairs of opposite sides that are parallel. 24. If a figure is a square, then the figure has four

congruent sides. If a figure is a square, then the figure has four right angles. In Exercises 25–28, state the law of logic that is illustrated. 25. If you do your homework, then you can watch TV. If

you watch TV, then you can watch your favorite show. If you do your homework, then you can watch your favorite show.

26. If you miss practice the day before a game, then you

37. REASONING The table shows the average weights

will not be a starting player in the game.

of several subspecies of tigers. What conjecture can you make about the relation between the weights of female tigers and the weights of male tigers? Explain your reasoning.

You miss practice on Tuesday. You will not start the game Wednesday. 27. If x > 12, then x + 9 > 20. The value of x is 14.

So, x + 9 > 20.

Weight of female (pounds)

Weight of male (pounds)

Amur

370

660

Bengal

300

480

South China

240

330

Sumatran

200

270

Indo-Chinese

250

400

28. If ∠1 and ∠2 are vertical angles, then ∠1 ≅ ∠2.

If ∠1 ≅ ∠2, then m∠1 = m∠2.

If ∠1 and ∠2 are vertical angles, then m∠1 = m∠2. In Exercises 29 and 30, use inductive reasoning to make a conjecture about the given quantity. Then use deductive reasoning to show that the conjecture is true. (See Example 6.) 29. the sum of two odd integers

38. HOW DO YOU SEE IT? Determine whether you

30. the product of two odd integers

can make each conjecture from the graph. Explain your reasoning.

In Exercises 31–34, decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. (See Example 7.)

Number of participants (thousands)

U.S. High School Girls’ Lacrosse

31. Each time your mom goes to the store, she buys milk.

So, the next time your mom goes to the store, she will buy milk. 32. Rational numbers can be written as fractions.

Irrational numbers cannot be written as fractions. So, —12 is a rational number. 33. All men are mortal. Mozart is a man, so Mozart

y 140 100 60 20 1

2

3

4

5

6

7x

Year

is mortal.

a. More girls will participate in high school lacrosse in Year 8 than those who participated in Year 7.

34. Each time you clean your room, you are allowed to

go out with your friends. So, the next time you clean your room, you will be allowed to go out with your friends.

b. The number of girls participating in high school lacrosse will exceed the number of boys participating in high school lacrosse in Year 9.

ERROR ANALYSIS In Exercises 35 and 36, describe and

correct the error in interpreting the statement. 35. If a figure is a rectangle, then the figure has four sides.

39. MATHEMATICAL CONNECTIONS Use inductive

reasoning to write a formula for the sum of the first n positive even integers.

A trapezoid has four sides.



Using the Law of Detachment, you can conclude that a trapezoid is a rectangle.

40. FINDING A PATTERN The following are the first nine

Lucas numbers. 2, 1, 3, 4, 7, 11, 18, 29, 47, . . .

36. Each day, you get to school before your friend.



a. Make a conjecture about each of the Lucas numbers after the first two.

Using deductive reasoning, you can conclude that you will arrive at school before your friend tomorrow.

b. Write the next three numbers in the pattern. c. Name another sequence with this same pattern.

Section 9.2

Inductive and Deductive Reasoning



41. MAKING AN ARGUMENT Which argument is correct?

Explain your reasoning.

45. DRAWING CONCLUSIONS Decide whether each

conclusion is valid. Explain your reasoning.

Argument 1: If two angles measure 30° and 60°, then the angles are complementary. ∠ 1 and ∠ 2 are complementary. So, m∠ 1 = 30° and m∠ 2 = 60°.

• Yellowstone is a national park in Wyoming.

Argument 2: If two angles measure 30° and 60°, then the angles are complementary. The measure of ∠ 1 is 30° and the measure of ∠ 2 is 60°. So, ∠ 1 and ∠ 2 are complementary.

• When you go camping, you go canoeing.

• You and your friend went camping at Yellowstone National Park. • If you go on a hike, your friend goes with you. • You go on a hike. • There is a 3-mile-long trail near your campsite.

42. THOUGHT PROVOKING The first two terms of a

sequence are —14 and —12 . Describe three different possible patterns for the sequence. List the first five terms for each sequence.

43. MATHEMATICAL CONNECTIONS Use the table to

make a conjecture about the relationship between x and y. Then write an equation for y in terms of x. Use the equation to test your conjecture for other values of x. x

0

1

2

3

4

y

2

5

8

11

14

a. You went camping in Wyoming. b. Your friend went canoeing. c. Your friend went on a hike. d. You and your friend went on a hike on a 3-mile-long trail. 46. CRITICAL THINKING Geologists use the Mohs’ scale

to determine a mineral’s hardness. Using the scale, a mineral with a higher rating will leave a scratch on a mineral with a lower rating. Testing a mineral’s hardness can help identify the mineral.

Mineral 44. REASONING Use the pattern below. Each figure is

made of squares that are 1 unit by 1 unit.

1

2

3

4

Gypsum

Calcite

Fluorite

1

2

3

4

Mohs’ rating

5

a. Find the perimeter of each figure. Describe the pattern of the perimeters. b. Predict the perimeter of the 20th figure.

Maintaining Mathematical Proficiency Determine which postulate is illustrated by the statement. 47. AB + BC = AC

Talc

a. The four minerals are randomly labeled A, B, C, and D. Mineral A is scratched by Mineral B. Mineral C is scratched by all three of the other minerals. What can you conclude? Explain your reasoning. b. What additional test(s) can you use to identify all the minerals in part (a)?

Reviewing what you learned in previous grades and lessons

(Section 8.2 and Section 8.5)

48. m∠ DAC = m∠ DAE + m∠ EAB

D

49. AD is the absolute value of the difference of the coordinates of A and D. 50. m∠ DAC is equal to the absolute value of the difference between the

real numbers matched with !!!⃗ AD and !!!⃗ AC on a protractor.

Find the mean, median, mode, range, and standard deviation of the data set. 51. 6, 7, 9, 2, 5, 10, 12

52. 20, 28, 26, 32, 18, 46, 35

53. 1, 2, 5, 7, 7, 9, 11, 12

54. 14, 15, 14, 14, 16, 19, 17, 17

458

Chapter 9

Reasoning and Proofs

E A

B

(Section 7.1)

C



Postulates and Diagrams Essential Question

In a diagram, what can be assumed and what

needs to be labeled?

Looking at a Diagram Work with a partner. On a piece of paper, draw two perpendicular lines. Label them ⃖""⃗ AB and ⃖""⃗ CD. Look at the diagram from different angles. Do the lines appear perpendicular regardless of the angle at which you look at them? Describe all the angles at which you can look at the lines and have them appear perpendicular.

C B

A

C

B

D

A

ATTENDING TO PRECISION To be proficient in math, you need to state the meanings of the symbols you choose.

view from m upper rightt

D

view from above

Interpreting a Diagram Work with a partner. When you draw a diagram, you are communicating with others. It is important that you include sufficient information in the diagram. Use the diagram to determine which of the following statements you can assume to be true. Explain your reasoning.

A

D C

B

G

b. Points D, G, and I are collinear.

I

F

a. All the points shown are coplanar. E

H

c. Points A, C, and H are collinear. d. ⃖""⃗ EG and ⃖""⃗ AH are perpendicular. e. ∠BCA and ∠ACD are a linear pair. f. ⃖""⃗ AF and ⃖""⃗ BD are perpendicular.

g. ⃖""⃗ EG and ⃖""⃗ BD are parallel.

h. ⃖""⃗ AF and ⃖""⃗ BD are coplanar.

i. ⃖""⃗ EG and ⃖""⃗ BD do not intersect.

j. ⃖""⃗ AF and ⃖""⃗ BD intersect.

k. ⃖""⃗ EG and ⃖""⃗ BD are perpendicular.

l. ∠ACD and ∠BCF are vertical angles.

m. ⃖""⃗ AC and ⃖""⃗ FH are the same line.

Communicate Your Answer 3. In a diagram, what can be assumed and what needs to be labeled? 4. Use the diagram in Exploration 2 to write two statements you can assume to be

true and two statements you cannot assume to be true. Your statements should be different from those given in Exploration 2. Explain your reasoning. Section 9.3

Postulates and Diagrams



 Lesson

What You Will Learn Identify postulates using diagrams. Sketch and interpret diagrams.

Core Vocabul Vocabulary larry line perpendicular to a plane, p. 462 Previous postulate point line plane

Identifying Postulates Here are seven more postulates involving points, lines, and planes.

Postulates Point, Line, and Plane Postulates Postulate

Example

Two Point Postulate

Through any two points, there exists exactly one line.

B A

Line-Point Postulate

Through points A and B, there is exactly one lineℓ. Lineℓcontains at least two points.

A line contains at least two points. Line Intersection Postulate

If two lines intersect, then their intersection is exactly one point.

C m

Three Point Postulate

Through any three noncollinear points, there exists exactly one plane.

n

E D

R

Through points D, E, and F, there is exactly one plane, plane R. Plane R contains at least three noncollinear points.

R

Points D and E lie in plane R, so ⃖""⃗ DE lies in plane R.

F

Plane-Point Postulate

A plane contains at least three noncollinear points. Plane-Line Postulate

If two points lie in a plane, then the line containing them lies in the plane.

E D

F

Plane Intersection Postulate

If two planes intersect, then their intersection is a line.



Chapter 9

Reasoning and Proofs

The intersection of line m and line n is point C.

S T

The intersection of plane S and plane T is lineℓ.

Identifying a Postulate Using a Diagram State the postulate illustrated by the diagram. a.

b.

then

If

then

If

SOLUTION a. Line Intersection Postulate If two lines intersect, then their intersection is exactly one point. b. Plane Intersection Postulate If two planes intersect, then their intersection is a line.

Identifying Postulates from a Diagram Use the diagram to write examples of the Plane-Point Postulate and the Plane-Line Postulate. Q m B

C

A n

P

SOLUTION Plane-Point Postulate Plane P contains at least three noncollinear points, A, B, and C. Plane-Line Postulate Point A and point B lie in plane P. So, line n containing points A and B also lies in plane P.

Monitoring Progress

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1. Use the diagram in Example 2. Which postulate allows you to say that the

intersection of plane P and plane Q is a line? 2. Use the diagram in Example 2 to write an example of the postulate.

a. Two Point Postulate b. Line-Point Postulate c. Line Intersection Postulate

Section 9.3

Postulates and Diagrams



Sketching and Interpreting Diagrams Sketching a Diagram

— at point W, so that TW — ≅ WV —. Sketch a diagram showing ⃖""⃗ TV intersecting PQ SOLUTION Step 1 Draw ⃖""⃗ TV and label points T and V. Step 2

—. Draw point W at the midpoint of TV

P V

Mark the congruent segments.

W

T

— through W. Step 3 Draw PQ

Q

ANOTHER WAY In Example 3, there are many ways you can sketch the diagram. Another way is shown below. T

P

W

t p A

A line is a line perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point. In a diagram, a line perpendicular to a plane must be marked with a right angle symbol, as shown.

q

V Q

Interpreting a Diagram Which of the following statements cannot be assumed from the diagram?

T A

Points A, B, and F are collinear. Points E, B, and D are collinear.

C

B

⃖""⃗ AB ⊥ plane S

S D

E F

⃖""⃗ CD ⊥ plane T ⃖""⃗ AF intersects ⃖""⃗ BC at point B. SOLUTION

No drawn line connects points E, B, and D. So, you cannot assume they are collinear. With no right angle marked, you cannot assume ⃖""⃗ CD ⊥ plane T.

Monitoring Progress

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Refer back to Example 3.

— and QW — are congruent, how can you 3. If the given information states that PW indicate that in the diagram?

4. Name a pair of supplementary angles in the diagram. Explain.

Use the diagram in Example 4. 5. Can you assume that plane S intersects plane T at ⃖""⃗ BC? 6. Explain how you know that ⃖""⃗ AB ⊥ ⃖""⃗ BC.



Chapter 9

Reasoning and Proofs

Exercises



Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE Through any __________ noncollinear points, there exists exactly

one plane. 2. WRITING Explain why you need at least three noncollinear points to determine a plane.

Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, state the postulate illustrated by the diagram. (See Example 1.) 3.

A

W

A

then

If B

4.

In Exercises 13–20, use the diagram to determine whether you can assume the statement. (See Example 4.)

R A

then

C

N

J

H

K G

q

K

X

L

P

B

In Exercises 5–8, use the diagram to write an example of the postulate. (See Example 2.) p

M

J

B

If

Q

M

L

13. Planes W and X intersect at ⃖""⃗ KL. 14. Points K, L, M, and N are coplanar. 15. Points Q, J, and M are collinear.

⃖"""⃗ and ⃖""⃗ 16. MN RP intersect.

5. Line-Point Postulate

17. ⃖""⃗ JK lies in plane X.

6. Line Intersection Postulate

19. ∠NKL and ∠JKM are vertical angles.

7. Three Point Postulate

20. ∠NKJ and ∠JKM are supplementary angles.

8. Plane-Line Postulate

ERROR ANALYSIS In Exercises 21

In Exercises 9–12, sketch a diagram of the description. (See Example 3.)

18. ∠PLK is a right angle.

and 22, describe and correct the error in the statement made about the diagram.

A

D

M

B

9. plane P and line m intersecting plane P at a 90° angle

— in plane P, XY — bisected by point A, and point C not 10. XY — on XY

— intersecting WV — at point A, so that XA = VA 11. XY —, CD —, and EF — are all in plane P, and point X is the 12. AB midpoint of all three segments.

C

21.

22.

  Section 9.3

— and BD —. M is the midpoint of AC — intersects BD — at a 90° angle, AC — — so AC ⊥ BD .

Postulates and Diagrams



In Exercises 27 and 28, (a) rewrite the postulate in if-then form. Then (b) write the converse, inverse, and contrapositive and state which ones are true.

23. ATTENDING TO PRECISION Select all the statements

about the diagram that you cannot conclude. T

27. Two Point Postulate

A

S

B

H

28. Plane-Point Postulate C

D

29. REASONING Choose the correct symbol to go

between the statements. F

number of points to determine a line

A A, B, and C are coplanar. ○




31. MAKING AN ARGUMENT Your friend claims that even

AF ⊥ ⃖""⃗ CD. H ⃖""⃗ ○

though two planes intersect in a line, it is possible for three planes to intersect in a point. Is your friend correct? Explain your reasoning.

24. HOW DO YOU SEE IT? Use the diagram of line m

and point C. Make a conjecture about how many planes can be drawn so that line m and point C lie in the same plane. Use postulates to justify your conjecture.

32. MAKING AN ARGUMENT Your friend claims that

by the Plane Intersection Postulate, any two planes intersect in a line. Is your friend’s interpretation of the Plane Intersection Postulate correct? Explain your reasoning.

C m

33. ABSTRACT REASONING Points E, F, and G all lie

25. MATHEMATICAL CONNECTIONS One way to graph a

linear equation is to plot two points whose coordinates satisfy the equation and then connect them with a line. Which postulate guarantees this process works for any linear equation?

in plane P and in plane Q. What must be true about points E, F, and G so that planes P and Q are different planes? What must be true about points E, F, and G to force planes P and Q to be the same plane? Make sketches to support your answers. 34. THOUGHT PROVOKING The postulates in this book

represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. A plane is the surface of the sphere. Find a postulate on page 460 that is not true in spherical geometry. Explain your reasoning.

26. MATHEMATICAL CONNECTIONS A way to solve

a system of two linear equations that intersect is to graph the lines and find the coordinates of their intersection. Which postulate guarantees this process works for any two linear equations?

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Solve the equation. Tell which algebraic property of equality you used.

Chapter 9

≥

intersect in exactly one point by the Line Intersection Postulate. Do the two lines have to be in the same plane? Draw a picture to support your answer. Then explain your reasoning.

E Plane T ⊥ plane S. ○



=

30. CRITICAL THINKING If two lines intersect, then they

D H, F, and D are coplanar. ○

35. t − 6 = −4

≤

number of points to determine a plane

36. 3x = 21

Reasoning and Proofs

37. 9 + x = 13

(Section 1.1) x 7

38. — = 5

o

What Did You Learn?

Core Vocabulary conditional statement, p. 442 if-then form, p. 442 hypothesis, p. 442 conclusion, p. 442 negation, p. 442 converse, p. 443

inverse, p. 443 contrapositive, p. 443 equivalent statements, p. 443 biconditional statement, p. 445 truth value, p. 446 truth table, p. 446

inductive reasoning, p. 452 counterexample, p. 453 deductive reasoning, p. 454 line perpendicular to a plane, p. 462

Core Concepts Section 9.1 Conditional Statement, p. 442 Negation, p. 442 Related Conditionals, p. 443

Biconditional Statement, p. 445 Making a Truth Table, p. 446

Section 9.2 Inductive Reasoning, p. 452 Counterexample, p. 453

Deductive Reasoning, p. 454 Laws of Logic, p. 454

Section 9.3 Point, Line, and Plane Postulates, p. 460 Identifying Postulates, p. 461 Sketching and Interpreting Diagrams, p. 462

Mathematical Practices 1.

Provide a counterexample for each false conditional statement in Exercises 17–24 on page 447. (You do not need to consider the converse, inverse, and contrapositive statements.)

2.

Create a truth table for each of your answers to Exercise 59 on page 450.

3.

For Exercise 32 on page 464, write a question you would ask your friend about his or her interpretation.

Completing Homework k Efficientlyy Before doing homework, review the Core Concepts and examples. Use the tutorials at BigIdeasMath.com m for additional help. Complete homework as though you are also preparing for a quiz. Memorize different types of problems, vocabulary, rules, and so on.

  

o

Quiz

Rewrite the conditional statement in if-then form. Then write the converse, inverse, and contrapositive of the conditional statement. Decide whether each statement is true or false. (Section 9.1) 1. An angle measure of 167° is an obtuse angle. 2. You are in a physics class, so you always have homework. 3. I will take my driving test, so I will get my driver’s license.

Find a counterexample to show that the conjecture is false. (Section 9.2) 4. The sum of a positive number and a negative number is always positive. 5. If a figure has four sides, then it is a rectangle.

Use inductive reasoning to make a conjecture about the given quantity. Then use deductive reasoning to show that the conjecture is true. (Section 9.2) 6. the sum of two negative integers 7. the difference of two even integers

Use the diagram to determine whether you can assume the statement. (Section 9.3) 8. Points D, B, and C are coplanar. 9. Plane EAF is parallel to plane DBC. 10. 11.

D

m

B

E

G

A

Line m intersects line ⃖""⃗ AB at point A. ⃖""⃗ lies in plane DBC. Line DC

C

F

12. m∠DBG = 90°

13. You and your friend are bowling. Your friend claims

that the statement “If I got a strike, then I used the green ball” can be written as a true biconditional statement. Is your friend correct? Explain your reasoning. (Section 9.1)

Females

Males

06:43

05:41

07:22

06:07

07:04

05:13

06:39

05:21

06:56

06:01

14. The table shows the 1-mile running times of the

members of a high school track team. (Section 9.2) a. What conjecture can you make about the running times of females and males? b. What type of reasoning did you use? Explain.

A B

15. List five of the seven Point, Line, and Plane

Postulates on page 460 that the diagram of the house demonstrates. Explain how the postulate is demonstrated in the diagram. (Section 9.3)




Chapter 9

Reasoning and Proofs

Y

C m G

X

E

n

D



Proving Statements about Segments and Angles Essential Question

How can you prove a mathematical statement?

A proof is a logical argument that uses deductive reasoning to show that a statement is true.

Writing Reasons in a Proof Work with a partner. Four steps of a proof are shown. Write the reasons for each statement.

REASONING ABSTRACTLY To be proficient in math, you need to know and be able to use algebraic properties.

Given AC = AB + AB

A

Prove AB = BC

B

STATEMENTS

REASONS

1. AC = AB + AB

1. Given

2. AB + BC = AC

2.

3. AB + AB = AB + BC

3.

4. AB = BC

4.

C

Writing Steps in a Proof Work with a partner. Six steps of a proof are shown. Complete the statements that correspond to each reason. Given

m∠1 = m∠3

Prove

m∠EBA = m∠CBD

D

E C

123

A

B

STATEMENTS

REASONS

1.

1. Given

2. m∠EBA = m∠2 + m∠3

2. Angle Addition Postulate

3. m∠EBA = m∠2 + m∠1

3. Substitution Property of Equality

4. m∠EBA =

4. Commutative Property of Addition

5. m∠1 + m∠2 =

5. Angle Addition Postulate

6.

6. Transitive Property of Equality

Communicate Your Answer 3. How can you prove a mathematical statement? 4. Use the given information and the figure to write a proof for the statement.

—. Given B is the midpoint of AC —. C is the midpoint of BD Prove AB = CD

Section 9.4

A

B

C

Proving Statements about Segments and Angles

D



 Lesson

What You Will Learn Use properties of equality involving segment lengths and angle measures. Write two-column proofs.

Core Vocabul Vocabulary larry

Name and prove properties of congruence.

proof, p. 470 two-column proof, p. 470

Using Other Properties of Equality The following properties of equality are true for all real numbers. Segment lengths and angle measures are real numbers, so these properties of equality are true for all segment lengths and angle measures.

Previous theorem

Core Concept Reflexive, Symmetric, and Transitive Properties of Equality Real Numbers

Segment Lengths

Angle Measures

a=a

AB = AB

m∠A = m∠A

If a = b, then b = a.

If AB = CD, then CD = AB.

If m∠A = m∠B, then m∠B = m∠A.

If a = b and b = c, then a = c.

If AB = CD and CD = EF, then AB = EF.

If m∠A = m∠B and m∠B = m∠C, then m∠A = m∠C.

Reflexive Property Symmetric Property Transitive Property

Using Properties of Equality with Angle Measures You reflect the beam of a spotlight off a mirror lying flat on a stage, as shown. Determine whether m∠DBA = m∠EBC. D

E 1

SOLUTION Equation

CONNECTIONS TO ALGEBRA Because angle measures are real numbers, you can use the properties of equality from Section 1.1.

2

C

B

Reason

m∠1 = m∠3

Marked in diagram.

Given

m∠DBA = m∠3 + m∠2

Add measures of adjacent angles.

Angle Addition Postulate

m∠DBA = m∠1 + m∠2

Substitute m∠1 for m∠3.

Substitution Property of Equality

m∠1 + m∠2 = m∠EBC

Add measures of adjacent angles.

Angle Addition Postulate

m∠DBA = m∠EBC

Both measures are equal to the sum m∠1 + m∠2.

Transitive Property of Equality

Help in English and Spanish at BigIdeasMath.com

Name the property of equality that the statement illustrates. 1. If m∠6 = m∠7, then m∠7 = m∠6.

2. 34° = 34°

3. m∠1 = m∠2 and m∠2 = m∠5. So, m∠1 = m∠5.

Chapter 9

A

Explanation

Monitoring Progress



3

Reasoning and Proofs

Modeling with Mathematics A park, a shoe store, a pizza shop, and a movie theater are located in order on a city street. The distance between the park and the shoe store is the same as the distance between the pizza shop and the movie theater. Show that the distance between the park and the pizza shop is the same as the distance between the shoe store and the movie theater.

SOLUTION 1. Understand the Problem You know that the locations lie in order and that the distance between two of the locations (park and shoe store) is the same as the distance between the other two locations (pizza shop and movie theater). You need to show that two of the other distances are the same. 2. Make a Plan Draw and label a diagram to represent the situation. park

shoe store

pizza shop

movie theater

Modify your diagram by letting the points P, S, Z, and M represent the park, the shoe store, the pizza shop, and the movie theater, respectively. Show any mathematical relationships. P

S

Z

M

Use the Segment Addition Postulate to show that PZ = SM. 3. Solve the Problem Equation

CONNECTIONS TO ALGEBRA Because segment lengths are real numbers, you can use the properties of equality from Section 1.1.

Explanation

Reason

PS = ZM

Marked in diagram. Given

PZ = PS + SZ

Add lengths of adjacent segments.

Segment Addition Postulate

SM = SZ + ZM

Add lengths of adjacent segments.

Segment Addition Postulate

PS + SZ = ZM + SZ

Add SZ to each side Addition Property of Equality of PS = ZM.

PZ = SM

Substitute PZ for PS + SZ and SM for ZM + SZ.

Substitution Property of Equality

4. Look Back Reread the problem. Make sure your diagram is drawn precisely using the given information. Check the steps in your solution.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Name the property of equality that the statement illustrates. 4. If JK = KL and KL = 16, then JK = 16. 5. PQ = ST, so ST = PQ. 6. ZY = ZY 7. In Example 2, a hot dog stand is located halfway between the shoe store and the

pizza shop, at point H. Show that PH = HM.

Section 9.4

Proving Statements about Segments and Angles



Writing Two-Column Proofs A proof is a logical argument that uses deductive reasoning to show that a statement is true. There are several formats for proofs. A two-column proof has numbered statements and corresponding reasons that show an argument in a logical order. In a two-column proof, each statement in the left-hand column is either given information or the result of applying a known property or fact to statements already made. Each reason in the right-hand column is the explanation for the corresponding statement.

Writing a Two-Column Proof Write a two-column proof for the situation in Example 1. Given m∠l = m∠3

D

Prove m∠DBA = m∠EBC

E 2

1 C

3

B

A

STATEMENTS

REASONS

1. m∠1 = m∠3

1. Given

2. m∠DBA = m∠3 + m∠2

2. Angle Addition Postulate

3. m∠DBA = m∠1 + m∠2

3. Substitution Property of Equality

4. m∠1 + m∠2 = m∠EBC

4. Angle Addition Postulate

5. m∠DBA = m∠EBC

5. Transitive Property of Equality

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

8. Six steps of a two-column proof are shown. Copy and complete the proof.

—. Given T is the midpoint of SU

S

7x

Prove x = 5 T

3x + 20 U

STATEMENTS

—. 1. T is the midpoint of SU

— ≅ TU — 2. ST



Chapter 9

REASONS 1. ________________________________ 2. Definition of midpoint

3. ST = TU

3. Definition of congruent segments

4. 7x = 3x + 20

4. ________________________________

5. ________________________

5. Subtraction Property of Equality

6. x = 5

6. ________________________________

Reasoning and Proofs

Using Properties of Congruence The reasons used in a proof can include definitions, properties, postulates, and theorems. Recall that a theorem is a statement that can be proven. Once you have proven a theorem, you can use the theorem as a reason in other proofs.

Theorems Properties of Segment Congruence Segment congruence is reflexive, symmetric, and transitive. Reflexive Symmetric Transitive

— ≅ AB —. For any segment AB, AB

— ≅ CD —, then CD — ≅ AB —. If AB

— ≅ CD — and CD — ≅ EF —, then AB — ≅ EF —. If AB

Proofs Ex. 35, p. 475; Example 5, p. 471; Chapter Review 9.4 Example, p. 490

Properties of Angle Congruence Angle congruence is reflexive, symmetric, and transitive. Reflexive

For any angle A, ∠A ≅ ∠A.

Symmetric

If ∠A ≅ ∠B, then ∠B ≅ ∠A.

Transitive

If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.

Proofs Ex. 21, p. 490; 9.4 Concept Summary, p. 472; Ex. 36, p. 475

Naming Properties of Congruence Name the property that the statement illustrates. a. If ∠T ≅ ∠V and ∠V ≅ ∠R, then ∠T ≅ ∠R.

— ≅ YZ —, then YZ — ≅ JL —. b. If JL SOLUTION

a. Transitive Property of Angle Congruence b. Symmetric Property of Segment Congruence

STUDY TIP When writing a proof, organize your reasoning by copying or drawing a diagram for the situation described. Then identify the Given and Prove statements.

In this lesson, most of the proofs involve showing that congruence and equality are equivalent. You may find that what you are asked to prove seems to be obviously true. It is important to practice writing these proofs to help you prepare for writing more-complicated proofs in later chapters.

Proving a Symmetric Property of Congruence Write a two-column proof for the Symmetric Property of Segment Congruence.

— ≅ NP — Given LM Prove

— ≅ LM — NP

L

STATEMENTS

REASONS

1.

1. Given

— ≅ NP — LM

M

N

2. LM = NP

2. Definition of congruent segments

3. NP = LM

3. Symmetric Property of Equality

4. NP ≅ LM

4. Definition of congruent segments

— —

Section 9.4

P

Proving Statements about Segments and Angles



Writing a Two-Column Proof

—, prove Prove this property of midpoints: If you know that M is the midpoint of AB that AB is two times AM and AM is one-half AB. Given

—. M is the midpoint of AB

Prove

AB = 2AM, AM = —12 AB

STATEMENTS

—. 1. M is the midpoint of AB

— ≅ MB — 2. AM

A

M

B

REASONS 1. Given 2. Definition of midpoint

3. AM = MB

3. Definition of congruent segments

4. AM + MB = AB

4. Segment Addition Postulate

5. AM + AM = AB

5. Substitution Property of Equality

6. 2AM = AB

6. Distributive Property

1

7. AM = —2 AB

7. Division Property of Equality

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Name the property that the statement illustrates.

— ≅ GH — 9. GH

10. If ∠K ≅ ∠P, then ∠P ≅ ∠K.

11. Look back at Example 6. What would be different if you were proving that

⋅

AB = 2 MB and that MB = —12 AB instead?

Concept Summary Writing a Two-Column Proof In a proof, you make one statement at a time until you reach the conclusion. Because you make statements based on facts, you are using deductive reasoning. Usually the first statement-and-reason pair you write is given information.

1

2

Proof of the Symmetric Property of Angle Congruence

Given ∠1 ≅ ∠2 Prove ∠2 ≅ ∠1 statements based on facts that you know or on conclusions from deductive reasoning

STATEMENTS

REASONS

1. ∠1 ≅ ∠2

1. Given

2. m∠1 = m∠2

2. Definition of congruent angles

3. m∠2 = m∠1

3. Symmetric Property of Equality

4. ∠2 ≅ ∠1

4. Definition of congruent angles

The number of statements will vary.



Chapter 9

Copy or draw diagrams and label given information to help develop proofs. Do not mark or label the information in the Prove statement on the diagram.

Reasoning and Proofs

Remember to give a reason for the last statement.

definitions, postulates, or proven theorems that allow you to state the corresponding statement



Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY The statement “The measure of an angle is equal to itself” is true because of

what property? 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers.

What property justifies the following statement? If c = d, then d = c.

If JK = LM, then LM = JK.

If e = f and f = g, then e = g.

If m∠ R = m∠ S, then m∠ S = m∠ R.

3. WRITING How is a theorem different from a postulate? 4. COMPLETE THE SENTENCE In a two-column proof, each ______ is on the left and each _____ is on

the right.

Monitoring Progress and Modeling with Mathematics In Exercises 5 –10, name the property of equality that the statement illustrates. 5. If AM = MB, then AM + 5 = MB + 5.

15. Subtraction Property of Equality:

If LM = XY, then LM − GH = ____. 16. Distributive Property:

If 5(AB + 8) = 2, then ____ + ____ = 2.

6. m∠Z = m∠Z 7. If m∠A = 29° and m∠B = 29°, then m∠A = m∠B.

17. Transitive Property of Equality:

If m∠1 = m∠2 and m∠2 = m∠3, then ____.

8. If AB = LM, then LM = AB. 9. If BC = XY and XY = 8, then BC = 8. 10. If m∠Q = m∠R, then m∠R = m∠Q.

In Exercises 11–18, use the property to copy and complete the statement.

18. Reflexive Property of Equality:

m∠ABC = ____. 19. ANALYZING RELATIONSHIPS In the diagram,

m∠ABD = m∠CBE. Show that m∠1 = m∠3. (See Example 1.) A

11. Substitution Property of Equality:

If AB = 20, then AB + CD = ____. B

12. Symmetric Property of Equality:

If m∠1 = m∠2, then ____.

1 2 3

C D

E

13. Addition Property of Equality:

If AB = CD, then AB + EF = ____.

20. ANALYZING RELATIONSHIPS In the diagram,

AC = BD. Show that AB = CD. (See Example 2.)

14. Multiplication Property of Equality:

⋅

A

If AB = CD, then 5 AB = ____.

Section 9.4

B

C

Proving Statements about Segments and Angles

D



REASONING In Exercises 22 and 23, show that the

21. ANALYZING RELATIONSHIPS Copy and complete the

table to show that m∠2 = m∠3.

perimeter of △ABC is equal to the perimeter of △ADC. A

22. F

1

2 3

E

D

4

H

G

Equation

A

23.

Given

D

C

24. ERROR ANALYSIS Describe and correct the error in

m∠EHF = m∠GHF

the reasoning.

m∠EHF = m∠1 + m∠2 m∠GHF = m∠3 + m∠4



m∠1 + m∠2 = m∠3 + m∠4 Substitution Property of Equality

If m∠1 = 30° and m∠2 = 30°, then m∠1 = m∠2 by the Symmetric Property of Equality.

m∠2 = m∠3 In Exercises 25 and 26, copy and complete the proof. (See Example 3.) 25. Given PQ = RS

P

Prove PR = QS

R

Q

S

STATEMENTS

REASONS

1. PQ = RS

1. ___________________________

2. PQ + QR = RS + QR

2. ___________________________

3. ___________________

3. Segment Addition Postulate

4. RS + QR = QS

4. Segment Addition Postulate

5. PR = QS

5. ___________________________

26. Given ∠1 is a complement of ∠2.

∠2 ≅ ∠3

Prove ∠1 is a complement of ∠3.

1

2 3

STATEMENTS

REASONS

1. ∠1 is a complement of ∠2.

1. Given

2. ∠2 ≅ ∠3

2. ___________________________

3. m∠1 + m∠2 = 90°

3. ___________________________

4. m∠2 = m∠3

4. Definition of congruent angles

5. ______________________

5. Substitution Property of Equality

6. ∠1 is a complement of ∠3.

6. ___________________________

Chapter 9

B

Reason

m∠1 = m∠4, m∠EHF = 90°, m∠GHF = 90°



B

C

Reasoning and Proofs

1 2

In Exercises 27–32, name the property that the statement illustrates. (See Example 4.)

39. MODELING WITH MATHEMATICS The distance from

the restaurant to the shoe store is the same as the distance from the café to the florist. The distance from the shoe store to the movie theater is the same as the distance from the movie theater to the cafe, and from the florist to the dry cleaners.

— ≅ ST — and ST — ≅ UV —, then PQ — ≅ UV —. 27. If PQ

28. ∠F ≅ ∠F 29. If ∠G ≅ ∠H, then ∠H ≅ ∠G.

— ≅ DE — 30. DE 31.

Flowers SHOE STORE

—, then UV — ≅ XY —. — ≅ UV If XY

restaurant shoe movie store theater

32. If ∠L ≅ ∠M and ∠M ≅ ∠N, then ∠L ≅ ∠N. 33. ERROR ANALYSIS

— ≅ PN —. Describe and correct the error in and LQ the reasoning.



Reflexive Property of Segment Congruence.

L

P

florist

dry cleaners

a. State what is given and what is to be proven for the situation.

M

Q

café

Use the steps below to prove that the distance from the restaurant to the movie theater is the same as the distance from the café to the dry cleaners.

— ≅ LQ — In the diagram, MN

— ≅ LQ — Because MN — — and LQ ≅ PN , then — ≅ PN — by the MN

DRY CLEANERS

b. Write a two-column proof.

N

40. ANALYZING RELATIONSHIPS The bar graph shows

34. WRITING Compare the Reflexive Property of Segment

the number of hours each employee works at a grocery store. Give an example of the Reflexive, Symmetric, and Transitive Properties of Equality.

Congruence with the Symmetric Property of Segment Congruence. How are the properties similar? How are they different? Hours

Hours Worked

PROOF In Exercises 35 and 36, write a two-column

proof for the property. (See Example 5.) 35. Reflexive Property of Segment Congruence 36. Transitive Property of Angle Congruence

70 60 50 40 30 20 10 0

1

2

3

4

5

6

7

Employee

PROOF In Exercises 37 and 38, write a two-column

proof. (See Example 6.)

41. ATTENDING TO PRECISION Which of the following

37. Given ∠GFH ≅ ∠GHF

Prove ∠EFG and ∠GHF are supplementary. G

E F

38. Given

H

— ≅ FG —, AB

— and DG —. ⃖##⃗ BF bisects AC

— ≅ DF — Prove BC

A D B F C

statements illustrate the Symmetric Property of Equality? Select all that apply.

A ○ B ○ C ○ D ○ E ○ F ○

If AC = RS, then RS = AC. If AB = 9, then 9 = AB. If AD = BC, then DA = CB. AB = BA If AB = LM and LM = RT, then AB = RT. If XY = EF, then FE = XY.

42. WRITING Write examples from your everyday life

to help you remember the Reflexive, Symmetric, and Transitive Properties of Equality. Justify your answers.

G

Section 9.4

Proving Statements about Segments and Angles



43. REASONING In the sculpture

47. REASONING Fold two corners of a piece of paper

shown, ∠1 ≅ ∠2 and ∠2 ≅ ∠3. Classify the triangle and justify your answer.

so their edges match, as shown. 3 1

a. What do you notice about the angle formed at the top of the page by the folds?

2

— ≅ QR —. Your friend claims that, because of this, AC — — by the Transitive Property of Segment CB ≅ AC Congruence. Is your friend correct? Explain your reasoning.

48. THOUGHT PROVOKING The distance from Springfield

Q

C

B

S

to Lakewood City is equal to the distance from Springfield to Bettsville. Janisburg is 50 miles farther from Springfield than Bettsville. Moon Valley is 50 miles farther from Springfield than Lakewood City is. Use line segments to draw a diagram that represents this situation.

R

45. WRITING Explain why you do not use inductive

reasoning when writing a proof.

49. MATHEMATICAL CONNECTIONS Solve for x using the

46. HOW DO YOU SEE IT? Use the figure to write Given

given information. Justify each step.

and Prove statements for each conclusion.

— ≅ PQ —, RS — ≅ PQ — Given QR

J

P

K

N

Q 2x + 5 R

M

L

S 10 ] 3x

50. MATHEMATICAL CONNECTIONS In the figure,

— ≅ XW —, ZX = 5x + 17, YW = 10 − 2x, and ZY YX = 3. Find ZY and XW.

a. The acute angles of a right triangle are complementary.

V

b. A segment connecting the midpoints of two sides of a triangle is half as long as the third side.

Y X

Z

Maintaining Mathematical Proficiency Use the figure.

W

Reviewing what you learned in previous grades and lessons

(Section 8.6)

51. ∠1 is a complement of ∠4,

1

52. ∠3 is a supplement of ∠2,

and m∠1 = 33°. Find m∠4.

2 3

and m∠2 = 147°. Find m∠3.

4

53. Name a pair of vertical angles. 54. The double box-and-whisker plot represents the quiz scores of two students. Identify the shape of

each distribution. Which student’s scores are more spread out?

(Section 7.2)

Student A Student B 55



Chapter 9

60

65

2

2

b. Write a two-column proof to show that the angle measure is always the same no matter how you make the folds.

— ≅ CB — and 44. MAKING AN ARGUMENT In the figure, SR

A

1 1

70

Reasoning and Proofs

75

80

85

90

95

100

Quiz score



Proving Geometric Relationships Essential Question

How can you use a flowchart to prove a

mathematical statement?

Matching Reasons in a Flowchart Proof Work with a partner. Match each reason with the correct step in the flowchart.

MODELING WITH MATHEMATICS To be proficient in math, you need to map relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas.

Given AC = AB + AB

B

A

Prove AB = BC

C

AC = AB + AB

AB + AB = AB + BC

AB + BC = AC

AB = BC

A. Segment Addition Postulate

B. Given

C. Transitive Property of Equality

D. Subtraction Property of Equality

Matching Reasons in a Flowchart Proof Work with a partner. Match each reason with the correct step in the flowchart. Given m∠1 = m∠3

D

E

Prove m∠EBA = m∠CBD

C

m∠ 1 = m∠ 3

123

A

B

m∠ EBA = m∠ 2 + m∠ 3

m∠ EBA = m∠ 2 + m∠ 1

m∠ EBA = m∠ 1 + m∠ 2

m∠ 1 + m∠ 2 = m∠ CBD

m∠ EBA = m∠ CBD

A. Angle Addition Postulate

B. Transitive Property of Equality

C. Substitution Property of Equality

D. Angle Addition Postulate

E. Given

F. Commutative Property of Addition

Communicate Your Answer 3. How can you use a flowchart to prove a mathematical statement? 4. Compare the flowchart proofs above with the two-column proofs in the

Section 9.4 Explorations. Explain the advantages and disadvantages of each. Section 9.5

Proving Geometric Relationships



 Lesson

What You Will Learn Write flowchart proofs to prove geometric relationships. Write paragraph proofs to prove geometric relationships.

Core Vocabul Vocabulary larry flowchart proof, or flow proof, p. 478 paragraph proof, p. 480

Writing Flowchart Proofs Another proof format is a flowchart proof, or flow proof, which uses boxes and arrows to show the flow of a logical argument. Each reason is below the statement it justifies. A flowchart proof of the Right Angles Congruence Theorem is shown in Example 1. This theorem is useful when writing proofs involving right angles.

Theorem Right Angles Congruence Theorem All right angles are congruent. Proof Example 1, p. 478

STUDY TIP When you prove a theorem, write the hypothesis of the theorem as the Given statement. The conclusion is what you must Prove.

Proving the Right Angles Congruence Theorem Use the given flowchart proof to write a two-column proof of the Right Angles Congruence Theorem. Given ∠1 and ∠2 are right angles. Prove ∠1 ≅ ∠2

2

1

Flowchart Proof ∠1 and ∠2 are right angles. Given

m∠1 = 90°, m∠2 = 90°

m∠l = m∠2

Definition of right angle

Transitive Property of Equality

∠l ≅ ∠2 Definition of congruent angles

Two-Column Proof STATEMENTS

REASONS

1. ∠1 and ∠2 are right angles.

1. Given

2. m∠1 = 90°, m∠2 = 90°

2. Definition of right angle

3. m∠1 = m∠2

3. Transitive Property of Equality

4. ∠1 ≅ ∠2

4. Definition of congruent angles

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1. Copy and complete the flowchart proof.

C

Then write a two-column proof.

— ⊥ BC —, DC — ⊥ BC — Given AB Prove ∠B ≅ ∠C

A

— ⊥ BC —, DC — ⊥ BC — AB Given



Chapter 9

Reasoning and Proofs

B

∠B ≅ ∠C Definition of ⊥ lines

D

Theorems Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. 2

1

If ∠1 and ∠2 are supplementary and ∠3 and ∠2 are supplementary, then ∠1 ≅ ∠3.

3

Proof Example 2, p. 479 (case 1); Ex. 20, p. 485 (case 2)

Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent. If ∠4 and ∠5 are complementary and ∠6 and ∠5 are complementary, then ∠4 ≅ ∠6.

4

5

6

Proof Ex. 19, p. 484 (case 1); Ex. 22, p. 485 (case 2) To prove the Congruent Supplements Theorem, you must prove two cases: one with angles supplementary to the same angle and one with angles supplementary to congruent angles. The proof of the Congruent Complements Theorem also requires two cases.

Proving a Case of Congruent Supplements Theorem Use the given two-column proof to write a flowchart proof that proves that two angles supplementary to the same angle are congruent. Given ∠1 and ∠2 are supplementary. ∠3 and ∠2 are supplementary. Prove ∠1 ≅ ∠3

1

2

3

Two-Column Proof STATEMENTS

REASONS

1. ∠1 and ∠2 are supplementary.

1. Given

2. m∠1 + m∠2 = 180°,

2. Definition of supplementary angles

3. m∠1 + m∠2 = m∠3 + m∠2

3. Transitive Property of Equality

4. m∠1 = m∠3

4. Subtraction Property of Equality

5. ∠1 ≅ ∠3

5. Definition of congruent angles

∠3 and ∠2 are supplementary.

m∠3 + m∠2 = 180°

Flowchart Proof ∠1 and ∠2 are supplementary. Given

Definition of supplementary angles

∠3 and ∠2 are supplementary. Given

m∠1 + m∠2 = 180°

m∠3 + m∠2 = 180° i i off Defifinition supplementary angles

m∠1 + m∠2 = m∠3 + m∠2 Transitive Property of Equality

m∠1 = m∠3 Subtraction Property of Equality

∠1 ≅ ∠3 Defi D finition i i off congruent angles

Section 9.5

Proving Geometric Relationships



Writing Paragraph Proofs Another proof format is a paragraph proof, which presents the statements and reasons of a proof as sentences in a paragraph. It uses words to explain the logical flow of the argument. Two intersecting lines form pairs of vertical angles and linear pairs. The Linear Pair Postulate formally states the relationship between linear pairs. You can use this postulate to prove the Vertical Angles Congruence Theorem.

Postulate and Theorem Linear Pair Postulate If two angles form a linear pair, then they are supplementary. ∠1 and ∠2 form a linear pair, so ∠1 and ∠2 are supplementary and m∠l + m∠2 = 180°.

1

2

Vertical Angles Congruence Theorem Vertical angles are congruent. 1

2 4

3

∠1 ≅ ∠3, ∠2 ≅ ∠4

Proof Example 3, p. 480

Proving the Vertical Angles Congruence Theorem Use the given paragraph proof to write a two-column proof of the Vertical Angles Congruence Theorem. Given ∠5 and ∠7 are vertical angles.

STUDY TIP In paragraph proofs, transitional words such as so, then, and therefore help make the logic clear.

7 5 6

Prove ∠5 ≅ ∠7

Paragraph Proof ∠5 and ∠7 are vertical angles formed by intersecting lines. As shown in the diagram, ∠5 and ∠6 are a linear pair, and ∠6 and ∠7 are a linear pair. Then, by the Linear Pair Postulate, ∠5 and ∠6 are supplementary and ∠6 and ∠7 are supplementary. So, by the Congruent Supplements Theorem, ∠5 ≅ ∠7.

JUSTIFYING STEPS You can use information labeled in a diagram in your proof.

Two-Column Proof STATEMENTS

REASONS

1. ∠5 and ∠7 are vertical angles.

1. Given

2. ∠5 and ∠6 are a linear pair.

2. Definition of linear pair,

∠6 and ∠7 are a linear pair.

3. ∠5 and ∠6 are supplementary.

3. Linear Pair Postulate

4. ∠5 ≅ ∠7

4. Congruent Supplements Theorem

∠6 and ∠7 are supplementary.



Chapter 9

as shown in the diagram

Reasoning and Proofs

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

2. Copy and complete the two-column proof. Then write a flowchart proof.

Given Prove

AB = DE, BC = CD

— ≅ CE — AC

A

B

C

D

E

STATEMENTS

REASONS

1. AB = DE, BC = CD

1. Given

2. AB + BC = BC + DE

2. Addition Property of Equality

3. ___________________________

3. Substitution Property of Equality

4. AB + BC = AC, CD + DE = CE

4. _____________________________

5. ___________________________

5. Substitution Property of Equality

— ≅ CE — 6. AC

6. _____________________________

3. Rewrite the two-column proof in Example 3 without using the Congruent

Supplements Theorem. How many steps do you save by using the theorem?

Using Angle Relationships Find the value of x.

Q

SOLUTION ∠TPS and ∠QPR are vertical angles. By the Vertical Angles Congruence Theorem, the angles are congruent. Use this fact to write and solve an equation. m∠TPS = m∠QPR 148° = (3x + 1)° 147 = 3x

(3x + 1)°

T

148° P R

S

Definition of congruent angles Substitute angle measures. Subtract 1 from each side.

49 = x

Divide each side by 3.

So, the value of x is 49.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use the diagram and the given angle measure to find the other three angle measures. 4

4. m∠1 = 117°

1 3

2

5. m∠2 = 59° 6. m∠4 = 88° 7. Find the value of w.

(5w + 3)° 98°

Section 9.5

Proving Geometric Relationships



Using the Vertical Angles Congruence Theorem Write a paragraph proof. Given ∠1 ≅ ∠4 Prove ∠2 ≅ ∠3

1

2

3

4

Paragraph Proof ∠1 and ∠4 are congruent. By the Vertical Angles Congruence Theorem, ∠1 ≅ ∠2 and ∠3 ≅ ∠4. By the Transitive Property of Angle Congruence, ∠2 ≅ ∠4. Using the Transitive Property of Angle Congruence once more, ∠2 ≅ ∠3.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

8. Write a paragraph proof.

Given ∠1 is a right angle. 2

Prove ∠2 is a right angle.

Concept Summary Types of Proofs Symmetric Property of Angle Congruence

Given ∠1 ≅ ∠2 Prove ∠2 ≅ ∠1 1

2

Two-Column Proof STATEMENTS

REASONS

1. ∠1 ≅ ∠2

1. Given

2. m∠1 = m∠2

2. Definition of congruent angles

3. m∠2 = m∠1

3. Symmetric Property of Equality

4. ∠2 ≅ ∠1

4. Definition of congruent angles

Flowchart Proof

∠1 ≅ ∠2

m∠1 = m∠2

m∠2 = m∠1

∠2 ≅ ∠1

Given

Definition of congruent angles

Symmetric Property of Equality

Definition of congruent angles

Paragraph Proof

∠1 is congruent to ∠2. By the definition of congruent angles, the measure of ∠1 is equal to the measure of ∠2. The measure of ∠2 is equal to the measure of ∠1 by the Symmetric Property of Equality. Then by the definition of congruent angles, ∠2 is congruent to ∠1. 

Chapter 9

Reasoning and Proofs

1

Exercises



Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Explain why all right angles are congruent. 2. VOCABULARY What are the two types of angles that are formed by intersecting lines?

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, identify the pair(s) of congruent angles in the figures. Explain how you know they are congruent. (See Examples 1, 2, and 3.)

In Exercises 11–14, find the values of x and y. (See Example 4.) 11.

P 50°

N 50°

5y°

Q

S

(7y − 12)°

(6y + 8)°

(6x − 26)°

R

13. F

J

G

4x°

(7y − 34)°

(9x − 4)°

M

4.

12.

(8x + 7)°

3.

H

45° 44°

K

L W

58° 32° X Z

M

5.

Y

(18y − 18)°

L

(6x + 50)° 6(x + 2)°

M

ERROR ANALYSIS In Exercises 15 and 16, describe and correct the error in using the diagram to find the value of x.

J

K

2(5x − 5)° (5y + 5)° (7y − 9)°

16y°

H G

14.

(10x − 4)°

(13x + 45)°

6. ∠ABC is supplementary to ∠CBD.

(6x + 2)°

(12x − 40)° (19x + 3)°

∠CBD is supplementary to ∠DEF. F

15. A

B

E

D



C

32x + 48 = 180 32x = 132

In Exercises 7–10, use the diagram and the given angle measure to find the other three measures. (See Example 3.) 7. m∠1 = 143° 8. m∠3 = 159° 9. m∠2 = 34°

(13x + 45)° + (19x + 3)° = 180°

4

1 3

x = 4.125 16.

2



(13x + 45)° + (12x − 40)° = 90° 25x + 5 = 90 25x = 85 x = 3.4

10. m∠4 = 29°

Section 9.5

Proving Geometric Relationships



17. PROOF Copy and complete the flowchart proof. Then write a two-column proof.

(See Example 1.) Given ∠1 ≅ ∠3

1

Prove ∠2 ≅ ∠4

3

2 4

∠1 ≅ ∠3 Given

∠2 ≅ ∠4

∠2 ≅ ∠3

∠l ≅ ∠2, ∠3 ≅ ∠4 Vertical Angles Congruence Theorem

18. PROOF Copy and complete the two-column proof. Then write a flowchart proof.

(See Example 2.) Given ∠ABD is a right angle. ∠CBE is a right angle.

C

A

Prove ∠ABC ≅ ∠DBE D

B

E

STATEMENTS

REASONS

1. ∠ABD is a right angle.

1. ________________________________

2. ∠ABC and ∠CBD are complementary.

2. Definition of complementary angles

3. ∠DBE and ∠CBD are complementary.

3. ________________________________

4. ∠ABC ≅ ∠DBE

4. ________________________________

∠CBE is a right angle.

19. PROVING A THEOREM Copy and complete the paragraph proof for the

Congruent Complements Theorem. Then write a two-column proof. (See Example 3.) Given ∠1 and ∠2 are complementary. ∠1 and ∠3 are complementary. Prove ∠2 ≅ ∠3

3 1

2

∠1 and ∠2 are complementary, and ∠1 and ∠3 are complementary. By the definition of _____________ angles, m∠1 + m∠2 = 90° and ____________ = 90°. By the __________________________, m∠1 + m∠2 = m∠1 + m∠3. By the Subtraction Property of Equality, ___________________. So, ∠2 ≅ ∠3 by the definition of ______________.



Chapter 9

Reasoning and Proofs

20. PROVING A THEOREM Copy and complete the two-column proof for the Congruent Supplement

Theorem. Then write a paragraph proof. (See Example 5.) Given ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. ∠1 ≅ ∠4

1

2

3

4

Prove ∠2 ≅ ∠3 STATEMENTS

REASONS

1. ∠1 and ∠2 are supplementary.

∠3 and ∠4 are supplementary. ∠1 ≅ ∠4

1. Given

2. m∠1 + m∠2 = 180°,

2. ________________________________

3. _________ = m∠3 + m∠4

3. Transitive Property of Equality

4. m∠1 = m∠4

4. Definition of congruent angles

5. m∠1 + m∠2 = _________

5. Substitution Property of Equality

6. m∠2 = m∠3

6. ________________________________

7. ________________________________

7. ________________________________

m∠3 + m∠4 = 180°

PROOF In Exercises 21–24, write a proof using

any format.

23. Given ∠AEB ≅ ∠DEC

Prove ∠AEC ≅ ∠DEB

21. Given ∠QRS and ∠PSR are supplementary.

Prove ∠QRL ≅ ∠PSR

A B L C

Q

R

E

D

M

— ⊥ JM —, KL — ⊥ ML —, 24. Given JK P

S

∠J ≅ ∠M, ∠K ≅ ∠L —⊥— —⊥— Prove JM ML and JK KL

N

K

22. Given ∠1 and ∠3 are complementary.

∠2 and ∠4 are complementary. Prove ∠1 ≅ ∠4

3

K

M

L

25. MAKING AN ARGUMENT You overhear your friend

4 2

J

discussing the diagram shown with a classmate. Your classmate claims ∠1 ≅ ∠4 because they are vertical angles. Your friend claims they are not congruent because he can tell by looking at the diagram. Who is correct? Support your answer with definitions or theorems.

1 2

Section 9.5

1 4

3

Proving Geometric Relationships



26. THOUGHT PROVOKING Draw three lines all

28. WRITING How can you save time writing proofs?

intersecting at the same point. Explain how you can give two of the angle measures so that you can find the remaining four angle measures.

29. MATHEMATICAL CONNECTIONS Find the measure of

each angle in the diagram.

27. CRITICAL THINKING Is the converse of the Linear Pair

Postulate true? If so, write a biconditional statement. Explain your reasoning.

(3y + 11)°

10y °

(4x − 22)°

(7x + 4)°

30. HOW DO YOU SEE IT? Use the student’s two-column proof.

Given ∠1 ≅ ∠2 ∠1 and ∠2 are supplementary. Prove _________________ STATEMENTS

REASONS

1. ∠1 ≅ ∠2

1. Given

2. m∠l = m∠2

2. Definition of congruent angles

3. m∠l + m∠2 = 180°

3. Definition of supplementary angles

4. m∠l + m∠1 = 180°

4. Substitution Property of Equality

5. 2m∠1 = 180°

5. Simplify.

6. m∠1 = 90°

6. Division Property of Equality

7. m∠2 = 90°

7. Transitive Property of Equality

8. _______________________________

8. ___________________________________

∠l and ∠2 are supplementary.

a. What is the student trying to prove? b. Your friend claims that the last line of the proof should be ∠1 ≅ ∠2, because the measures of the angles are both 90°. Is your friend correct? Explain.

Maintaining Mathematical Proficiency Use the cube.

Reviewing what you learned in previous grades and lessons

(Section 8.1) B

31. Name three collinear points. 32. Name the intersection of plane ABF and plane EHG. 33.

—. Name two planes containing BC

I

A

34. Name three planes containing point D.

D F

G

35. Name three points that are not collinear. 36. Name two planes containing point J.



Chapter 9

Reasoning and Proofs

E

C

J

H

o

What Did You Learn?

Core Vocabulary proof, p. 470 two-column proof, p. 470

flowchart proof, or flow proof, p. 478 paragraph proof, p. 480

Core Concepts Section 9.4 Reflexive, Symmetric, and Transitive Properties of Equality, p. 468 Writing Two-Column Proofs, p. 470 Properties of Segment Congruence Theorem, p. 471 Properties of Angle Congruence Theorem, p. 471

Section 9.5 Writing Flowchart Proofs, p. 478 Right Angles Congruence Theorem, p. 478 Congruent Supplements Theorem, p. 479 Congruent Complements Theorem, p. 479

Writing Paragraph Proofs, p. 480 Linear Pair Postulate, p. 480 Vertical Angles Congruence Theorem, p. 480

Mathematical Practices 1.

Explain the purpose of justifying each step in Exercises 25 and 26 on page 474.

2.

Create a diagram to model each statement in Exercises 27–32 on page 475.

3.

Explain why you would not be able to prove the statement in Exercise 21 on page 485 if you were not provided with the given information or able to use any postulates or theorems.

Performance Task:

Reasoning at the Zoo People use both inductive and deductive reasoning to help them make decisions. When you visit the zoo, what conjectures do you make about the animals? What types of reasoning could be used to support or reject your conjectures? To explore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.









9

Chapter Review 9.1

Dynamic Solutions available at BigIdeasMath.com

Conditional Statements (pp. 441–450)

Write the if-then form, the converse, the inverse, the contrapositive, and the biconditional of the conditional statement “A leap year is a year with 366 days.” If-then form: If it is a leap year, then it is a year with 366 days. Converse: If it is a year with 366 days, then it is a leap year. Inverse: If it is not a leap year, then it is not a year with 366 days. Contrapositive: If it is not a year with 366 days, then it is not a leap year. Biconditional: It is a leap year if and only if it is a year with 366 days. Write the if-then form, the converse, the inverse, the contrapositive, and the biconditional of the conditional statement. 1. Two lines intersect in a point. 2. 4x + 9 = 21 because x = 3. 3. Supplementary angles sum to 180°. 4. Right angles are 90°.

9.2

Inductive and Deductive Reasoning (pp. 451–458)

What conclusion can you make about the sum of any two even integers? Step 1

Look for a pattern in several examples. Use inductive reasoning to make a conjecture. 2+4=6

6 + 10 = 16

12 + 16 = 28

−2 + 4 = 2

6 + (−10) = −4

−12 + (−16) = −28

Conjecture Even integer + Even integer = Even integer Step 2

Let n and m each be any integer. Use deductive reasoning to show that the conjecture is true. 2n and 2m are even integers because any integer multiplied by 2 is even. 2n + 2m represents the sum of two even integers. 2n + 2m = 2(n + m) by the Distributive Property. 2(n + m) is the product of 2 and an integer (n + m). So, 2(n + m) is an even integer.

The sum of any two even integers is an even integer. 5. What conclusion can you make about the difference of any two odd integers? 6. What conclusion can you make about the product of an even and an odd integer? 7. Use the Law of Detachment to make a valid conclusion.

If an angle is a right angle, then the angle measures 90°. ∠B is a right angle. 8. Use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements: If x = 3, then 2x = 6. If 4x = 12, then x = 3.




Chapter 9


Reasoning and Proofs

9.3

Postulates and Diagrams (pp. 459–464)

Use the diagram to make three statements that can be concluded and three statements that cannot be concluded. Justify your answers.

C M A F P

H

J D

G

B

You can conclude: 1. Points A, B, and C are coplanar because they lie in plane M.

⃖""⃗ lies in plane P by the Plane-Line Postulate. 2. FG ⃖""⃗ and ⃖""⃗ 3. CD FH intersect at point H by the Line Intersection Postulate. You cannot conclude: 1. ⃖""⃗ CD ⊥ to plane P because no right angle is marked. 2. Points A, F, and G are coplanar because point A lies in plane M and point G lies in plane P. 3. Points G, D, and J are collinear because no drawn line connects the points. Use the diagram to determine whether you can assume the statement. M H F P

9. Points A, B, C, and E are coplanar. 11. Points F, B, and G are collinear.

B

A

C

G

E

10. ⃖""⃗ HC ⊥ ⃖""⃗ GE

12. ⃖""⃗ AB % ⃖""⃗ GE

Sketch a diagram of the description. 13. ∠ABC, an acute angle, is bisected by """⃗ BE.

14. ∠CDE, a straight angle, is bisected by ⃖""⃗ DK.

—

15. Plane P and plane R intersect perpendicularly in ⃖""⃗ XY . ZW lies in plane P.

Chapter 9

Chapter Review



9.4

Proving Statements about Segments and Angles (pp. 467–476)

Write a two-column proof for the Transitive Property of Segment Congruence.

— ≅ CD —, CD —≅— Given AB EF

— Prove AB ≅ EF —

STATEMENTS

— ≅ CD —, CD —≅— 1. AB EF

REASONS 1. Given

2. AB = CD, CD = EF

2. Definition of congruent segments

3. AB = EF

3. Transitive Property of Equality

— ≅ EF — 4. AB

4. Definition of congruent segments

Name the property that the statement illustrates. 16. If LM = RS and RS = 25, then LM = 25.

17. AM = AM

18. If ∠DEF ≅ ∠JKL, then ∠JKL ≅ ∠DEF. 19. ∠C ≅ ∠C

— ≅ PQ — and PQ — ≅ RS —, then MN — ≅ RS —. 20. If MN 21. Write a two-column proof for the Reflexive Property of Angle Congruence.

9.5

Proving Geometric Relationships (pp. 477–486)

Rewrite the two-column proof into a paragraph proof. Given ∠2 ≅ ∠3

23 1

Prove ∠3 ≅ ∠6

7

Two-Column Proof STATEMENTS

REASONS

1. ∠2 ≅ ∠3

1. Given

2. ∠2 ≅ ∠6

2. Vertical Angles Congruence Theorem

3. ∠3 ≅ ∠6

3. Transitive Property of Angle Congruence

Paragraph Proof ∠2 and ∠3 are congruent. By the Vertical Angles Congruence Theorem, ∠2 ≅ ∠6. So, by the Transitive Property of Angle Congruence, ∠3 ≅ ∠6. 22. Write a proof using any format.

Given ∠3 and ∠2 are complementary. m∠1 + m∠2 = 90° Prove 


Chapter 9


∠3 ≅ ∠1

Reasoning and Proofs

4 5 6

9

Chapter Test

Use the diagram to determine whether you can assume the statement. Explain your reasoning. M H F P

G

B C

A E

1. ⃖""⃗ AB ⊥ plane M

2. Points F, G, and A are coplanar.

3. Points E, C, and G are collinear.

4. Planes M and P intersect at ⃖""⃗ BC.

5. ⃖""⃗ FA lies in plane P.

⃖""⃗ intersects ⃖""⃗ 6. FG AB at point B.

Name the property that the statement illustrates. 7. ∠B ≅ ∠B

8. If PQ = RS, then RS = PQ.

Write the if-then form, the converse, the inverse, the contrapositive, and the biconditional of the conditional statement. 9. Two planes intersect at a line.

10. A relation that pairs each input with exactly

one output is a function. Use inductive reasoning to make a conjecture about the given quantity. Then use deductive reasoning to show that the conjecture is true. 11. the sum of three odd integers 12. the product of three even integers 13. Give an example of two statements for which the Law of Detachment does not apply. 14. You visit the zoo and notice the following.

• The elephants, giraffes, lions, tigers, and zebras are located along a straight walkway. • The giraffes are halfway between the elephants and the lions. • The tigers are halfway between the lions and the zebras. • The lions are halfway between the giraffes and the tigers. Draw and label a diagram that represents this information. Then prove that the distance between the elephants and the giraffes is equal to the distance between the tigers and the zebras. Use any proof format. 15. Write a proof using any format.

X

Given ∠2 ≅ ∠3

"""⃗ TV bisects ∠UTW. Prove ∠1 ≅ ∠3

T

3 Z

Y

W

1 2 V

U

Chapter 9

Chapter Test



9

Cumulative Assessment

1. List all segment bisectors given x = 3. k

m

x+1 A

6(4 − x) B

n

4x − 6

2(5x − 7) − 8

C

D

3(5 − x) + 2 E

F

2. Select all the functions whose average rate of change from x = 1 to x = 3 is greater

than 2. f (x) = 2x + 1

f (x) = −3x + 6

f (x) = 3x − 1

f (x) = 3x − 6

f (x) = 2x + 1

f (x) = 4x + 2

3. Find the distance between each pair of points. Then order each line

segment from longest to shortest. a. A(−6, 1), B(−1, 6)

b. C(−5, 8), D(5, 8)

c. E(2, 7), F(4, −2)

d. G(7, 3), H(7, −1)

e. J(−4, −2), K(1, −5)

f. L(3, −8), M(7, −5)

4. Which interpretation corresponds to the information in the graphing calculator display? LinReg y=ax+b a=-2.14 b=47.06 r2=.9969529275 r=-.9984753014

A The data have a strong positive correlation. ○ B The data have a weak positive correlation. ○ C The data have a strong negative correlation. ○ D The data have a weak negative correlation. ○ 5. You conduct a survey that asks students whether your school should include healthier

options in its vending machines. The results are shown in the two-way table. Find and interpret the marginal frequencies.

Class

Healthier Options




Chapter 9


Yes

No

Freshman

58

21

Sophomore

43

26

Reasoning and Proofs

6. The endpoints of a line segment are (−6, 13) and (11, 5). Which choice shows the

correct midpoint and distance between these two points?

A ○

( —, 4 ); 18.8 units

B ○

( —, 9 ); 18.8 units

C ○

( —, 4 ); 9.4 units

D ○

( —, 9 ); 9.4 units

5 2 5 2 5 2 5 2

7. Find the perimeter and area of the figure shown. 4

Q

y

R

2

−4

−2

2

4 x

−2

T

−4

S

8. Classify each system of equations by the number of solutions.

y = 6x + 9

7x + 4y = 12

2x + 4y = −2

8y − 12 = −14x

10x + 4y = −2

3x + y = 5

y − 2x = —32

y = −3x + 5

−15 + 3y + 9x = 0

−3 + 2y = 4x

y = −3x + 9

y=

− —16 x

+9

9. Consider the equation y = mx + b. Fill in values for m and b so that each statement is true.

a. When m = ______ and b = ______, the graph of the equation passes through the point (−1, 4). b. When m = ______ and b = ______, the graph of the equation has a positive slope and passes through the point (−2, −5). c. When m = ______ and b = ______, the graph of the equation is perpendicular to the graph of y = 4x − 3 and passes through the point (1, 6).

Chapter 9

Cumulative Assessment



10 10.1 10.2 10.3 10.4 10.5

Parallel and Perpendicular Lines Pairs of Lines and Angles Parallel Lines and Transversals Proofs with Parallel Lines Proofs with Perpendicular Lines Using Parallel and Perpendicular Lines

Bike Path (p. 532)

(p 526) Crosswalk (p.

Kiteboarding (p. 515)

SEE the Big Idea

Gymnastics (p. 502)

Tree T ree H House ouse ((p. p. 50 502) 2)

Maintaining Mathematical Proficiency Finding Angle Measures Example 1

D

Find the measure of each angle.

(8x − 6)° (9x + 16)° A

B

C

Step 1 Use the fact that the sum of the measures of supplementary angles is 180°. m∠ABD + m∠DBC = 180° (8x − 6)° + (9x + 16)° = 180° 17x + 10 = 180 x = 10

Write an equation. Substitute angle measures. Combine like terms. Solve for x.

Step 2 Evaluate the given expressions when x = 10.

⋅ m∠DBC = (9x + 16)° = (9 ⋅ 10 + 16)° = 106° m∠ABD = (8x − 6)° = (8 10 − 6)° = 74°

So, m∠ABD = 74° and m∠DBC = 106°.

Find the measure of each angle. 1.

2.

H

(4x + 2)°

(3x + 2)° (5x − 6)° E

F

3.

(5x + 7)° L

K

G

J

M

Q (4x − 5)° (2x − 7)° N

O

P

Writing Equations of Lines Example 2 Write an equation of the line that passes through the point (−4, 5) and has a slope of —34. y = mx + b

Write the slope-intercept form.

5 = —34 (−4) + b

Substitute —34 for m, −4 for x, and 5 for y.

5 = −3 + b

Simplify.

8=b

Solve for b.

So, an equation is y = —34 x + 8.

Write an equation of the line that passes through the given point and has the given slope. 4. (6, 1); m = −3 7. (2, −4); m =

1 —2

5. (−3, 8); m = −2 8. (−8, −5); m =

1 −—4

6. (−1, 5); m = 4 2

9. (0, 9); m = —3

10. ABSTRACT REASONING How can you write an equation of a line that passes through the point

(a, b) and has an undefined slope? Dynamic Solutions available at BigIdeasMath.com

495

Mathematical Practices

Mathematically proficient students use technological tools to explore concepts.

Characteristics of Lines in a Coordinate Plane

Core Concept Lines in a Coordinate Plane 1.

In a coordinate plane, two lines are parallel if and only if they are both vertical lines or they both have the same slope.

2.

In a coordinate plane, two lines are perpendicular if and only if one is vertical and the other is horizontal or the slopes of the lines are negative reciprocals of each other.

3.

In a coordinate plane, two lines are coincident if and only if their equations are equivalent.

Classifying Pairs of Lines Here are some examples of pairs of lines in a coordinate plane. a.

2x + y = 2 x−y=4

These lines are not parallel or perpendicular. They intersect at (2, −2).

b.

2x + y = 2 These lines are coincident 4x + 2y = 4 because their equations are equivalent.

4

4

−6

6

−6

6

−4

c.

2x + y = 2 2x + y = 4

−4

These lines are parallel. Each line has a slope of m = −2.

d. 2x + y = 2 x − 2y = 4

4

These lines are perpendicular. They have slopes of m1 = −2 and m2 = —12 . 4

−6

6

−6

6

−4

−4

Monitoring Progress Use a graphing calculator to graph the pair of lines. Use a square viewing window. Classify the lines as parallel, perpendicular, coincident, or nonperpendicular intersecting lines. Justify your answer. 1. x + 2y = 2

2x − y = 4 496

Chapter 10

2. x + 2y = 2

3. x + 2y = 2

2x + 4y = 4 Parallel and Perpendicular Lines

x + 2y = −2

4. x + 2y = 2

x − y = −4

10.1 Pairs of Lines and Angles Essential Question

What does it mean when two lines are parallel, intersecting, coincident, or skew? Points of Intersection Work with a partner. Write the number of points of intersection of each pair of coplanar lines. a. parallel lines

b. intersecting lines

c. coincident lines

Classifying Pairs of Lines Work with a partner. The figure shows a right rectangular prism. All its angles are right angles. Classify each of the following pairs of lines as parallel, intersecting, coincident, or skew. Justify your answers. (Two lines are skew lines when they do not intersect and are not coplanar.)

B D

A F E

Pair of Lines

C

Classification

G I

H

Reason

a. ⃖""⃗ AB and ⃖""⃗ BC b. ⃖""⃗ AD and ⃖""⃗ BC c. ⃖""⃗ EI and ⃖""⃗ IH d. ⃖""⃗ BF and ⃖""⃗ EH

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.

e. ⃖""⃗ EF and ⃖""⃗ CG f. ⃖""⃗ AB and ⃖""⃗ GH

Identifying Pairs of Angles Work with a partner. In the figure, two parallel lines are intersected by a third line called a transversal. a. Identify all the pairs of vertical angles. Explain your reasoning. b. Identify all the linear pairs of angles. Explain your reasoning.

5 6 1 2 8 7 4 3

Communicate Your Answer 4. What does it mean when two lines are parallel, intersecting, coincident, or skew? 5. In Exploration 2, find three more pairs of lines that are different from those

given. Classify the pairs of lines as parallel, intersecting, coincident, or skew. Justify your answers. Section 10.1

Pairs of Lines and Angles

497

10.1 Lesson

What You Will Learn Identify lines and planes. Identify parallel and perpendicular lines.

Core Vocabul Vocabulary larry

Identify pairs of angles formed by transversals.

skew lines, p. 498 parallel planes, p. 498 transversal, p. 500 corresponding angles, p. 500 alternate interior angles, p. 500 alternate exterior angles, p. 500 consecutive interior angles, p. 500 Previous parallel lines perpendicular lines

Identifying Lines and Planes

Core Concept Parallel Lines, Skew Lines, and Parallel Planes Two lines that do not intersect are either parallel lines or skew lines. Recall that two lines are parallel lines when they do not intersect and are coplanar. Two lines are skew lines when they do not intersect and are not coplanar. Also, two planes that do not intersect are parallel planes. Lines m and n are parallel lines (m $ n).

k m T

n

Lines m and k are skew lines. Planes T and U are parallel planes (T $ U ). Lines k and n are intersecting lines, and there is a plane (not shown) containing them.

U

Small directed arrows, as shown in red on lines m and n above, are used to show that lines are parallel. The symbol $ means “is parallel to,” as in m $ n. Segments and rays are parallel when they lie in parallel lines. A line is parallel to a plane when the line is in a plane parallel to the given plane. In the diagram above, line n is parallel to plane U.

Identifying Lines and Planes

REMEMBER Recall that if two lines intersect to form a right angle, then they are perpendicular lines.

Think of each segment in the figure as part of a line. Which line(s) or plane(s) appear to fit the description? a. line(s) parallel to ⃖""⃗ CD and containing point A

C D

A F

b. line(s) skew to ⃖""⃗ CD and containing point A c. line(s) perpendicular to ⃖""⃗ CD and containing point A

B

E

G H

d. plane(s) parallel to plane EFG and containing point A

SOLUTION a. ⃖""⃗ AB, ⃖""⃗ HG, and ⃖""⃗ EF all appear parallel to ⃖""⃗ CD, but only ⃖""⃗ AB contains point A. b. Both ⃖""⃗ AG and ⃖""⃗ AH appear skew to ⃖""⃗ CD and contain point A. c. ⃖""⃗ BC, ⃖""⃗ AD, ⃖""⃗ DE, and ⃖""⃗ FC all appear perpendicular to ⃖""⃗ CD, but only ⃖""⃗ AD contains point A. d. Plane ABC appears parallel to plane EFG and contains point A.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1. Look at the diagram in Example 1. Name the line(s) through point F that appear

skew to ⃖""⃗ EH.

498

Chapter 10

Parallel and Perpendicular Lines

Identifying Parallel and Perpendicular Lines Two distinct lines in the same plane either are parallel, like lineℓ and line n, or intersect in a point, like line j and line n.

k

j

P

n

Through a point not on a line, there are infinitely many lines. Exactly one of these lines is parallel to the given line, and exactly one of them is perpendicular to the given line. For example, line k is the line through point P perpendicular to lineℓ, and line n is the line through point P parallel to lineℓ.

Postulates Parallel Postulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.

P

There is exactly one line through P parallel toℓ.

Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.

P

There is exactly one line through P perpendicular toℓ.

Identifying Parallel and Perpendicular Lines ne Pay

The given line markings show how the roads in a town are related to one another.

B

t tree

b. Name a pair of perpendicular lines. Is ⃖""⃗ FE $ ⃖""⃗ AC? Explain.

429

265

C

384 ay Tra Seaw

R lck Wa

9t h

Ave

M

d

A

Pay

D

il

SOLUTION

Nash Rd

c.

Ave

S ver Oli

a. Name a pair of parallel lines.

ne Ave

ver Oli

a. ⃖"""⃗ MD $ ⃖""⃗ FE

ee t Str

⃖"""⃗ ⊥ ⃖""⃗ b. MD BF

eatf Wh

265

E

F Ave

Monitoring Progress

429

yne Pa

c. ⃖""⃗ FE is not parallel to ⃖""⃗ AC, because ⃖"""⃗ MD is parallel to ⃖""⃗ FE, and by the Parallel Postulate, there is exactly one line parallel to ⃖""⃗ FE through M.

384 38

St ield

Help in English and Spanish at BigIdeasMath.com

2. In Example 2, can you use the Perpendicular Postulate to show that ⃖""⃗ AC is not

perpendicular to ⃖""⃗ BF? Explain why or why not. Section 10.1

Pairs of Lines and Angles

499

Identifying Pairs of Angles A transversal is a line that intersects two or more coplanar lines at different points.

Core Concept Angles Formed by Transversals t 2

t 4

6

5

Two angles are corresponding angles when they have corresponding positions. For example, ∠2 and ∠6 are above the lines and to the right of the transversal t.

Two angles are alternate interior angles when they lie between the two lines and on opposite sides of the transversal t.

t

1

t 3 5

8

Two angles are alternate exterior angles when they lie outside the two lines and on opposite sides of the transversal t.

Two angles are consecutive interior angles when they lie between the two lines and on the same side of the transversal t.

Identifying Pairs of Angles Identify all pairs of angles of the given type. a. b. c. d.

corresponding alternate interior alternate exterior consecutive interior

5 6 7 8

1 2 3 4

SOLUTION a. ∠ l and ∠ 5 ∠ 2 and ∠ 6 ∠ 3 and ∠ 7 ∠ 4 and ∠ 8

b. ∠ 2 and ∠ 7 ∠ 4 and ∠ 5

Monitoring Progress

c. ∠ l and ∠ 8 ∠ 3 and ∠ 6

d. ∠ 2 and ∠ 5 ∠ 4 and ∠ 7

Help in English and Spanish at BigIdeasMath.com

Classify the pair of numbered angles. 3.

1

5

4.

5.

2

5 4 7

500

Chapter 10

Parallel and Perpendicular Lines

10.1 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE Two lines that do not intersect and are also not parallel

are ________ lines. 2. WHICH ONE DOESN’T BELONG? Which angle pair does not belong with the other three?

Explain your reasoning. ∠2 and ∠3

∠4 and ∠5

∠1 and ∠8

∠2 and ∠7

1 2 3 4 5 6 7 8

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, think of each segment in the diagram as part of a line. All the angles are right angles. Which line(s) or plane(s) contain point B and appear to fit the description? (See Example 1.) C

B

D

9. Is ⃖""⃗ PN $ ⃖"""⃗ KM? Explain. 10. Is ⃖""⃗ PR ⊥ ⃖""⃗ NP? Explain.

In Exercises 11–14, identify all pairs of angles of the given type. (See Example 3.)

A

1 2 3 4 G

F

5 6 7 8

H

E

11. corresponding

3.

line(s) parallel to ⃖""⃗ CD

4.

line(s) perpendicular to ⃖""⃗ CD

5.

line(s) skew to ⃖""⃗ CD

12. alternate interior 13. alternate exterior 14. consecutive interior

6. plane(s) parallel to plane CDH

USING STRUCTURE In Exercises 15–18, classify the

In Exercises 7–10, use the diagram. (See Example 2.)

angle pair as corresponding, alternate interior, alternate exterior, or consecutive interior angles.

N

M L

K

9 10 11 12

1 2 3 4

S

Q

5 6 7 8

P R

13 14 15 16

7. Name a pair of parallel lines.

15. ∠5 and ∠1

16. ∠11 and ∠13

8. Name a pair of perpendicular lines.

17. ∠6 and ∠13

18. ∠2 and ∠11

Section 10.1

Pairs of Lines and Angles

501

ERROR ANALYSIS In Exercises 19 and 20, describe and correct the error in the conditional statement about lines. 19.



20.



24. HOW DO YOU SEE IT? Think of each segment in the

figure as part of a line. a. Which lines are ⃖""⃗? parallel to NQ

If two lines do not intersect, then they are parallel.

K N

b. Which lines intersect ⃖""⃗ NQ?

R P

d. Should you have named all the lines on the cube in parts (a)–(c) except ⃖""⃗ NQ? Explain.

21. MODELING WITH MATHEMATICS Use the photo to

In Exercises 25–28, copy and complete the statement. List all possible correct answers.

decide whether the statement is true or false. Explain your reasoning.

G

E D

D C

S

Q

c. Which lines are skew to ⃖""⃗ NQ?

If there is a line and a point not on the line, then there is exactly one line through the point that intersects the given line.

L M

A

B A

F

J C

H

B

25. ∠BCG and ____ are corresponding angles. 26. ∠BCG and ____ are consecutive interior angles. 27. ∠FCJ and ____ are alternate interior angles.

a. The plane containing the floor of the tree house is parallel to the ground.

28. ∠FCA and ____ are alternate exterior angles.

b. The lines containing the railings of the staircase, such as ⃖""⃗ AB, are skew to all lines in the plane containing the ground.

29. MAKING AN ARGUMENT Your friend claims the

uneven parallel bars in gymnastics are not really parallel. She says one is higher than the other, so they cannot be in the same plane. Is she correct? Explain.

c. All the lines containing the balusters, such as ⃖""⃗ CD, are perpendicular to the plane containing the floor of the tree house. 22. THOUGHT PROVOKING If two lines are intersected by

a third line, is the third line necessarily a transversal? Justify your answer with a diagram. 23. MATHEMATICAL CONNECTIONS Two lines are cut by

a transversal. Is it possible for all eight angles formed to have the same measure? Explain your reasoning.

Maintaining Mathematical Proficiency Use the diagram to find the measures of all the angles. (Section 9.5) 30. m∠1 = 76°

1

Chapter 10

2 4

31. m∠2 = 159°

502

Reviewing what you learned in previous grades and lessons

Parallel and Perpendicular Lines

3

10.2 Parallel Lines and Transversals Essential Question

When two parallel lines are cut by a transversal, which of the resulting pairs of angles are congruent? Exploring Parallel Lines Work with a partner. Use dynamic geometry software to draw two parallel lines. Draw a third line that intersects both parallel lines. Find the measures of the eight angles that are formed. What can you conclude?

6

D

5 4

B

3

E

1

2

F

6 8 7

5

2

4 3

1

A

0 −3

ATTENDING TO PRECISION To be proficient in math, you need to communicate precisely with others.

−2

−1

0

1

2

3

4

C 5

6

Writing Conjectures Work with a partner. Use the results of Exploration 1 to write conjectures about the following pairs of angles formed by two parallel lines and a transversal. a. corresponding angles

1 2 4 3

5 6 8 7

1 2 4 3

c. alternate exterior angles

1 2 4 3

b. alternate interior angles

5 6 8 7

d. consecutive interior angles

5 6 8 7

1 2 4 3

5 6 8 7

Communicate Your Answer 3. When two parallel lines are cut by a transversal, which of the resulting pairs of

angles are congruent? 4. In Exploration 2, m∠1 = 80°. Find the other angle measures.

Section 10.2

Parallel Lines and Transversals

503

What You Will Learn

10.2 Lesson

Use properties of parallel lines. Prove theorems about parallel lines.

Core Vocabul Vocabulary larry

Solve real-life problems.

Previous corresponding angles parallel lines supplementary angles vertical angles

Using Properties of Parallel Lines

Theorems Corresponding Angles Theorem If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

t 1 2 3 4

Examples In the diagram at the left, ∠2 ≅ ∠6 and ∠3 ≅ ∠7. p

Proof Ex. 36, p. 550

Alternate Interior Angles Theorem 5 6 7 8

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. q

Examples In the diagram at the left, ∠3 ≅ ∠6 and ∠4 ≅ ∠5. Proof Example 4, p. 506

Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Examples In the diagram at the left, ∠1 ≅ ∠8 and ∠2 ≅ ∠7. Proof Ex. 15, p. 508

Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Examples In the diagram at the left, ∠3 and ∠5 are supplementary, and

∠4 and ∠6 are supplementary.

ANOTHER WAY There are many ways to solve Example 1. Another way is to use the Corresponding Angles Theorem to find m∠5 and then use the Vertical Angles Congruence Theorem to find m∠4 and m∠8.

Proof Ex. 16, p. 508

Identifying Angles The measures of three of the numbered angles are 120°. Identify the angles. Explain your reasoning.

SOLUTION

120º 2 3 4

5 6 7 8

By the Alternate Exterior Angles Theorem, m∠8 = 120°. ∠5 and ∠8 are vertical angles. Using the Vertical Angles Congruence Theorem, m∠5 = 120°. ∠5 and ∠4 are alternate interior angles. By the Alternate Interior Angles Theorem, ∠4 = 120°. So, the three angles that each have a measure of 120° are ∠4, ∠5, and ∠8. 504

Chapter 10

Parallel and Perpendicular Lines

Using Properties of Parallel Lines Find the value of x. 115° 4 (x + 5)°

a b

SOLUTION By the Vertical Angles Congruence Theorem, m∠4 = 115°. Lines a and b are parallel, so you can use the theorems about parallel lines. Check 115° + (x + 5)° = 180° ? 115 + (60 + 5) = 180 180 = 180

m∠4 + (x + 5)° = 180°

Consecutive Interior Angles Theorem

115° + (x + 5)° = 180°

Substitute 115° for m∠4.

x + 120 = 180



Combine like terms.

x = 60

Subtract 120 from each side.

So, the value of x is 60.

Using Properties of Parallel Lines Find the value of x. 1

136° c (7x + 9)° d

SOLUTION By the Linear Pair Postulate, m∠1 = 180° − 136° = 44°. Lines c and d are parallel, so you can use the theorems about parallel lines. m∠1 = (7x + 9)°

Check 44° = (7x + 9)° ? 44 = 7(5) + 9 44 = 44



Alternate Exterior Angles Theorem

44° = (7x + 9)°

Substitute 44° for m∠1.

35 = 7x

Subtract 9 from each side.

5=x

Divide each side by 7.

So, the value of x is 5.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use the diagram. 1. Given m∠1 = 105°, find m∠4, m∠5, and

m∠8. Tell which theorem you use in each case. 2. Given m∠3 = 68° and m∠8 = (2x + 4)°,

1 2 3 4

5 6 7 8

what is the value of x? Show your steps.

Section 10.2

Parallel Lines and Transversals

505

Proving Theorems about Parallel Lines Proving the Alternate Interior Angles Theorem Prove that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

SOLUTION

STUDY TIP Before you write a proof, identify the Given and Prove statements for the situation described or for any diagram you draw.

t

Draw a diagram. Label a pair of alternate interior angles as ∠1 and ∠2. You are looking for an angle that is related to both ∠1 and ∠2. Notice that one angle is a vertical angle with ∠2 and a corresponding angle with ∠1. Label it ∠3.

p

1 2

q

3

Given p ! q Prove ∠1 ≅ ∠2 STATEMENTS

REASONS

1. p ! q

1. Given

2. ∠1 ≅ ∠3

2. Corresponding Angles Theorem

3. ∠3 ≅ ∠2

3. Vertical Angles Congruence Theorem

4. ∠1 ≅ ∠2

4. Transitive Property of Congruence

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

3. In the proof in Example 4, if you use the third statement before the second

statement, could you still prove the theorem? Explain.

Solving Real-Life Problems Solving a Real-life Problem When sunlight enters a drop of rain, different colors of light leave the drop at different angles. This process is what makes a rainbow. For violet light, m∠2 = 40°. What is m∠1? How do you know?

2 1

SOLUTION Because the Sun’s rays are parallel, ∠1 and ∠2 are alternate interior angles. By the Alternate Interior Angles Theorem, ∠1 ≅ ∠2. So, by the definition of congruent angles, m∠1 = m∠2 = 40°.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

4. WHAT IF? In Example 5, yellow light leaves a drop at an angle of m∠2 = 41°.

What is m∠1? How do you know? 506

Chapter 10

Parallel and Perpendicular Lines

10.2 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING How are the Alternate Interior Angles Theorem and the Alternate Exterior Angles Theorem

alike? How are they different? 2. WHICH ONE DOESN’T BELONG? Which pair of angle measures does not belong with the

other three? Explain.

m∠1 and m∠3

1

m∠2 and m∠4

m∠2 and m∠3

2 4

m∠1 and m∠5

3 5

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, find m∠1 and m∠2. Tell which theorem you use in each case. (See Example 1.) 3.

4. 117°

10.

(8x + 6)°

150°

1

1 2

2

5.

6. 1 2

140°

122°

118° 4

2

1

In Exercises 11 and 12, find m∠1, m∠2, and m∠3. Explain your reasoning. 11. 1 2 80° 3

In Exercises 7–10, find the value of x. Show your steps. (See Examples 2 and 3.) 7.

12. 1

8. 2x° 128°

72° (7x + 24)°

133° 2

3

13. ERROR ANALYSIS Describe and correct the error in

the student’s reasoning.

9.

65º

5 (11x − 17)º



9 10

Section 10.2

∠9 ≅ ∠10 by the Corresponding Angles Theorem.

Parallel Lines and Transversals

507

14. HOW DO YOU SEE IT?

A D

a. b.

19. CRITICAL THINKING Is it possible for consecutive

B

Use the diagram.

interior angles to be congruent? Explain.

C

— and Name two pairs of congruent angles when AD — BC are parallel. Explain your reasoning. — Name two pairs of supplementary angles when AB — and DC are parallel. Explain your reasoning.

PROVING A THEOREM In Exercises 15 and 16, prove the

theorem. (See Example 4.)

20. THOUGHT PROVOKING The postulates and theorems

in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, is it possible that a transversal intersects two parallel lines? Explain your reasoning. MATHEMATICAL CONNECTIONS In Exercises 21 and 22,

write and solve a system of linear equations to find the values of x and y.

15. Alternate Exterior Angles Theorem

21.

16. Consecutive Interior Angles Theorem

2y °

17. PROBLEM SOLVING

A group of campers tie up their food between two parallel trees, as shown. The rope is pulled taut, forming a straight line. Find m∠2. Explain your reasoning. (See Example 5.)

(14x − 10)° 22.

4x °

2y °

(2x + 12)° (y + 6)°

5x °

23. MAKING AN ARGUMENT During a game of pool,

your friend claims to be able to make the shot shown in the diagram by hitting the cue ball so that m∠1 = 25°. Is your friend correct? Explain your reasoning.

76°

2

1

18. DRAWING CONCLUSIONS You are designing a box

like the one shown. 65°

1

2 3

A 1

B 3 2

C

— bisects 24. REASONING In the diagram, ∠4 ≅ ∠5 and SE ∠RSF. Find m∠1. Explain your reasoning. E

a. The measure of ∠1 is 70°. Find m∠2 and m∠3. c. If m∠1 is 60°, will ∠ABC still be a straight angle? Will the opening of the box be more steep or less steep? Explain.

Maintaining Mathematical Proficiency

1 T

25. If two angles are vertical angles, then they are congruent. 26. If you go to the zoo, then you will see a tiger. 27. If two angles form a linear pair, then they are supplementary.

508

Chapter 10

Parallel and Perpendicular Lines

2

3

5

S

Reviewing what you learned in previous grades and lessons

Write the converse of the conditional statement. Decide whether it is true or false.

28. If it is warm outside, then we will go to the park.

4

F

b. Explain why ∠ABC is a straight angle.

(Section 9.1)

R

10.3 Proofs with Parallel Lines Essential Question

For which of the theorems involving parallel lines and transversals is the converse true? Exploring Converses

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.

Work with a partner. Write the converse of each conditional statement. Draw a diagram to represent the converse. Determine whether the converse is true. Justify your conclusion. a. Corresponding Angles Theorem If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Converse

b. Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Converse

c. Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Converse

d. Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Converse

1 2 4 3

5 6 8 7

1 2 4 3

5 6 8 7

1 2 4 3

5 6 8 7

1 2 4 3

5 6 8 7

Communicate Your Answer 2. For which of the theorems involving parallel lines and transversals is

the converse true? 3. In Exploration 1, explain how you would prove any of the theorems

that you found to be true. Section 10.3

Proofs with Parallel Lines

509

10.3 Lesson

What You Will Learn Use the Corresponding Angles Converse.

Core Vocabul Vocabulary larry

Construct parallel lines.

Previous converse parallel lines transversal corresponding angles congruent alternate interior angles alternate exterior angles consecutive interior angles

Use the Transitive Property of Parallel Lines.

Prove theorems about parallel lines.

Using the Corresponding Angles Converse The theorem below is the converse of the Corresponding Angles Theorem. Similarly, the other theorems about angles formed when parallel lines are cut by a transversal have true converses. Remember that the converse of a true conditional statement is not necessarily true, so you must prove each converse of a theorem.

Theorem Corresponding Angles Converse If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.

2 j

6

k

Proof Ex. 36, p. 550

j!k

Using the Corresponding Angles Converse Find the value of x that makes m ! n.

(3x + 5)° m 65° n

SOLUTION Lines m and n are parallel when the marked corresponding angles are congruent. (3x + 5)° = 65° 3x = 60 x = 20

Use the Corresponding Angles Converse to write an equation. Subtract 5 from each side. Divide each side by 3.

So, lines m and n are parallel when x = 20.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1. Is there enough information in the diagram to conclude that m ! n? Explain. 75° m 105°

n

2. Explain why the Corresponding Angles Converse is the converse of the

Corresponding Angles Theorem. 510

Chapter 10

Parallel and Perpendicular Lines

Constructing Parallel Lines The Corresponding Angles Converse justifies the construction of parallel lines, as shown below.

Constructing Parallel Lines Use a compass and straightedge to construct a line through point P that is parallel to line m.

P m

SOLUTION Step 1

Step 2 P

Q

Step 3

P m

Draw a point and line Start by drawing point P and line m. Choose a point Q anywhere on line m and draw ⃖""⃗ QP.

A QB

C

Step 4

P A

m

Draw arcs Draw an arc with center Q that crosses ⃖""⃗ QP and line m. Label points A and B. Using the same compass setting, draw an arc with center P. Label point C.

C

P

D

QB

C

A

m

D m

QB

Copy angle Draw an arc with radius AB and center A. Using the same compass setting, draw an arc with center C. Label the intersection D.

Draw parallel lines

⃖""⃗. This line is Draw PD parallel to line m.

Theorems Alternate Interior Angles Converse If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.

4

5

j k

j$k

Proof Example 2, p. 512

Alternate Exterior Angles Converse

1

If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.

j

8

Proof Ex. 11, p. 514

k

j$k

Consecutive Interior Angles Converse If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel.

3 5

j k

If ∠3 and ∠5 are supplementary, then j $ k.

Proof Ex. 12, p. 514 Section 10.3

Proofs with Parallel Lines

511

Proving Theorems about Parallel Lines Proving the Alternate Interior Angles Converse Prove that if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.

SOLUTION

1

Given ∠4 ≅ ∠5

4

5

Prove g " h

g

h

STATEMENTS

REASONS

1. ∠4 ≅ ∠5

1. Given

2. ∠1 ≅ ∠4

2. Vertical Angles Congruence Theorem

3. ∠1 ≅ ∠5

3. Transitive Property of Congruence

4. g " h

4. Corresponding Angles Converse

Determining Whether Lines Are Parallel r

In the diagram, r " s and ∠1 is congruent to ∠3. Prove p " q.

s p

3 2

1

q

SOLUTION Look at the diagram to make a plan. The diagram suggests that you look at angles 1, 2, and 3. Also, you may find it helpful to focus on one pair of lines and one transversal at a time. Plan for Proof

a. Look at ∠1 and ∠2. ∠1 ≅ ∠2 because r " s. b. Look at ∠2 and ∠3. If ∠2 ≅ ∠3, then p " q.

Plan for Action a. It is given that r " s, so by the Corresponding Angles Theorem, ∠1 ≅ ∠2. b. It is also given that ∠1 ≅ ∠3. Then ∠2 ≅ ∠3 by the Transitive Property of Congruence. So, by the Alternate Interior Angles Converse, p " q.

Monitoring Progress

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3. If you use the diagram below to prove the Alternate Exterior Angles Converse,

what Given and Prove statements would you use? 1 j

8

k

4. Copy and complete the following paragraph proof of the Alternate Interior

Angles Converse using the diagram in Example 2. It is given that ∠4 ≅ ∠5. By the ______, ∠1 ≅ ∠4. Then by the Transitive Property of Congruence, ______. So, by the ______, g " h. 512

Chapter 10

Parallel and Perpendicular Lines

Using the Transitive Property of Parallel Lines

Theorem Transitive Property of Parallel Lines p

If two lines are parallel to the same line, then they are parallel to each other.

Proof Ex. 39, p. 516; Ex. 25, p. 532

q

r

If p ! q and q ! r, then p ! r.

Using the Transitive Property of Parallel Lines The flag of the United States has 13 alternating red and white stripes. Each stripe is parallel to the stripe immediately below it. Explain why the top stripe is parallel to the bottom stripe. s1 s3 s5 s7 s9 s11 s13

s2 s4 s6 s8 s10 s12

SOLUTION You can name the stripes from top to bottom as sl, s2, s3, . . . , s13. Each stripe is parallel to the one immediately below it, so s1 ! s2, s2 ! s3, and so on. Then s1 ! s3 by the Transitive Property of Parallel Lines. Similarly, because s3 ! s4, it follows that s1 ! s4. By continuing this reasoning, s1 ! s13. So, the top stripe is parallel to the bottom stripe.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

5. Each step is parallel to the step immediately

above it. The bottom step is parallel to the ground. Explain why the top step is parallel to the ground. 6. In the diagram below, p ! q and q ! r. Find m∠8.

Explain your reasoning. s 115° p q 8

r

Section 10.3

Proofs with Parallel Lines

513

10.3 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept p Check 1. VOCABULARY Two lines are cut by a transversal. Which angle pairs must be congruent for the lines

to be parallel? 2. WRITING Use the theorems from Section 10.2 and the converses of those theorems in this

section to write three biconditional statements about parallel lines and transversals.

Monitoring Progress and Modeling with Mathematics In Exercises 3–8, find the value of x that makes m ! n. Explain your reasoning. (See Example 1.) 3.

4.

m

(2x + 15)°

n

3x °

5.

135°

m

120°

6.

m

m

n

(3x − 15)°

n

13.

m

m

14.

n

n r

r

n

(180 − x)°

150°

In Exercises 13–18, decide whether there is enough information to prove that m ! n. If so, state the theorem you would use. (See Example 3.)

15.

16.

r

m

n

x°

r

m m

7.

n

8.

2x°

x°

n

n

m

17.

(2x + 20)°

r

18.

s

3x°

In Exercises 9 and 10, use a compass and straightedge to construct a line through point P that is parallel to line m. 9.

P

10.

P m

r

m

m

n

n

ERROR ANALYSIS In Exercises 19 and 20, describe and

correct the error in the reasoning. 19.

m



a x° x°

b y° c

y°

Conclusion: a ! b

PROVING A THEOREM In Exercises 11 and 12, prove

the theorem. (See Example 2.) 11. Alternate Exterior Angles Converse 12. Consecutive Interior Angles Converse

20.



2

a

1 b

Conclusion: a ! b 514

Chapter 10

Parallel and Perpendicular Lines

s

In Exercises 21–24, are ⃖""⃗ AC and ⃖""⃗ DF parallel? Explain your reasoning. 21.

A 57° B

22.

D 123° E C F

28. REASONING Use the diagram. Which rays are

parallel? Which rays are not parallel? Explain your reasoning.

143° A B

37°

D

F

C E

E 62º

F

58º

24.

62° A

B 62°

C

D

E

F

115° B C 65° 65° E F

A D

59º B

A

23.

G

H 61º

C

D

29. ATTENDING TO PRECISION Use the diagram. Which

theorems allow you to conclude that m $ n? Select all that apply. Explain your reasoning. m

25. ANALYZING RELATIONSHIPS The map shows part of

Denver, Colorado. Use the markings on the map. Are the numbered streets parallel to one another? Explain your reasoning. (See Example 4.)

n

E. 20th Ave.

A Corresponding Angles Converse ○

E. 19th Ave.

B Alternate Interior Angles Converse ○ C Alternate Exterior Angles Converse ○

Pa rk Av e.

Franklin St.

Ogden St.

Downing St.

D Consecutive Interior Angles Converse ○ Clarkson St.

Washington St.

Pearl St.

E. 17th Ave.

Pennsylvania St.

E. 18th Ave.

26. ANALYZING RELATIONSHIPS Each

rung of the ladder is parallel to the rung directly above it. Explain why the top rung is parallel to the bottom rung.

30. MODELING WITH MATHEMATICS One way to build

stairs is to attach triangular blocks to an angled support, as shown. The sides of the angled support are parallel. If the support makes a 32° angle with the floor, what must m∠1 be so the top of the step will be parallel to the floor? Explain your reasoning.

triangular block

1

27. MODELING WITH MATHEMATICS The diagram of

the control bar of the kite shows the angles formed between the control bar and the kite lines. How do you know that n is parallel to m?

2

32°

31. ABSTRACT REASONING In the diagram, how many m 108° n

angles must be given to determine whether j $ k? Give four examples that would allow you to conclude that j $ k using the theorems from this lesson. j

k

108° 1 2 3 4

Section 10.3

5 6 7 8 t

Proofs with Parallel Lines

515

32. THOUGHT PROVOKING Draw a diagram of at least

37. MAKING AN ARGUMENT Your classmate decided

that ⃖$$⃗ AD ! ⃖$$⃗ BC based on the diagram. Is your classmate correct? Explain your reasoning.

two lines cut by at least one transversal. Mark your diagram so that it cannot be proven that any lines are parallel. Then explain how your diagram would need to change in order to prove that lines are parallel.

A

B

PROOF In Exercises 33–36, write a proof. 33. Given m∠1 = 115°, m∠2 = 65°

D

C

Prove m ! n 38. HOW DO YOU SEE IT? Are the markings on the 1

diagram enough to conclude that any lines are parallel? If so, which ones? If not, what other information is needed?

m 2

n

q

34. Given ∠1 and ∠3 are supplementary.

p

Prove m ! n

r 1 2

3

s

4 1 2 3

m

39. PROVING A THEOREM Use these steps to prove

n

the Transitive Property of Parallel Lines Theorem. a. Copy the diagram with the Transitive Property of Parallel Lines Theorem on page 513.

35. Given ∠1 ≅ ∠2, ∠3 ≅ ∠4

— ! CD — Prove AB

b. Write the Given and Prove statements. A 1

c. Use the properties of angles formed by parallel lines cut by a transversal to prove the theorem.

D 2 E

40. MATHEMATICAL CONNECTIONS Use the diagram.

3

B

4

r

C

s

(2x + 2)° p

36. Given a ! b, ∠2 ≅ ∠3

(x + 56)° (y + 7)°

Prove c ! d c

(3y − 17)°

q

d 1

3

2 4

a

a. Find the value of x that makes p ! q.

b

b. Find the value of y that makes r ! s. c. Can r be parallel to s and can p be parallel to q at the same time? Explain your reasoning.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Use the Distance Formula to find the distance between the two points.

(Section 8.3)

41. (1, 3) and (−2, 9)

42. (−3, 7) and (8, −6)

43. (5, −4) and (0, 8)

44. (13, 1) and (9, −4)

516

Chapter 10

Parallel and Perpendicular Lines

10.1–10.3

What Did You Learn?

Core Vocabulary skew lines, p. 498 parallel planes, p. 498 transversal, p. 500 corresponding angles, p. 500

alternate interior angles, p. 500 alternate exterior angles, p. 500 consecutive interior angles, p. 500

Core Concepts Section 10.1 Parallel Lines, Skew Lines, and Parallel Planes, p. 498 Parallel Postulate, p. 499

Perpendicular Postulate, p. 499 Angles Formed by Transversals, p. 500

Section 10.2 Corresponding Angles Theorem, p. 504 Alternate Interior Angles Theorem, p. 504

Alternate Exterior Angles Theorem, p. 504 Consecutive Interior Angles Theorem, p. 504

Section 10.3 Corresponding Angles Converse, p. 510 Alternate Interior Angles Converse, p. 511 Alternate Exterior Angles Converse, p. 511

Consecutive Interior Angles Converse, p. 511 Transitive Property of Parallel Lines, p. 513

Mathematical Practices 1.

Draw the portion of the diagram that you used to answer Exercise 26 on page 502.

2.

In Exercise 40 on page 516, explain how you started solving the problem and why you started that way.

Visual Learners Draw a picture of a word problem. • Draw a picture of a word problem before starting g to solve the problem. You do not have to be an artist. st. • When making a review card for a word problem, include a picture. This will help you recall the information while taking a test. • Make sure your notes are visually neat for easy recall. 517 7

10.1–10.3

Quiz

Think of each segment in the diagram as part of a line. Which line(s) or plane(s) contain point G and appear to fit the description? (Section 10.1) 1. line(s) parallel to ⃖##⃗ EF

2. line(s) perpendicular to ⃖##⃗ EF

3. line(s) skew to ⃖##⃗ EF

4. plane(s) parallel to plane ADE

B C

A D F

Identify all pairs of angles of the given type. (Section 10.1)

G E

5. consecutive interior 1 3 2 4

5 7 68

H

6. alternate interior 7. corresponding 8. alternate exterior

Find m∠1 and m∠2. Tell which theorem you use in each case. (Section 10.2) 9.

10.

11.

2 1

123° 2 1

138°

57° 1 2

Decide whether there is enough information to prove that m ! n. If so, state the theorem you would use. (Section 10.3) 12.

m

13. 69° 111°

n

14.

m n

m

ℓ! m andℓ! n

n

15. Cellular phones use bars like the ones shown to indicate how much signal

2

strength a phone receives from the nearest service tower. Each bar is parallel to the bar directly next to it. (Section 10.3) a. Explain why the tallest bar is parallel to the shortest bar. b. Imagine that the left side of each bar extends infinitely as a line. If m∠1 = 58°, then what is m∠2? 1

16. The diagram shows lines formed on a tennis court. q p

1 2

k

n

m

518

Chapter 10

Parallel and Perpendicular Lines

(Section 10.1 and Section 10.3) a. Identify two pairs of parallel lines so that each pair is in a different plane. b. Identify two pairs of perpendicular lines. c. Identify two pairs of skew lines. d. Prove that ∠1 ≅ ∠2.

10.4 Proofs with Perpendicular Lines Essential Question

What conjectures can you make about

perpendicular lines?

Writing Conjectures Work with a partner. Fold a piece of paper in half twice. Label points on the two creases, as shown.

D

— and CD —. a. Write a conjecture about AB Justify your conjecture.

— and OB —. b. Write a conjecture about AO Justify your conjecture.

A

O

B

C

Exploring a Segment Bisector Work with a partner. Fold and crease a piece of paper, as shown. Label the ends of the crease as A and B.

A

a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles?

B

Writing a Conjecture

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.

Work with a partner.

—, as shown. a. Draw AB

A

b. Draw an arc with center A on each —. Using the same compass side of AB setting, draw an arc with center B —. Label the on each side of AB intersections of the arcs C and D.

O

C

—. Label its intersection c. Draw CD — with AB as O. Write a conjecture about the resulting diagram. Justify your conjecture.

D

B

Communicate Your Answer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, find AO and OB when AB = 4 units.

Section 10.4

Proofs with Perpendicular Lines

519

10.4 Lesson

What You Will Learn Find the distance from a point to a line.

Core Vocabul Vocabulary larry

Construct perpendicular lines.

distance from a point to a line, p. 520 perpendicular bisector, p. 521

Solve real-life problems involving perpendicular lines.

Prove theorems about perpendicular lines.

Finding the Distance from a Point to a Line The distance from a point to a line is the length of the perpendicular segment from the point to the line. This perpendicular segment is the shortest distance between the point and the line. For example, the distance between point A and line k is AB. A k B distance from a point to a line

Finding the Distance from a Point to a Line Find the distance from point A to ⃖""⃗ BD. y

4

A(−3, 3)

D(2, 0) −4

REMEMBER Recall that if A(x1, y1) and C(x2, y2) are points in a coordinate plane, then the distance between A and C is

———

AC = √ (x2 − x1)2 + (y2 − y1)2 .

4x

C(1, −1)

B(−1, −3) −4

SOLUTION

— ⊥ ⃖""⃗ Because AC BD, the distance from point A to ⃖""⃗ BD is AC. Use the Distance Formula. ———

—

—

AC = √ (−3 − 1)2 + [3 − (−1)]2 = √ (−4)2 + 42 = √ 32 ≈ 5.7 So, the distance from point A to ⃖""⃗ BD is about 5.7 units.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1. Find the distance from point E to ⃖""⃗ FH. 4

−4

2 −2

E(−4, −3)

520

Chapter 10

Parallel and Perpendicular Lines

y

F(0, 3) G(1, 2) H(2, 1) x

Constructing Perpendicular Lines Constructing a Perpendicular Line P

Use a compass and straightedge to construct a line perpendicular to line m through point P, which is not on line m.

m

SOLUTION Step 1

Step 2

Step 3 P P

P m

B

A

m

A

B

m

Q

Q

Draw arc with center P Place the compass at point P and draw an arc that intersects the line twice. Label the intersections A and B.

Draw intersecting arcs Draw an arc with center A. Using the same radius, draw an arc with center B. Label the intersection of the arcs Q.

Draw perpendicular line Draw ⃖""⃗ PQ. This line is perpendicular to line m.

— is the line n with the following The perpendicular bisector of a line segment PQ two properties.

n

P

B

A

M

Q

— • n ⊥ PQ

—. • n passes through the midpoint M of PQ

Constructing a Perpendicular Bisector Use a compass and straightedge to construct —. the perpendicular bisector of AB

A

B

SOLUTION Step 2

Step 3 13

14

in.

15

Step 1

8

9

2

10

11

12

1 4 3

A

M

B

2

B

cm

1

A

5

B

4

5

6

7

3

A

6

Draw an arc Place the compass at A. Use a compass setting that is —. greater than half the length of AB Draw an arc.

Draw a second arc Keep the same compass setting. Place the compass at B. Draw an arc. It should intersect the other arc at two points.

Section 10.4

Bisect segment Draw a line through the two points of intersection. This line is the —. It perpendicular bisector of AB passes through M, the midpoint —. So, AM = MB. of  AB

Proofs with Perpendicular Lines

521

Proving Theorems about Perpendicular Lines

Theorems Linear Pair Perpendicular Theorem If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.

g 1

If ∠l ≅ ∠2, then g ⊥ h.

2

h

Proof Ex. 13, p. 525

Perpendicular Transversal Theorem In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.

j h k

If h ! k and j ⊥ h, then j ⊥ k. Proof Example 2, p. 522; Question 2, p. 522

Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

m

n p

If m ⊥ p and n ⊥ p, then m ! n. Proof Ex. 14, p. 525; Ex. 24, p. 532

Proving the Perpendicular Transversal Theorem Use the diagram to prove the Perpendicular Transversal Theorem.

j

SOLUTION Given h ! k, j ⊥ h Prove j ⊥ k

2 1 3 4

h

5 6 7 8

k

STATEMENTS

REASONS

1. h ! k, j ⊥ h

1. Given

2. m∠2 = 90°

2. Definition of perpendicular lines

3. ∠2 ≅ ∠6

3. Corresponding Angles Theorem

4. m∠2 = m∠6

4. Definition of congruent angles

5. m∠6 = 90°

5. Transitive Property of Equality

6. j ⊥ k

6. Definition of perpendicular lines

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

2. Prove the Perpendicular Transversal Theorem using the diagram in Example 2

and the Alternate Exterior Angles Theorem. 522

Chapter 10

Parallel and Perpendicular Lines

Solving Real-Life Problems Proving Lines Are Parallel The photo shows the layout of a neighborhood. Determine which lines, if any, must be parallel in the diagram. Explain your reasoning. s

t

u

p

q

SOLUTION Lines p and q are both perpendicular to s, so by the Lines Perpendicular to a Transversal Theorem, p ! q. Also, lines s and t are both perpendicular to q, so by the Lines Perpendicular to a Transversal Theorem, s ! t. So, from the diagram you can conclude p ! q and s ! t.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use the lines marked in the photo. a

b

c

d

3. Is b ! a? Explain your reasoning. 4. Is b ⊥ c? Explain your reasoning.

Section 10.4

Proofs with Perpendicular Lines

523

10.4 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept p Check 1. COMPLETE THE SENTENCE The perpendicular bisector of a segment is the line that passes through

the ________ of the segment at a ________ angle. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

Find the distance from point X to line ⃖""⃗ WZ.

X(−3, 3)

4

y

Z(4, 4)

Find XZ.

—. Find the length of XY

Y(3, 1) −4

−2

2 −2

Find the distance from lineℓto point X.

4

x

W(2, −2)

−4

Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, find the distance from point A to ⃖""⃗ XZ. (See Example 1.) 3.

y

Z(2, 7)

6

CONSTRUCTION In Exercises 5–8, trace line m and point

P. Then use a compass and straightedge to construct a line perpendicular to line m through point P. 5.

6.

P

P

4

m m

Y(0, 1)

A(3, 0)

−2

4

X(−1, −2)

4.

x

7.

y 3

−3

−1

x 1

Z(4, −1) −3 Y(2, −1.5) X(−4, −3)

m

—

CONSTRUCTION In Exercises 9 and 10, trace AB . Then

use a compass and straightedge to construct the —. perpendicular bisector of AB 9.

10.

A B

524

Chapter 10

P

m

A(3, 3)

1

8.

P

Parallel and Perpendicular Lines

A

B

ERROR ANALYSIS In Exercises 11 and 12, describe and

correct the error in the statement about the diagram.

In Exercises 17–22, determine which lines, if any, must be parallel. Explain your reasoning. (See Example 3.)

11.

17.



y

z

v

18.

w

x

a

x

b

y

c

Lines y and z are parallel. 19. 12.

m

a

20.

n

b c

p



C 12 cm A

d

q

8 cm B

The distance from point C to ⃖""⃗ AB is 12 centimeters.

n

21.

p

z

22. v

y

m w

PROVING A THEOREM In Exercises 13 and 14, prove the

theorem. (See Example 2.)

x

k

13. Linear Pair Perpendicular Theorem 14. Lines Perpendicular to a Transversal Theorem PROOF In Exercises 15 and 16, use the diagram to write

a proof of the statement.

23. USING STRUCTURE Find all the unknown angle

measures in the diagram. Justify your answer for each angle measure.

15. If two intersecting lines are perpendicular, then they

intersect to form four right angles. Given a ⊥ b Prove ∠1, ∠2, ∠3, and ∠4 are right angles. b 1

2 3 4

a

16. If two sides of two adjacent acute angles are

perpendicular, then the angles are complementary. Given """⃗ BA ⊥ """⃗ BC Prove ∠1 and ∠2 are complementary.

B

2

3 40°

5

30°

4

24. MAKING AN ARGUMENT Your friend claims that

because you can find the distance from a point to a line, you should be able to find the distance between any two lines. Is your friend correct? Explain your reasoning. 25. MATHEMATICAL CONNECTIONS Find the value of x

when a⊥ b and b % c. b

A

a (9x + 18)° 1

1

c [5(x + 7) + 15]°

2 C

Section 10.4

Proofs with Perpendicular Lines

525

CONSTRUCTION In Exercises 29 and 30, use the segment

26. HOW DO YOU SEE IT? You are trying to cross a

shown.

stream from point A. Which point should you jump to in order to jump the shortest distance? Explain your reasoning.

A

B

29. Construct a square of side length AB. A

30. Inscribe a square in a circle with radius AB. 31. ANALYZING RELATIONSHIPS The painted line

B

C

D

segments that form the path of a crosswalk are usually perpendicular to the crosswalk. Sketch what the segments in the photo would look like if they were perpendicular to the crosswalk. Which type of line segment requires less paint? Explain your reasoning.

E

27. ATTENDING TO PRECISION In which of the following

— " BD — and AC — ⊥ CD —? Select all diagrams is AC that apply. A ○

C ○

A

B

C

D

A

C

B

E ○

A

B ○

A

C

D

D ○

D

A

B

C

D

B

32. ABSTRACT REASONING Two lines, a and b, are C

perpendicular to line c. Line d is parallel to line c. The distance between lines a and b is x meters. The distance between lines c and d is y meters. What shape is formed by the intersections of the four lines?

D

28. THOUGHT PROVOKING The postulates and theorems

in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, how many right angles are formed by two perpendicular lines? Justify your answer.

33. WRITING Describe how you would find the distance

from a point to a plane. Can you find the distance from a line to a plane? Explain your reasoning.

Maintaining Mathematical Proficiency Simplify the ratio.

(Skills Review Handbook)

6 − (−4) 8−3

35. —

34. —

3−5 4−1

Reviewing what you learned in previous grades and lessons

8 − (−3) 7 − (−2)

36. —

Find the slope and the y-intercept of the graph of the linear equation. 38. y = 3x + 9

526

Chapter 10

1 2

39. y = −— x + 7

1 6

40. y = — x − 8

Parallel and Perpendicular Lines

13 − 4 2 − (−1)

37. —

(Section 3.5) 41. y = −8x − 6

10.5 Using Parallel and Perpendicular Lines Essential Question

How can you find the distance between two

parallel lines?

Finding the Distance Between Two Parallel Lines 15

Work with a partner. 12

13

.

14

in

10

11

1

9

2

7

8

a. Draw a line and label itℓ. Draw a point not on lineℓ and label it P.

6

3

5

P

2

3

4

4

cm

1

5

6

b. Construct a line through point P perpendicular to lineℓ. c. Use a centimeter ruler to measure the distance from point P to lineℓ.

in

10

11

1

12

13

.

14

15

d. Construct a line through point P parallel to lineℓ and label it m. 7

8

9

2

6

3

5 2

3

4

4

m

1 cm 6

To be proficient in math, you need to justify your conclusions and communicate them to others.

P

5

CONSTRUCTING VIABLE ARGUMENTS

e. Choose any point except point P on line ℓ or line m and label it Q. Describe how to find the distance from point Q to the other line. f. Find the distance from point Q to the other line. Compare this distance to the distance from point P to lineℓ. g. Is the distance from any point on lineℓ to line m constant? Explain your reasoning.

Communicate Your Answer 2. How can you find the distance between two parallel lines? 3. Use centimeter graph paper and a centimeter ruler to find the distance between the

two parallel lines. a. y = 2x + 2

b. y = −x + 4

y = 2x − 7

y = −x − 5

Section 10.5

Using Parallel and Perpendicular Lines

527

10.5 Lesson

What You Will Learn Prove the slope criteria for parallel lines. Find the distance from a point to a line.

Core Vocabul Vocabulary larry

Find the distance between two parallel lines.

Previous parallel lines perpendicular lines

Proving the Slope Criteria for Parallel Lines In the coordinate plane, the x-axis and the y-axis are perpendicular. Horizontal lines are parallel to the x-axis, and vertical lines are parallel to the y-axis.

Theorems CONNECTIONS TO ALGEBRA The information you learned about parallel and perpendicular lines in Section 4.3 can be stated as theorems.

Slopes of Parallel Lines In a coordinate plane, two distinct nonvertical lines are parallel if and only if they have the same slope.

y

Any two vertical lines are parallel.

x

Proof p. 528; Ex. 26, p. 532

m1 = m 2

Slopes of Perpendicular Lines In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is −1.

y

Horizontal lines are perpendicular to vertical lines. Proof Ex. 28, p. 532; p. 636

x

⋅

m1 m2 = −1

Part of Slopes of Parallel Lines Theorem Given Two distinct nonvertical lines in a coordinate plane Prove The lines are parallel if and only if they have the same slope.

CONNECTIONS TO ALGEBRA In Section 5.4, you determined the number of solutions of a linear system by looking at its graph. One solution: The lines intersect. No solution: The lines are parallel. Infinitely many solutions: The lines are the same.

528

Chapter 10

Paragraph Proof Consider the system of linear equations at the right, where Equations 1 and 2 represent two distinct nonvertical lines in a coordinate plane. The lines are parallel if and only if the system has no solution.

y = m1x + b1

Equation 1

y = m2x + b2

Equation 2

Substitute m2x + b2 for y in Equation 1. y = m1x + b1 Equation 1 m2x + b2 = m1x + b1

Substitute m2x + b2 for y.

m2x – m1x = b1 – b2

Isolate like terms.

(m2 – m1)x = b1 – b2

Factor.

When m1 ≠ m2, you can divide each side of the equation above by m2 − m1 to find the value of x and then substitute the value of x into Equation 1 or 2 to find the value of y. So, the system has a solution when m1 ≠ m2. When m1 = m2, the equation simplifies to b2 = b1 for all values of x. This is false because b1 and b2 must be different for the lines to be distinct. So, the system has no solution when m1 = m2. Because two distinct nonvertical lines are parallel if and only if their system has no solution and their system has no solution if and only if they have the same slope, two distinct nonvertical lines in a coordinate plane are parallel if and only if they have the same slope.

Parallel and Perpendicular Lines

Finding the Distance from a Point to a Line Recall that the distance from a point to a line is the length of the perpendicular segment from the point to the line.

Finding the Distance from a Point to a Line y

y = −x + 3

2 2

(1, 0)

4

x

Find the distance from the point (1, 0) to the line y = −x + 3.

SOLUTION Step 1 Find an equation of the line perpendicular to the line y = −x + 3 that passes through the point (1, 0). First, find the slope m of the perpendicular line. The line y = −x + 3 has a slope of −1. Use the Slopes of Perpendicular Lines Theorem.

⋅

−1 m = −1 m=1

The product of the slopes of ⊥ lines is −1. Divide each side by −1.

Then find the y-intercept b by using m = 1 and (x, y) = (1, 0). y = mx + b

Use slope-intercept form.

0 = 1(1) + b

Substitute for x, y, and m.

−1 = b

Solve for b.

Because m = 1 and b = −1, an equation of the line is y = x − 1. Step 2 Use the two equations to write and solve a system of equations to find the point where the two lines intersect. y = −x + 3

Equation 1

y=x−1

Equation 2

Substitute −x + 3 for y in Equation 2. y=x−1 −x + 3 = x − 1 x=2

Equation 2 Substitute −x + 3 for y. Solve for x. y

Substitute 2 for x in Equation 1 and solve for y. y = −x + 3

Equation 1

y = −2 + 3

Substitute 2 for x.

y=1

Simplify.

y=x−1

2

So, the perpendicular lines intersect at (2, 1).

(2, (2 2 1)) 2

4

(1, 1, 0) 0 y = −x + 3

x

Step 3 Use the Distance Formula to find the distance from (1, 0) to (2, 1). ——

distance = √ (1 − 2)2 + (0 − 1)2 ≈ 1.4 So, the distance from the point (1, 0) to the line y = −x + 3 is about 1.4 units.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1. Find the distance from the point (6, 4) to the line y = x + 4. 2. Find the distance from the point (−1, 6) to the line y = −2x.

Section 10.5

Using Parallel and Perpendicular Lines

529

Finding the Distance Between Two Parallel Lines The distance between two parallel lines is the length of any perpendicular segment joining the two lines. For instance, the distance between line p and line m below is CD or EF. C

E m

D

p

F

Finding the Distance Between Two Parallel Lines 1 1 Find the distance between y = — x − 2 and y = — x + 1. 2 2

SOLUTION

y = −2x − 2 4

( −4

6 2 −5 , 5

)

y

1

y = 2x + 1

2

−2

⋅

m = −2

4 x

2

(0, −2) 1

−4

Step 1 Find an equation of a line perpendicular to the two parallel lines. The slopes 1 of the two parallel lines are both —. Use the Slopes of Perpendicular Lines 2 Theorem. 1 The product of the slopes of ⊥ lines is −1. — m = −1 2

y = 2x − 2

Multiply each side by 2.

Any line with a slope of −2 is perpendicular to the two parallel lines. Use y = −2x – 2 because it has the same y-intercept, (0, −2), as one of the two parallel lines. Step 2 The distance between the points where y = −2x − 2 intersects the parallel lines is the distance between the lines. You already know that one point of intersection is (0, –2). Use substitution to find the other point of intersection. 1 y = —x + 1 Write original equation. 2 1 −2x − 2 = —x + 1 Substitute −2x − 2 for y. 2 6 x = −— Solve for x. 5 6 Substituting x = −— into y = −2x − 2 and solving for y gives the y-value of 5 2 the other point of intersection, which is —. So, the points of intersection are 5 6 2 (0, −2) and −—, — . 5 5 6 2 Step 3 Use the Distance Formula to find the distance from (0, −2) to −—, — . 5 5

(

)

√(

(

)

———

) (

)

2 2 6 2 − — − 0 + — − (−2) ≈ 2.7 5 5 1 1 So, the distance between y = — x − 2 and y = — x + 1 is about 2.7 units. 2 2

distance =

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Find the distance between the parallel lines. 3. y = 2x − 4, y = 2x + 1

3 4

3 4

5. y = — x − 3, y = — x + 2

530

Chapter 10

Parallel and Perpendicular Lines

4. y = −3x − 2, y = −3x − 4

3 2

3 2

6. y = −— x − 6, y = −— x + 5

10.5 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept p Check 1. COMPLETE THE SENTENCE The distance between two parallel lines is the length of any ________

segment joining the two lines. 2. WRITING How is the distance between two parallel lines related to the distance between two points?

Monitoring Progress and Modeling with Mathematics In Exercises 3–8, find the distance from point A to the given line. (See Example 1.) 3. A(−1, 7)

4.

y

4

In Exercises 15–18, use the map shown. Each unit in the coordinate plane corresponds to 1 mile.

y 1

y = −2 x + 3

6

y = −4x

4

−4

−2

y = 3x

2 −2

−2

4x

2

4

2

stadium (−6, 6)

6x

y

Pine Ave. school (9, 6) Oak Ave.

y = 2x − 12

A(6, −2)

x

1

−4

y = −2 x − 2 Hazel St.

y = 2x + 2

5. A(−9, −3), y = x − 6

Cherry St.

6. A(−4, 4), y = −2x + 1 7. A(−1, 4), line with a slope of −3 that passes through

(2, −4)

16. Find the distance from the stadium to Pine Avenue.

8. A(15, −21), line for which f(4) = −8 and f(8) = −18

In Exercises 9–14, find the distance between the parallel lines. (See Example 2.) 9. y = 2x + 7

15. Find the distance from the school to Cherry Street.

4

10.

y

y = −3x + 4

x

18. Find the distance between Hazel Street and

Cherry Street.

4 2

Pine Avenue.

y

2 −2

17. Find the distance between Oak Avenue and

19. PROBLEM SOLVING A gazebo is being built near a

y = −3x + 1

−2

y = 2x − 2

−4

−2

2

4x

nature trail. An equation of the line representing the nature trail is y = —13 x − 4. Each unit in the coordinate plane corresponds to 10 feet. Approximately how far is the gazebo from the nature trail?

−2 y

gazebo (−6, 4)

11. y = −4x + 5, y = −4x + 7

−12 −8

12. y = 3x + 6, y = 3x − 2

−4

4 4

12 x

2

13. y = −—3 x + 8, parallel line that passes through (0, 0) 5

14. y = −—6 x − 1, parallel line that passes through (6, −4)

Section 10.5

Using Parallel and Perpendicular Lines

531

20. PROBLEM SOLVING A bike path is being constructed

perpendicular to Washington Boulevard starting at point P(2, 2). An equation of the line representing 2 Washington Boulevard is y = −—3 x. Each unit in the coordinate plane corresponds to 10 feet. Approximately how far is the starting point from Washington Boulevard?

23. MAKING AN ARGUMENT Your classmate claims that

no two nonvertical parallel lines can have the same y-intercept. Is your classmate correct? Explain. PROVING A THEOREM In Exercises 24 and 25, use

the slopes of lines to write a paragraph proof of the theorem. 24. Lines Perpendicular to a Transversal Theorem: In a

plane, if two lines are perpendicular to the same line, then they are parallel to each other. 25. Transitive Property of Parallel Lines Theorem: If

two lines are parallel to the same line, then they are parallel to each other. 26. PROOF Prove the statement: If two lines are vertical,

then they are parallel. 27. PROOF Prove the statement: If two lines are

21. ERROR ANALYSIS Describe and correct the error in

horizontal, then they are parallel.

finding the distance between y = x + 2 and y = x + 1.



28. PROOF Prove that horizontal lines are perpendicular

distance = (x + 2) − (x + 1) =x+2−x−1 =1 The distance between y = x + 2 and y = x + 1 is 1 unit.

to vertical lines. 29. MATHEMATICAL CONNECTIONS Find the area of

△ABC.

4

y

A(4, 3)

2

22. HOW DO YOU SEE IT? Determine whether

quadrilateral JKLM is a square. Explain your reasoning.

−2

4

C(−4, −1)

y

K(0, n)

B(1, −3)

−4

L(n, n)

x

30. THOUGHT PROVOKING Find a formula for the J(0, 0)

distance from the point (x0, y0) to the line ax + by = 0. Verify your formula using a point and a line.

M(n, 0) x

Maintaining Mathematical Proficiency Plot the point in a coordinate plane.

Reviewing what you learned in previous grades and lessons

(Skills Review Handbook)

31. A(3, 6)

32. B(0, −4)

33. C(5, 0)

34. D(−1, −2)

Copy and complete the table. 35.

x

−2

−1

(Skills Review Handbook) 0

1

2

36.

3 y=x−— 4

y=x+9

532

Chapter 10

x

Parallel and Perpendicular Lines

−2

−1

0

1

2

10.4–10.5

What Did You Learn?

Core Vocabulary distance from a point to a line, p. 520 perpendicular bisector, p. 521

Core Concepts Section 10.4 Finding the Distance from a Point to a Line, p. 520 Constructing Perpendicular Lines, p. 521 Linear Pair Perpendicular Theorem, p. 522 Perpendicular Transversal Theorem, p. 522 Lines Perpendicular to a Transversal Theorem, p. 522

Section 10.5 Slopes of Parallel Lines, p. 528 Slopes of Perpendicular Lines, p. 528 Finding the Distance from a Point to a Line, p. 529 Finding the Distance Between Two Parallel Lines, p. 530

Mathematical Practices 1.

Compare the effectiveness of the argument in Exercise 24 on page 525 with the argument “You can find the distance between any two parallel lines.” What flaw(s) exist in the argument(s)? Does either argument use correct reasoning? Explain.

2.

Look back at your construction of a square in Exercise 29 on page 526. How would your construction change if you were to construct a rectangle?

3.

In Exercise 23 on page 532, another classmate makes the same claim about the x-intercept. Respond to your classmate’s argument by adapting your original answer.

Performance Task:

Squaring a Treehouse When builders construct any structure, they make sure it is plumb, level, and square. What do these terms mean? How are they related to the concepts of geometry? What relationships between them can you support? To explore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.

533

10

Chapter Review 10.1

Dynamic Solutions available at BigIdeasMath.com

Pairs of Lines and Angles (pp. 497–502)

Think of each segment in the figure as part of a line. a.

B

Which line(s) appear perpendicular to ⃖""⃗ AB? ⃖""⃗ appear perpendicular to ⃖""⃗ ⃖""⃗ BD, ⃖""⃗ AC, ⃖""⃗ BH, and AG AB.

D C

A

b. Which line(s) appear parallel to ⃖""⃗ AB?

H

⃖""⃗ CD, ⃖""⃗ GH, and ⃖""⃗ EF appear parallel to ⃖""⃗ AB.

G

F E

c. Which line(s) appear skew to ⃖""⃗ AB?

⃖""⃗ ⃖""⃗, and ⃖""⃗ CF, ⃖""⃗ CE, ⃖""⃗ DF, FH EG appear skew to ⃖""⃗ AB. d. Which plane(s) appear parallel to plane ABC? K

Plane EFG appears parallel to plane ABC. J

Think of each segment in the figure as part of a line. Which line(s) or plane(s) appear to fit the description? 1. line(s) perpendicular to ⃖""⃗ QR 3. line(s) skew to ⃖""⃗ QR

10.2

2. line(s) parallel to ⃖""⃗ QR 4. plane(s) parallel to plane LMQ

Parallel Lines and Transversals

L M

P N

Q R

(pp. 503–508)

Find the value of x. By the Vertical Angles Congruence Theorem, m∠6 = 50°. (x − 5)° + m∠6 = 180° (x − 5)° + 50° = 180° x + 45 = 180 x = 135

Consecutive Interior Angles Theorem Substitute 50° for m∠6.

(x − 5)° 6

Combine like terms.

50°

Subtract 45 from each side.

So, the value of x is 135. Find the values of x and y. 5.

6. 35º

yº

48º

xº

(5x − 17)º

7.

8. 2y º

534

Chapter 10

yº

58º 2x º

Parallel and Perpendicular Lines

(5y − 21)º (6x + 32)º 116º

10.3

Proofs with Parallel Lines (pp. 509–516)

Find the value of x that makes m ! n. By the Alternate Interior Angles Converse, m ! n when the marked angles are congruent.

m

(5x + 8)° 53°

n

(5x + 8)° = 53° 5x = 45 x=9 The lines m and n are parallel when x = 9. Find the value of x that makes m ! n. 9.

10.

147°

m

x° 73°

11.

12.

m

(7x − 11)°

(2x + 20)° m

10.4

n

(x + 14)°

n

(4x + 58)° n

n

3x °

m

Proofs with Perpendicular Lines (pp. 519–526)

Determine which lines, if any, must be parallel. Explain your reasoning. Lines a and b are both perpendicular to d, so by the Lines Perpendicular to a Transversal Theorem, a ! b. Also, lines c and d are both perpendicular to b, so by the Lines Perpendicular to a Transversal Theorem, c ! d.

a

b

c d

Determine which lines, if any, must be parallel. Explain your reasoning. 13.

x

14.

y

w

z

x

z

y

15.

a b

m

n

16.

a

m

n

b c

Chapter 10

Chapter Review

535

10.5

Using Parallel and Perpendicular Lines (pp. 527–532)

1 Find the distance from the point (4, 0) to the line y = — x + 1. 2 1 Step 1 Find an equation of the line perpendicular to the line y = — x + 1 that passes through the 2 point (4, 0). 1 1 First, find the slope m of the perpendicular line. The line y = — x + 1 has a slope of —. By 2 2 the Slopes of Perpendicular Lines Theorem, the slope of the perpendicular line is m = −2. Then find the y-intercept b by using m = −2 and (x, y) = (4, 0). y = mx + b

Use slope-intercept form.

0 = −2(4) + b

Substitute for x, y, and m.

8=b

Solve for b.

Because m = −2 and b = 8, an equation of the line is y = −2x + 8. Step 2

Use the two equations to write and solve a system of equations to find the point where the two lines intersect. 1 y = —x + 1 2 y = −2x + 8

Equation 1 Equation 2

1 Substitute — x + 1 for y in Equation 2. 2 y = −2x + 8 Equation 2 1 2

y

1 Substitute — x + 1 for y. 2

— x + 1 = −2x + 8

4

14 x=— Solve for x. 5 14 Substitute — for x in Equation 1 and solve for y. 5 1 y = —x + 1 Equation 1 2 1 14 14 y=— — +1 Substitute — for x. 5 2 5 12 y=— Simplify. 5 14 12 So, the perpendicular lines intersect at —, — . 5 5

y=

1 x 2

+1

(4, 0) −2

( )

(

( 145, 125 ) 2

−2

y = −2x + 8

)

(

)

14 12 Step 3 Use the Distance Formula to find the distance from (4, 0) to —, — . 5 5 —— 2 2 14 12 distance = — − 4 + — − 0 ≈ 2.7 5 5 1 So, the distance from the point (4, 0) to the line y = — x + 1 is about 2.7 units. 2

√(

) (

)

Find the distance from point A to the given line. 17. A(2, −1), y = −x + 4

3 4

18. A(−6, 4), y = −— x + 1

Find the distance between the parallel lines. 19. y = 6x − 4 and y = 6x + 10

536

Chapter 10

Parallel and Perpendicular Lines

1 4

1 4

20. y = −— x − 3 and y = −— x + 4

x

10

Chapter Test

Find the values of x and y. State which theorem(s) you used. 1.

2. 61°

3.

y°

42°

(11y + 19)° 8x°

x°

(8x + 2)°

96°

[6(2y − 3)]°

Find the distance from point A to the given line. 1

4. A(3, 4), y = −x

5. A(−3, 7), y = —3 x − 2

Find the value of x that makes m " n. 6.

7.

m

8.

n

(11x + 33)°

8x° x°

m

97°

n

m

(4x + 24)°

(6x − 6)° n

Find the distance between the parallel lines. 2

9. y = −5x − 14, y = −5x + 1

2

10. y = —3 x − 9, y = —3 x + 7

j k

11. A student says, “Because j ⊥ k, j ⊥ℓ.” What missing information is the student

assuming from the diagram? Which theorem is the student trying to use?

12. You and your family are visiting some attractions while on vacation. You

and your mom visit the shopping mall while your dad and your sister visit the aquarium. You decide to meet at the intersection of lines q and p. Each unit in the coordinate plane corresponds to 50 yards. a. Find an equation of line q. b. Find an equation of line p. c. What are the coordinates of the meeting point? d. What is the distance from the meeting point to the subway? A

I

C

E

G

1 D

2 3

L

F H

q p

aquarium subway

shopping mall

x

B

13.

J

K

y

M

Identify an example on the puzzle cube of each description. Explain your reasoning. a. a pair of skew lines b. a pair of perpendicular lines c. a pair of parallel lines d. a pair of congruent corresponding angles e. a pair of congruent alternate interior angles

Chapter 10

Chapter Test

537

10

Cumulative Assessment

1. Use the steps in the construction to explain how you know that ⃖""⃗ CD is the perpendicular

—. bisector of AB

Step 2

Step 3 13

14

in.

15

Step 1

8

9

2

10

11

12

1

C

3

4

A

cm

1

2

B

5

A

B

4

5

6

7

3

A

6

2. The table shows the distances you travel over a 6-hour period.

M D

Hours, x

Create an equation that models the distance traveled as a function of the number of hours.

B

Distance (miles), y

1

62

2

123

3

184

4

245

5

306

6

367

3. Classify each pair of angles whose measurements are given.

a.

b.

D 44°

136°

A

c.

B

J

C

75°

E

d.

K I

F 18° G 23° H

P

Q

75° 42°

M

L

N

48° R

4. Your school is installing new turf on the football field. A coordinate plane has been superimposed

on a diagram of the football field where 1 unit = 20 feet. a. What is the length of the field?

y

b. What is the perimeter of the field? c. Turf costs $2.69 per square foot. Your school has a $150,000 budget. Does the school have enough money to purchase new turf for the entire field? x

538

Chapter 10

Parallel and Perpendicular Lines

y

5. The graph shows the function f(x) = 2(3)x.

a. Is the function increasing or decreasing for increasing values of x?

16 12

b. Identify any x- and y-intercepts.

8 4

−4

−2

2

4 x

6. Which of the following is true when ⃖""⃗ AB and ⃖""⃗ CD are skew?

AB and ⃖""⃗ CD are parallel. A ⃖""⃗ ○

AB and ⃖""⃗ CD are perpendicular. C ⃖""⃗ ○

AB and ⃖""⃗ CD intersect. B ⃖""⃗ ○ D A, B, and C are noncollinear. ○

7. Select all the statements that are valid

Attending School Concert Yes No Class

for the data in the two-way table. Freshman Sophomore

28 22

36 14

Of the students who are attending, 56% are freshmen.

Of the students who are freshmen, about 64% are attending.

Of the students who are not attending, 28% are sophomores.

Of the students who are sophomores, about 50% are not attending.

y

8. You and your friend walk to school together every day. You meet at the

ffriend’s house

halfway point between your houses first and then walk to school. Each unit in the coordinate plane corresponds to 50 yards. a. What are the coordinates of the midpoint of the line segment joining the two houses?

school

b. What is the distance that the two of you walk together?

your house yo

x

9. Which of the following is not true for the diagram shown?

A ∠ABF and ∠EFH are ○

B ∠DBC and ∠HFE are ○

C ∠ABF and ∠CBF are ○

D ∠ABF and ∠EFB are ○

corresponding angles. alternate interior angles.

D

alternate exterior angles. consecutive interior angles.

A

E

C

B

F

G H

Chapter 10

Cumulative Assessment

539

11 11.1 11.2 11.3 11.4

Transformations Translations Reflections Rotations Congruence and Transformations

(p 576) Stained Glass Window (p.

Tessellation (p. 575)

SEE the Big Idea

(p 568) Kaleidoscope (p.

Revolving R l i D Door ((p. 567)

Chess (p. 549)

Maintaining Mathematical Proficiency Identifying Transformations Example 1 Tell whether the red figure is a translation, reflection, or rotation of the blue figure. a.

The blue figure b. turns to form the red figure, so it is a rotation.

The red figure is a mirror image of the blue figure, so it is a reflection.

Tell whether the red figure is a translation, reflection, or rotation of the blue figure. 1.

2.

3.

Finding Perimeter and Area in the Coordinate Plane Example 2

Find the perimeter and area of rectangle ABCD with vertices A (−3, 4), B (5, 4), C (5, −3), and D (−3, −3). Step 1 Draw the rectangle in a coordinate plane.

y

A(−3, 4)

Step 2 Use the Ruler Postulate to find the length and width of the rectangle. Length AB = ∣ 5 − (−3) ∣ = 8 Width AD = ∣ 4 − (−3) ∣ = 7 The length is 8 units and the width is 7 units.

B(5, 4)

2

−2

4

2

x

−2

D(−3, −3)

C(5, −3)

Step 3 Substitute the values for the length and width into the formulas for the perimeter and area of a rectangle. Perimeter P = 2ℓ + 2w = 2(8) + 2(7) = 16 + 14 = 30 Area A =ℓw = (8)(7) = 56 So, the perimeter of rectangle ABCD is 30 units and the area is 56 square units.

Find the perimeter and area of the polygon with the given vertices. 4. D(1, 7), E(1, 1), F(−7, 1)

5. J(−4, 4), K(2, 4), L(2, −2), M(−4, −2)

6. P(0, 6), Q(3, 6), R(3, −3), S(0, −3)

7. X(0, −8), Y(−3, −4), Z(3, −4)

8. ABSTRACT REASONING Do the trapezoids in Example 1(a) have the same size and shape?

Do the triangles in Example 1(b) have the same size and shape? If so, how can you change the red figures so that they do not have the same size and shape as the blue figures? Dynamic Solutions available at BigIdeasMath.com

541

Mathematical Practices

Mathematically proficient students use dynamic geometry software strategically.

Using Dynamic Geometry Software

Core Concept Using Dynamic Geometry Software Dynamic geometry software allows you to create geometric drawings, including: • • • •

drawing a point drawing a line drawing a line segment drawing an angle

• • • •

measuring an angle measuring a line segment drawing a circle drawing an ellipse

• • • •

drawing a perpendicular line drawing a polygon copying and sliding an object reflecting an object in a line

Finding Side Lengths and Angle Measures Use dynamic geometry software to draw a triangle with vertices at A(−2, 1), B(2, 1), and C(2, −2). Find the side lengths and angle measures of the triangle.

SOLUTION Using dynamic geometry software, you can create △ABC, as shown.

Sample 2

A

B

1

0 −2

−1

0

1

2

3

−1

−2

C

−3

Points A(−2, 1) B(2, 1) C(2, −2) Segments AB = 4 BC = 3 AC = 5 Angles m∠A = 36.87° m∠B = 90° m∠C = 53.13°

From the display, the side lengths are AB = 4 units, BC = 3 units, and AC = 5 units. The angle measures, rounded to two decimal places, are m∠A ≈ 36.87°, m∠B = 90°, and m∠C ≈ 53.13°.

Monitoring Progress Use dynamic geometry software to draw the polygon with the given vertices. Use the software to find the side lengths and angle measures of the polygon. Round your answers to the nearest hundredth.

542

1. A(0, 2), B(3, −1), C(4, 3)

2. A(−2, 1), B(−2, −1), C(3, 2)

3. A(1, 1), B(−3, 1), C(−3, −2), D(1, −2)

4. A(1, 1), B(−3, 1), C(−2, −2), D(2, −2)

5. A(−3, 0), B(0, 3), C(3, 0), D(0, −3)

6. A(0, 0), B(4, 0), C(1, 1), D(0, 3)

Chapter 11

Transformations

11.1 Translations Essential Question

How can you translate a figure in a

coordinate plane?

Translating a Triangle in a Coordinate Plane Work with a partner. a. Use dynamic geometry software to draw any triangle and label it △ABC. b. Copy the triangle and translate (or slide) it to form a new figure, called an image, △A′B′C′ (read as “triangle A prime, B prime, C prime”). c. What is the relationship between the coordinates of the vertices of △ABC and those of △A′B′C′? d. What do you observe about the side lengths and angle measures of the two triangles?

USING TOOLS STRATEGICALLY To be proficient in math, you need to use appropriate tools strategically, including dynamic geometry software.

Sample 4

A′

3

A

B′

B

2

1

C′

0 −1

0

1

2

3

4

5

6

7

C

−1

−2

Points A(−1, 2) B(3, 2) C(2, −1) Segments AB = 4 BC = 3.16 AC = 4.24 Angles m∠A = 45° m∠B = 71.57° m∠C = 63.43°

Translating a Triangle in a Coordinate Plane y

B

4

A

(

2 −4

−2

4 −2 −4

Work with a partner. a. The point (x, y) is translated a units horizontally and b units vertically. Write a rule to determine the coordinates of the image of (x, y). (x, y) → , b. Use the rule you wrote in part (a) to translate △ABC 4 units left and 3 units down. What are the coordinates of the vertices of the image, △A′B′C′? c. Draw △A′B′C′. Are its side lengths the same as those of △ABC ? Justify your answer.

C

x

)

Comparing Angles of Translations Work with a partner. a. In Exploration 2, is △ABC a right triangle? Justify your answer. b. In Exploration 2, is △A′B′C′ a right triangle? Justify your answer. c. Do you think translations always preserve angle measures? Explain your reasoning.

Communicate Your Answer 4. How can you translate a figure in a coordinate plane? 5. In Exploration 2, translate △A′B′C ′ 3 units right and 4 units up. What are the

coordinates of the vertices of the image, △A″B ″C ″? How are these coordinates related to the coordinates of the vertices of the original triangle, △ABC ? Section 11.1

Translations

543

What You Will Learn

11.1 Lesson

Perform translations. Perform compositions.

Core Vocabul Vocabulary larry

Solve real-life problems involving compositions.

vector, p. 544 initial point, p. 544 terminal point, p. 544 horizontal component, p. 544 vertical component, p. 544 component form, p. 544 transformation, p. 544 image, p. 544 preimage, p. 544 translation, p. 544 rigid motion, p. 546 composition of transformations, p. 546

Performing Translations A vector is a quantity that has both direction and magnitude, or size, and is represented in the coordinate plane by an arrow drawn from one point to another.

Core Concept Vectors The diagram shows a vector. The initial point, or starting point, of the vector is P, and the terminal point, or ending point, is Q. The vector is named PQ , which is read as “vector PQ.” The horizontal component of PQ is 5, and the vertical component is 3. The component form of a vector combines the horizontal and vertical components. So, the component form of PQ is 〈5, 3〉.

⃑

⃑

Q 3 units up P 5 units right

⃑

Identifying Vector Components K

In the diagram, name the vector and write its component form.

SOLUTION

⃑

The vector is JK . To move from the initial point J to the terminal point K, you move 3 units right and 4 units up. So, the component form is 〈3, 4〉. J

A transformation is a function that moves or changes a figure in some way to produce a new figure called an image. Another name for the original figure is the preimage. The points on the preimage are the inputs for the transformation, and the points on the image are the outputs.

Core Concept Translations

STUDY TIP You can use prime notation to name an image. For example, if the preimage is point P, then its image is point P′, read as “point P prime.”

544

Chapter 11

A translation moves every point of a figure the same distance in the same direction. More specifically, a translation maps, or moves, the points P and Q of a plane figure along a vector 〈a, b〉 to the points P′ and Q′, so that one of the following statements is true. • •

y

P′(x1 + a, y1 + b)

P(x1, y1) Q′(x2 + a, y2 + b) Q(x2, y2)

— % QQ′ —, or PP′ = QQ′ and PP′ — — are collinear. PP′ = QQ′ and PP′ and QQ′

Translations map lines to parallel lines and segments to parallel segments. For — — instance, in the figure above, PQ % P′Q′ .

Transformations

x

Translating a Figure Using a Vector The vertices of △ABC are A(0, 3), B(2, 4), and C(1, 0). Translate △ABC using the vector 〈5, −1〉.

SOLUTION First, graph △ABC. Use 〈5, −1〉 to move each vertex 5 units right and 1 unit down. Label the image vertices. Draw △A′B′C′. Notice that the vectors drawn from preimage vertices to image vertices are parallel.

y

B B′(7, 3)

A 2

A′(5, 2) C

8

x

C′(6, −1)

You can also express a translation along the vector 〈a, b〉 using a rule, which has the notation (x, y) → (x + a, y + b).

Writing a Translation Rule Write a rule for the translation of △ABC to △A′B′C′.

y

A′

SOLUTION

A

3

To go from A to A′, you move 4 units left and 1 unit up, so you move along the vector 〈−4, 1〉.

B′ C

C′ 2

B

4

6

8 x

So, a rule for the translation is (x, y) → (x − 4, y + 1).

Translating a Figure in the Coordinate Plane Graph quadrilateral ABCD with vertices A(−1, 2), B(−1, 5), C(4, 6), and D(4, 2) and its image after the translation (x, y) → (x + 3, y − 1).

SOLUTION B

6 4

y

C C′ B′

Graph quadrilateral ABCD. To find the coordinates of the vertices of the image, add 3 to the x-coordinates and subtract 1 from the y-coordinates of the vertices of the preimage. Then graph the image, as shown at the left. (x, y) → (x + 3, y − 1)

A

A(−1, 2) → A′(2, 1) B(−1, 5) → B′(2, 4) C(4, 6) → C′(7, 5) D(4, 2) → D′(7, 1)

D D′

A′ 2

4

6

x

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1. Name the vector and write its component form. 2. The vertices of △LMN are L(2, 2), M(5, 3), and N(9, 1). Translate △LMN using

K

the vector 〈−2, 6〉.

B

3. In Example 3, write a rule to translate △A′B′C′ back to △ABC. 4. Graph △RST with vertices R(2, 2), S(5, 2), and T(3, 5) and its image after the

translation (x, y) → (x + 1, y + 2).

Section 11.1

Translations

545

Performing Compositions A rigid motion is a transformation that preserves length and angle measure. Another name for a rigid motion is an isometry. A rigid motion maps lines to lines, rays to rays, and segments to segments.

Postulate Translation Postulate A translation is a rigid motion.

Because a translation is a rigid motion, and a rigid motion preserves length and angle measure, the following statements are true for the translation shown.

E′ E D

D′ F

• DE = D′E′, EF = E′F′, FD = F′D′ F′

• m∠D = m∠D′, m∠E = m∠E′, m∠F = m∠F′ When two or more transformations are combined to form a single transformation, the result is a composition of transformations.

Theorem Composition Theorem The composition of two (or more) rigid motions is a rigid motion. Proof

Ex. 35, p. 550 Q″

The theorem above is important because it states that no matter how many rigid motions you perform, lengths and angle measures will be preserved in the final image. For instance, the composition of two or more translations is a translation, as shown.

m

po

sit

2

io

n

ion

lat

ns

tra

P″

co

Q′ Q

P

n1

P′

latio trans

Performing a Composition —

Graph RS with endpoints R(−8, 5) and S(−6, 8) and its image after the composition. Translation: (x, y) → (x + 5, y − 2) Translation: (x, y) → (x − 4, y − 2)

SOLUTION

—. Step 1 Graph RS — 5 units right and Step 2 Translate RS — has endpoints 2 units down. R′S′ R′(−3, 3) and S′(−1, 6). — 4 units left and Step 3 Translate R′S′ — has endpoints 2 units down. R″S″ R″(−7, 1) and S″(−5, 4). 546

Chapter 11

Transformations

S(−6, 8)

8

S′(−1, 6) R(−8, 5) S″(−5, 4) R′(−3, 3)

y

6 4 2

R″(−7, 1) −8

−6

−4

−2

x

Solving Real-Life Problems Modeling with Mathematics You are designing a favicon for a golf website. In an image-editing program, you move the red rectangle 2 units left and 3 units down. Then you move the red rectangle 1 unit right and 1 unit up. Rewrite the composition as a single translation.

y 14 12 10 8 6

SOLUTION 1. Understand the Problem You are given two translations. You need to rewrite the result of the composition of the two translations as a single translation.

4 2 2

4

6

8

10

12

14

x

2. Make a Plan You can choose an arbitrary point (x, y) in the red rectangle and determine the horizontal and vertical shift in the coordinates of the point after both translations. This tells you how much you need to shift each coordinate to map the original figure to the final image.

3. Solve the Problem Let A(x, y) be an arbitrary point in the red rectangle. After the first translation, the coordinates of its image are A′(x − 2, y − 3). The second translation maps A′(x − 2, y − 3) to A″(x − 2 + 1, y − 3 + 1) = A″(x − 1, y − 2). The composition of translations uses the original point (x, y) as the input and returns the point (x − 1, y − 2) as the output. So, the single translation rule for the composition is (x, y) → (x − 1, y − 2).

4. Look Back Check that the rule is correct by testing a point. For instance, (10, 12) is a point in the red rectangle. Apply the two translations to (10, 12). (10, 12) → (8, 9) → (9, 10) Does the final result match the rule you found in Step 3? (10, 12) → (10 − 1, 12 − 2) = (9, 10)

Monitoring Progress



Help in English and Spanish at BigIdeasMath.com

— with endpoints T(1, 2) and U(4, 6) and its image after the composition. 5. Graph TU Translation: (x, y) → (x − 2, y − 3) Translation: (x, y) → (x − 4, y + 5)

— with endpoints V(−6, −4) and W(−3, 1) and its image after the 6. Graph VW composition. Translation: (x, y) → (x + 3, y + 1) Translation: (x, y) → (x − 6, y − 4) 7. In Example 6, you move the gray square 2 units right and 3 units up. Then you

move the gray square 1 unit left and 1 unit down. Rewrite the composition as a single transformation. Section 11.1

Translations

547

11.1 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY Name the preimage and image of the transformation △ABC → △A′B′C ′. 2. COMPLETE THE SENTENCE A ______ moves every point of a figure the same distance in the

same direction.

Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, name the vector and write its component form. (See Example 1.)

12. M

−7

3. C

L

1

M′ −3

L′

y 1

N

3 x

N′

−5

D

4.

In Exercises 13–16, use the translation. (x, y) → (x − 8, y + 4)

S

13. What is the image of A(2, 6)? 14. What is the image of B(−1, 5)? T

15. What is the preimage of C ′(−3, −10)?

In Exercises 5–8, the vertices of △DEF are D(2, 5), E(6, 3), and F(4, 0). Translate △DEF using the given vector. Graph △DEF and its image. (See Example 2.) 5. 〈6, 0〉

6. 〈5, −1〉

7. 〈−3, −7〉

8. 〈−2, −4〉

In Exercises 9 and 10, find the component form of the vector that translates P(−3, 6) to P′. 9. P′(0, 1)

10. P′(−4, 8)

In Exercises 11 and 12, write a rule for the translation of △LMN to △L′M′N ′. (See Example 3.) 11. M′

4

L′ −4

y

N′ −2

16. What is the preimage of D′(4, −3)?

In Exercises 17–20, graph △PQR with vertices P(−2, 3), Q(1, 2), and R(3, −1) and its image after the translation. (See Example 4.) 17. (x, y) → (x + 4, y + 6) 18. (x, y) → (x + 9, y − 2) 19. (x, y) → (x − 2, y − 5) 20. (x, y) → (x − 1, y + 3)

In Exercises 21 and 22, graph △XYZ with vertices X(2, 4), Y(6, 0), and Z(7, 2) and its image after the composition. (See Example 5.) 21. Translation: (x, y) → (x + 12, y + 4)

M

Translation: (x, y) → (x − 5, y − 9) 6x

−2

L

N

22. Translation: (x, y) → (x − 6, y)

Translation: (x, y) → (x + 2, y + 7)

548

Chapter 11

Transformations

In Exercises 23 and 24, describe the composition of translations. 23.

4

A

2

C −4

27. PROBLEM SOLVING You are studying an amoeba

through a microscope. Suppose the amoeba moves on a grid-indexed microscope slide in a straight line from square B3 to square G7.

y

A′ C′

B′

1 2 3 4 5 6 7 8

B −2

A″

4

2

x

−2

B″

C″

ABCDEFGH

X

a. Describe the translation. 24.

b. The side length of each grid square is 2 millimeters. How far does the amoeba travel?

y

D

G D″

E

3

F E″ −1

c. The amoeba moves from square B3 to square G7 in 24.5 seconds. What is its speed in millimeters per second? D′

E′

5 x

28. MATHEMATICAL CONNECTIONS Translation A maps

−2

F″ G′

G″

(x, y) to (x + n, y + t). Translation B maps (x, y) to (x + s, y + m).

F′

a. Translate a point using Translation A, followed by Translation B. Write an algebraic rule for the final image of the point after this composition.

25. ERROR ANALYSIS Describe and correct the error in

graphing the image of quadrilateral EFGH after the translation (x, y) → (x − 1, y − 2).



5

y

E′

3

E

1

F′ H′

3

c. Compare the rules you wrote for parts (a) and (b). Does it matter which translation you do first? Explain your reasoning.

F G′

H 1

b. Translate a point using Translation B, followed by Translation A. Write an algebraic rule for the final image of the point after this composition.

5

G

9x

26. MODELING WITH MATHEMATICS In chess, the

knight (the piece shaped like a horse) moves in an L pattern. The board shows two consecutive moves of a black knight during a game. Write a composition of translations for the moves. Then rewrite the composition as a single translation that moves the knight from its original position to its ending position. (See Example 6.)

MATHEMATICAL CONNECTIONS In Exercises 29 and 30,

a translation maps the blue figure to the red figure. Find the value of each variable. 29.

3w° 162° r°

100° 2t

s

8

10

30.

b+6

20

a°

55° 4c − 6 14

Section 11.1

Translations

549

31. USING STRUCTURE Quadrilateral DEFG has vertices

D(−1, 2), E(−2, 0), F(−1, −1), and G(1, 3). A translation maps quadrilateral DEFG to quadrilateral D′E′F′G′. The image of D is D′(−2, −2). What are the coordinates of E′, F′, and G′?

35. PROVING A THEOREM Prove the Composition

Theorem. 36. PROVING A THEOREM Use properties of translations

to prove each theorem. a. Corresponding Angles Theorem

32. HOW DO YOU SEE IT? Which two figures represent

b. Corresponding Angles Converse

a translation? Describe the translation.

37. WRITING Explain how to use translations to draw

a rectangular prism. 38. MATHEMATICAL CONNECTIONS The vector

PQ = 〈4, 1〉 describes the translation of A(−1, w) onto A′(2x + 1, 4) and B(8y − 1, 1) onto B′(3, 3z). Find the values of w, x, y, and z.

7

1

4 6

3

— to 39. MAKING AN ARGUMENT A translation maps GH

8

2

—. Your friend claims that if you draw segments G′H′ connecting G to G′ and H to H′, then the resulting quadrilateral is a parallelogram. Is your friend correct? Explain your reasoning.

5 9

40. THOUGHT PROVOKING You are a graphic designer 33. REASONING The translation (x, y) → (x + m, y + n)

— to P′Q′ —. Write a rule for the translation of maps PQ — — P′Q′ to PQ . Explain your reasoning.

34. DRAWING CONCLUSIONS The vertices of a rectangle

for a company that manufactures floor tiles. Design a floor tile in a coordinate plane. Then use translations to show how the tiles cover an entire floor. Describe the translations that map the original tile to four other tiles.

are Q(2, −3), R(2, 4), S(5, 4), and T(5, −3).

41. REASONING The vertices of △ABC are A(2, 2),

a. Translate rectangle QRST 3 units left and 3 units down to produce rectangle Q′R′S′T ′. Find the area of rectangle QRST and the area of rectangle Q′R′S′T ′.

— is perpendicular to lineℓ. M′N′ — is the 42. PROOF MN

B(4, 2), and C(3, 4). Graph the image of △ABC after the transformation (x, y) → (x + y, y). Is this transformation a translation? Explain your reasoning.

— 2 units to the left. Prove that M′N′ — translation of MN is perpendicular toℓ.

b. Compare the areas. Make a conjecture about the areas of a preimage and its image after a translation.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Tell whether the figure can be folded in half so that one side matches the other. (Skills Review Handbook) 43.

44.

45.

46.

49. x − (12 − 5x)

50. x − (−2x + 4)

Simplify the expression. (Skills Review Handbook) 47. −(−x)

550

Chapter 11

48. −(x + 3)

Transformations

11.2 Reflections Essential Question

How can you reflect a figure in a

coordinate plane?

Reflecting a Triangle Using a Reflective Device Work with a partner. Use a straightedge to draw any triangle on paper. Label it △ABC. a. Use the straightedge to draw a line that does not pass through the triangle. Label it m. b. Place a reflective device on line m. c. Use the reflective device to plot the images of the vertices of △ABC. Label the images of vertices A, B, and C as A′, B′, and C′, respectively. d. Use a straightedge to draw △A′B′C′ by connecting the vertices.

LOOKING FOR STRUCTURE To be proficient in math, you need to look closely to discern a pattern or structure.

Reflecting a Triangle in a Coordinate Plane Work with a partner. Use dynamic geometry software to draw any triangle and label it △ABC. a. Reflect △ABC in the y-axis to form △A′B′C′. b. What is the relationship between the coordinates of the vertices of △ABC and those of △A′B′C′? c. What do you observe about the side lengths and angle measures of the two triangles? d. Reflect △ABC in the x-axis to form △A′B′C′. Then repeat parts (b) and (c).

Sample C A

C′

4

A′

3

2

1

0 −3

−2

B

−1

0 −1

1

2

3

4

B′

Points A(−3, 3) B(−2, −1) C(−1, 4) Segments AB = 4.12 BC = 5.10 AC = 2.24 Angles m∠A = 102.53° m∠B = 25.35° m∠C = 52.13°

Communicate Your Answer 3. How can you reflect a figure in a coordinate plane?

Section 11.2

Reflections

551

11.2 Lesson

What You Will Learn Perform reflections.

Core Vocabul Vocabulary larry

Perform glide reflections.

reflection, p. 552 line of reflection, p. 552 glide reflection, p. 554 line symmetry, p. 555 line of symmetry, p. 555

Solve real-life problems involving reflections.

Identify lines of symmetry.

Performing Reflections

Core Concept Reflections A reflection is a transformation that uses a line like a mirror to reflect a figure. The mirror line is called the line of reflection. A reflection in a line m maps every point P P in the plane to a point P′, so that for each point one of the following properties is true. • If P is not on m, then m is the —, or perpendicular bisector of PP′ m • If P is on m, then P = P′.

P

P′

P′ m

point P not on m

point P on m

Reflecting in Horizontal and Vertical Lines Graph △ABC with vertices A(1, 3), B(5, 2), and C(2, 1) and its image after the reflection described. a. In the line n: x = 3

b. In the line m: y = 1

SOLUTION a. Point A is 2 units left of line n, so its reflection A′ is 2 units right of line n at (5, 3). Also, B′ is 2 units left of line n at (1, 2), and C′ is 1 unit right of line n at (4, 1). 4 2

y

n

A

4

A′ B

B′ C

b. Point A is 2 units above line m, so A′ is 2 units below line m at (1, −1). Also, B′ is 1 unit below line m at (5, 0). Because point C is on line m, you know that C = C′. y

A

2

C

C′ 2

4

C′ 6

B

m

B′ 6

x

x

A′

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Graph △ABC from Example 1 and its image after a reflection in the given line.

552

Chapter 11

1. x = 4

2. x = −3

3. y = 2

4. y = −1

Transformations

Reflecting in the Line y = x

REMEMBER The product of the slopes of perpendicular lines is −1. Because the slope of y = x is 1 and 1(−1) = −1, the slope of FF′ is −1.

— with endpoints F(−1, 2) and G(1, 2) and its image after a reflection in the Graph FG line y = x. SOLUTION

— and the line y = x. The slope of Graph FG y = x is 1. The segment from F to its image, F′, is perpendicular to the line of reflection y = x, — will be −1. From F, move so the slope of FF′ 1.5 units right and 1.5 units down to y = x. From that point, move 1.5 units right and 1.5 units down to locate F′(2, −1).

y

4

F

y=x G G′

−2

4 x

— will also be −1. From G, move The slope of GG′ 0.5 unit right and 0.5 unit down to y = x. Then move 0.5 unit right and 0.5 unit down to locate G′(2, 1).

F′

−2

You can use coordinate rules to find the images of points reflected in four special lines.

Core Concept Coordinate Rules for Reflections • If (a, b) is reflected in the x-axis, then its image is the point (a, −b). • If (a, b) is reflected in the y-axis, then its image is the point (−a, b). • If (a, b) is reflected in the line y = x, then its image is the point (b, a). • If (a, b) is reflected in the line y = −x, then its image is the point (−b, −a).

Reflecting in the Line y = −x

— from Example 2 and its image after a reflection in the line y = −x. Graph FG SOLUTION

— and the line y = −x. Use the coordinate rule Graph FG for reflecting in the line y = −x to find the coordinates of the endpoints of the image. Then graph the image.

y

F

G

F′

(a, b) → (−b, −a)

2

F(−1, 2) → F′(−2, 1)

G′

G(1, 2) → G′(−2, −1)

Monitoring Progress

−2

x

y = −x

Help in English and Spanish at BigIdeasMath.com

The vertices of △JKL are J(1, 3), K(4, 4), and L(3, 1). 5. Graph △JKL and its image after a reflection in the x-axis. 6. Graph △JKL and its image after a reflection in the y-axis. 7. Graph △JKL and its image after a reflection in the line y = x. 8. Graph △JKL and its image after a reflection in the line y = −x.

— is perpendicular to y = −x. 9. In Example 3, verify that FF′ Section 11.2

Reflections

553

Performing Glide Reflections

Postulate Reflection Postulate A reflection is a rigid motion. m

E D

F

E′

Because a reflection is a rigid motion, and a rigid motion preserves length and angle measure, the following statements are true for the reflection shown.

F′

D′

• DE = D′E′, EF = E′F′, FD = F′D′ • m∠D = m∠D′, m∠E = m∠E′, m∠F = m∠F′ Because a reflection is a rigid motion, the Composition Theorem guarantees that any composition of reflections and translations is a rigid motion.

STUDY TIP The line of reflection must be parallel to the direction of the translation to be a glide reflection.

A glide reflection is a transformation involving a translation followed by a reflection in which every point P is mapped to a point P ″ by the following steps.

Q′

P′

Q″ P″

Step 1 First, a translation maps P to P′. Step 2 Then, a reflection in a line k parallel to the direction of the translation maps P′ to P ″.

Q

P

k

Performing a Glide Reflection Graph △ABC with vertices A(3, 2), B(6, 3), and C(7, 1) and its image after the glide reflection. Translation: (x, y) → (x − 12, y) Reflection: in the x-axis

SOLUTION Begin by graphing △ABC. Then graph △A′B′C′ after a translation 12 units left. Finally, graph △A″B″C″ after a reflection in the x-axis. y

B′(−6, 3) 2

A′(−9, 2)

B(6, 3) A(3, 2)

C′(−5, 1) −12

−10

−8

−6

−4

C(7, 1)

−2

C″(−5, −1) A″(−9, −2)

2

4

6

8

x

−2

B″(−6, −3)

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

10. WHAT IF? In Example 4, △ABC is translated 4 units down and then reflected in

the y-axis. Graph △ABC and its image after the glide reflection.

11. In Example 4, describe a glide reflection from △A″B″C ″ to △ABC.

554

Chapter 11

Transformations

Identifying Lines of Symmetry A figure in the plane has line symmetry when the figure can be mapped onto itself by a reflection in a line. This line of reflection is a line of symmetry, such as line m at the left. A figure can have more than one line of symmetry.

Identifying Lines of Symmetry

m

How many lines of symmetry does each hexagon have? a.

b.

c.

b. 6

c. 1

SOLUTION a. 2

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Determine the number of lines of symmetry for the figure. 12.

13.

14.

15. Draw a hexagon with no lines of symmetry.

Solving Real-Life Problems Finding a Minimum Distance You are going to buy books. Your friend is going to buy CDs. Where should you park to minimize the distance you both will walk? B

A m

SOLUTION Reflect B in line m to obtain B′. Then — draw— AB′ . Label the intersection of AB′ and m as C. Because AB′ is the shortest distance between A and B′ and BC = B′C, park at point C to minimize the combined distance, AC + BC, you both have to walk.

Monitoring Progress

B B′

A C

m

Help in English and Spanish at BigIdeasMath.com

16. Look back at Example 6. Answer the question by using a reflection of point A

instead of point B. Section 11.2

Reflections

555

11.2 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY A glide reflection is a combination of which two transformations? 2. WHICH ONE DOESN’T BELONG? Which transformation does not belong with the other three? Explain

your reasoning. y

y

y

y

6

2

2

2 4 4

2

2

−4

x

2

−4

x

−2

x

−2

−2 −2

−2

x

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, determine whether the coordinate plane shows a reflection in the x-axis, y-axis, or neither. 3.

11. J(2, 4), K(−4, −2), L(−1, 0); y = 1

4. 2 −4

A

y

D

12. J(3, −5), K(4, −1), L(0, −3); y = −3

D

In Exercises 13–16, graph the polygon and its image after a reflection in the given line. (See Examples 2 and 3.)

B

4x

A

−4

E

−2

−4

C

y

4

C

B E

F

−6

−4

5.

4x

13. y = x

F

4

C

−4

−2

E

4

B −2

A D

A 2

4x

4

C F x

−2

F

D

B

C

−4

E

6

D

15. y = −x

8. J(5, 3), K(1, −2), L(−3, 4); y-axis 9. J(2, −1), K(4, −5), L(3, 1); x = −1

Transformations

x

A

16. y = −x 4

A

y

4

2

2 2x

D

4 −2

A

7. J(2, −4), K(3, 7), L(6, −1); x-axis

Chapter 11

C

x −2

B

−4

In Exercises 7–12, graph △JKL and its image after a reflection in the given line. (See Example 1.)

556

y

y

2

B

4

2 4

2

14. y = x y

6. y

10. J(1, −1), K(3, 0), L(0, −4); x = 2

B

C −4

y

A

B

−2

4 −2 −4

C

6x

In Exercises 17–20, graph △RST with vertices R(4, 1), S(7, 3), and T(6, 4) and its image after the glide reflection. (See Example 4.) 17. Translation: (x, y) → (x, y − 1)

Reflection: in the y-axis

27. MODELING WITH MATHEMATICS You park at some

point K on line n. You deliver a pizza to House H, go back to your car, and deliver a pizza to House J. Assuming that you can cut across both lawns, how can you determine the parking location K that minimizes the distance HK + KJ ? (See Example 6.)

18. Translation: (x, y) → (x − 3, y)

Reflection: in the line y = −1 19. Translation: (x, y) → (x, y + 4)

J

H

Reflection: in the line x = 3

n

20. Translation: (x, y) → (x + 2, y + 2)

Reflection: in the line y = x

28. ATTENDING TO PRECISION Use the numbers and

In Exercises 21–24, determine the number of lines of symmetry for the figure. (See Example 5.) 21.

symbols to create the glide reflection resulting in the image shown.

22. C″(−1, 5)

6

y

A″(5, 6)

4

B(−1, 1) −4

23.

B″(4, 2)

2

A(3, 2)

−2

24.

4

2

6

8x

−2 −4

Translation: (x, y) → Reflection: in y = x

C(2, −4)

(

)

,

25. USING STRUCTURE Identify the line symmetry

(if any) of each word. a.

LOOK

b.

MOM

c.

OX

d.

DAD

1

x

26. ERROR ANALYSIS Describe and correct the error in

describing the transformation. A′

2

B″ −8

−6

−4

−2

2 −2

B

y

3

+

−

In Exercises 29–32, find point C on the x-axis so AC + BC is a minimum. 29. A(1, 4), B(6, 1)

y

A″

2

B′

4

30. A(4, −5), B(12, 3) 31. A(−8, 4), B(−1, 3)

6

A

8x

32. A(−1, 7), B(5, −4) 33. MATHEMATICAL CONNECTIONS The line y = 3x + 2



— to A″B″ — is a glide reflection. AB

is reflected in the line y = −1. What is the equation of the image?

Section 11.2

Reflections

557

34. HOW DO YOU SEE IT? Use Figure A.

35. CONSTRUCTION Follow these steps to construct a

reflection of △ABC in line m. Use a compass and straightedge.

y

m

Step 1 Draw △ABC and line m. Step 2 Use one compass setting to find two points that are equidistant from A on line m. Use the same compass setting to find a point on the other side of m that is the same distance from these two points. Label that point as A′.

x

Figure A y

y

A

C B

Step 3 Repeat Step 2 to find points B′ and C′. Draw △A′B′C′. 36. USING TOOLS Use a reflective device to verify your x

x

Figure 1

Figure 2

y

construction in Exercise 35. 37. MATHEMATICAL CONNECTIONS Reflect △MNQ in

the line y = −2x.

y

y = −2x

4

y

M

Q −5 x

x

a. Which figure is a reflection of Figure A in the line x = a? Explain. b. Which figure is a reflection of Figure A in the line y = b? Explain. c. Which figure is a reflection of Figure A in the line y = x? Explain. d. Is there a figure that represents a glide reflection? Explain your reasoning.

Maintaining Mathematical Proficiency Use the diagram to find the angle measure. 40. m∠AOC

44. m∠COD

45. m∠EOD

46. m∠COE

47. m∠AOB

48. m∠COB

49. m∠BOD

Chapter 11

translation and a reflection commutative? (In other words, do you obtain the same image regardless of the order in which you perform the transformations?) Justify your answer. 39. MATHEMATICAL CONNECTIONS Point B′(1, 4) is the

image of B(3, 2) after a reflection in line c. Write an equation for line c. Reviewing what you learned in previous grades and lessons

(Section 8.5)

Transformations

80 90 10 0 70 10 0 90 80 110 1 70 20 60 0 110 60 13 2 50 0 1 50 0 13

A

D

C

E O

170 180 60 0 1 20 10 0 15 0 30 14 0 4

43. m∠AOE

558

38. THOUGHT PROVOKING Is the composition of a

41. m∠AOD

42. m∠BOE

1x

−3

Figure 4

0 10 180 170 1 20 3 60 15 0 4 01 0 40

Figure 3

N

B

11.1–11.2

What Did You Learn?

Core Vocabulary vector, p. 544 initial point, p. 544 terminal point, p. 544 horizontal component, p. 544 vertical component, p. 544 component form, p. 544 transformation, p. 544

image, p. 544 preimage, p. 544 translation, p. 544 rigid motion, p. 546 composition of transformations, p. 546 reflection, p. 552

line of reflection, p. 552 glide reflection, p. 554 line symmetry, p. 555 line of symmetry, p. 555

Core Concepts Section 11.1 Vectors, p. 544 Translations, p. 544

Translation Postulate, p. 546 Composition Theorem, p. 546

Section 11.2 Reflections, p. 552 Coordinate Rules for Reflections, p. 553

Reflection Postulate, p. 554 Line Symmetry, p. 555

Mathematical Practices 1.

How could you determine whether your results make sense in Exercise 26 on page 549?

2.

Look back at the words in Exercise 25 on page 557. Find a word with 6 or more letters that has a horizontal line of symmetry. Find a word with 4 or more letters that has a vertical line of symmetry.

3.

State the meaning of the numbers and symbols you chose in Exercise 28 on page 557.

Keeping a Positive Attitude Ever feel frustrated or overwhelmed by math? You’re not alone. Just take a deep breath and assess the situation. Try to find a productive study environment, review your notes and examples in the textbook, and ask your teacher or peers for help.

559 59 9

11.1–11.2

Quiz

Name the vector and write its component form. (Section 11.1) 1.

2.

E

Y D

Z

Graph quadrilateral ABCD with vertices A(−4, 1), B(−3, 3), C(0, 1), and D(−2, 0) and its image after the translation. (Section 11.1) 3. (x, y) → (x + 4, y − 2)

4. (x, y) → (x − 1, y − 5)

5. (x, y) → (x + 3, y + 6)

— with endpoints D(3, −4) and E(2, 1) and its image after the Graph DE composition. (Section 11.1) 6. Translation: (x, y) → (x − 2, y + 5)

7. Translation: (x, y) → (x + 3, y − 3)

Translation: (x, y) → (x − 4, y − 1)

Translation: (x, y) → (x − 7, y + 6)

Graph the polygon with the given vertices and its image after a reflection in the given line. (Section 11.2) 9. D(−5, −1), E(−2, 1), F(−1, −3); y = x

8. A(−5, 6), B(−7, 8), C(−3, 11); x-axis 10. J(−1, 4), K(2, 5), L(5, 2), M(4, −1); x = 3

11. P(2, −4), Q(6, −1), R(9, −4), S(6, −6); y = −2

12. Translate △ABC 6 units left and then reflect it in the line y = x. Graph the image.

4

Is this an example of a glide reflection? Explain. (Section 11.2)

y

B

2

Graph △ABC with vertices A(2, −1), B(5, 2), and C(8, −2) and its image after the glide reflection. (Section 11.2) 13. Translation: (x, y) → (x, y + 6)

Reflection: in the y-axis

C

A −4

−2

2

4x

−2

14. Translation: (x, y) → (x − 9, y)

Reflection: in the line y = 1

−4

Determine the number of lines of symmetry for the figure. (Section 11.2) 15.

16.

17.

18.

y

19. The figure shows a game in which the object is to create solid rows

using the pieces given. Using only translations and reflections, describe the transformations for each piece at the top that will form two solid rows at the bottom. (Section 11.1 and Section 11.2) x

560

Chapter 11

Transformations

11.3 Rotations Essential Question

How can you rotate a figure in a

coordinate plane?

Rotating a Triangle in a Coordinate Plane Work with a partner. a. Use dynamic geometry software to draw any triangle and label it △ABC. b. Rotate the triangle 90° counterclockwise about the origin to form △A′B′C′. c. What is the relationship between the coordinates of the vertices of △ABC and those of △A′B′C′? d. What do you observe about the side lengths Sample and angle measures of the two triangles? B′

C′

4

A

3

B

2

1

A′

0 −3

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to use previously established results in constructing arguments.

5

y

B

A

−3

−1 −3 −5

−1

0

1

2

3

4

−1

Rotating a Triangle in a Coordinate Plane Work with a partner. a. The point (x, y) is rotated 90° counterclockwise about the origin. Write a rule to determine the coordinates of the image of (x, y). b. Use the rule you wrote in part (a) to rotate △ABC 90° counterclockwise about the origin. What are the coordinates of the vertices of the image, △A′B′C′? c. Draw △A′B′C′. Are its side lengths the same as those of △ABC? Justify your answer.

Rotating a Triangle in a Coordinate Plane

1 −5

−2

C D

Points A(1, 3) B(4, 3) C(4, 1) D(0, 0) Segments AB = 3 BC = 2 AC = 3.61 Angles m∠A = 33.69° m∠B = 90° m∠C = 56.31°

1

5x

C

Work with a partner. a. The point (x, y) is rotated 180° counterclockwise about the origin. Write a rule to determine the coordinates of the image of (x, y). Explain how you found the rule. b. Use the rule you wrote in part (a) to rotate △ABC (from Exploration 2) 180° counterclockwise about the origin. What are the coordinates of the vertices of the image, △A′B′C′?

Communicate Your Answer 4. How can you rotate a figure in a coordinate plane? 5. In Exploration 3, rotate △A′B′C′ 180° counterclockwise about the origin.

What are the coordinates of the vertices of the image, △A″B″C″? How are these coordinates related to the coordinates of the vertices of the original triangle, △ABC? Section 11.3

Rotations

561

11.3 Lesson

What You Will Learn Perform rotations. Perform compositions with rotations.

Core Vocabul Vocabulary larry

Identify rotational symmetry.

rotation, p. 562 center of rotation, p. 562 angle of rotation, p. 562 rotational symmetry, p. 565 center of symmetry, p. 565

Performing Rotations

Core Concept Rotations A rotation is a transformation in which a figure is turned about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and its image form the angle of rotation.

Q

40°

• If Q is not the center of rotation P, then QP = Q′P and m∠QPQ′ = x°, or

Q′

angle of rotation

center of rotation

• If Q is the center of rotation P, then Q = Q′.

Direction of rotation

R

R′

A rotation about a point P through an angle of x° maps every point Q in the plane to a point Q′ so that one of the following properties is true.

P

The figure above shows a 40° counterclockwise rotation. Rotations can be clockwise or counterclockwise. In this chapter, all rotations are counterclockwise unless otherwise noted.

Drawing a Rotation

clockwise

Draw a 120° rotation of △ABC about point P.

A C

counterclockwise

B

P

SOLUTION Step 1 Draw a segment from P to A.

Step 2 Draw a ray to form a 120° angle —. with PA

A

A 50 60 30 40 20 150 140 130 120 70 11 0 0 1 80 10 0 16 00 0 0 17 18

C

B

P

Step 3 Draw A′ so that PA′ = PA.

0 18 0

140 15 120 130 0 110 60 50 40 30 160 0 20 17 10 0 70 10 0 8 90 0 9

C

B

P

Step 4 Repeat Steps 1–3 for each vertex. Draw △A′B′C′. B′

A A′

562

Chapter 11

Transformations

120° P

C

A B

A′

C′

C P

B

USING ROTATIONS You can rotate a figure more than 360°. The effect, however, is the same as rotating the figure by the angle minus 360°.

You can rotate a figure more than 180°. The diagram shows rotations of point A 130°, 220°, and 310° about the origin. Notice that point A and its images all lie on the same circle. A rotation of 360° maps a figure onto itself.

y

A

A′

130°

You can use coordinate rules to find the coordinates of a point after a rotation of 90°, 180°, or 270° about the origin.

x

220°

A‴

310° A″

Core Concept Coordinate Rules for Rotations about the Origin When a point (a, b) is rotated counterclockwise about the origin, the following are true. • For a rotation of 90°, (a, b) → (−b, a).

y

(−b, a)

(a, b) 90°

180°

• For a rotation of 180°, (a, b) → (−a, −b).

x

• For a rotation of 270°, (a, b) → (b, −a).

270°

(−a, −b)

(b, −a)

Rotating a Figure in the Coordinate Plane Graph quadrilateral RSTU with vertices R(3, 1), S(5, 1), T(5, −3), and U(2, −1) and its image after a 270° rotation about the origin.

SOLUTION Use the coordinate rule for a 270° rotation to find the coordinates of the vertices of the image. Then graph quadrilateral RSTU and its image.

2

−4

(a, b) → (b, −a)

y

R

S

−2

6 x

U′

R(3, 1) → R′(1, −3)

U R′ T

S(5, 1) → S′(1, −5) T(5, −3) → T′(−3, −5)

T′

−6

U(2, −1) → U′(−1, −2)

Monitoring Progress

S′

Help in English and Spanish at BigIdeasMath.com

1. Trace △DEF and point P. Then draw a 50° rotation of △DEF about point P. E

D

F

P

2. Graph △JKL with vertices J(3, 0), K(4, 3), and L(6, 0) and its image after

a 90° rotation about the origin. Section 11.3

Rotations

563

Performing Compositions with Rotations

Postulate Rotation Postulate A rotation is a rigid motion.

D E

F′

D′ E′

F

Because a rotation is a rigid motion, and a rigid motion preserves length and angle measure, the following statements are true for the rotation shown. • DE = D′E′, EF = E′F′, FD = F′D′ • m∠D = m∠D′, m∠E = m∠E′, m∠F = m∠F′ Because a rotation is a rigid motion, the Composition Theorem guarantees that compositions of rotations and other rigid motions, such as translations and reflections, are rigid motions.

Performing a Composition

— with endpoints R(1, −3) and S(2, −6) and its image after the composition. Graph RS Reflection: in the y-axis Rotation: 90° about the origin

COMMON ERROR Unless you are told otherwise, perform the transformations in the  order given.

SOLUTION

—. Step 1 Graph RS

— in the y-axis. Step 2 Reflect RS — R′S′ has endpoints R′(−1, −3) and S′(−2, −6). — 90° about the Step 3 Rotate R′S′ — has endpoints origin. R″S″ R″(3, −1) and S″(6, −2).

Monitoring Progress

y −4

R″(3, −1)

−2

R′(−1, −3)

S′(−2, −6)

R(1, −3)

−6

8 x

S″(6, −2)

S(2, −6)

Help in English and Spanish at BigIdeasMath.com

— from Example 3. Perform the rotation first, followed by the reflection. 3. Graph RS Does the order of the transformations matter? Explain.

— — the origin. Graph RS and its image after the composition. — with endpoints A(−4, 4) and B(−1, 7) and its image after Graph AB

4. WHAT IF? In Example 3, RS is reflected in the x-axis and rotated 180° about 5.

the composition. Translation: (x, y) → (x − 2, y − 1) Rotation: 90° about the origin 6. Graph △TUV with vertices T(1, 2), U(3, 5), and V(6, 3) and its image after

the composition. Rotation: 180° about the origin Reflection: in the x-axis 564

Chapter 11

Transformations

Identifying Rotational Symmetry A figure in the plane has rotational symmetry when the figure can be mapped onto itself by a rotation of 180° or less about the center of the figure. This point is the center of symmetry. Note that the rotation can be either clockwise or counterclockwise. For example, the figure below has rotational symmetry, because a rotation of either 90° or 180° maps the figure onto itself (although a rotation of 45° does not). 0°

45°

90°

180°

The figure above also has point symmetry, which is 180° rotational symmetry.

Identifying Rotational Symmetry Does the figure have rotational symmetry? If so, describe any rotations that map the figure onto itself. a. parallelogram

b. regular octagon

c. trapezoid

SOLUTION a. The parallelogram has rotational symmetry. The center is the intersection of the diagonals. A 180° rotation about the center maps the parallelogram onto itself. b. The regular octagon has rotational symmetry. The center is the intersection of the diagonals. Rotations of 45°, 90°, 135°, or 180° about the center all map the octagon onto itself. c. The trapezoid does not have rotational symmetry because no rotation of 180° or less maps the trapezoid onto itself.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. 7. rhombus

8. octagon

9. right triangle

Section 11.3

Rotations

565

11.3 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE When a point (a, b) is rotated counterclockwise about the origin,

(a, b) → (b, −a) is the result of a rotation of ______.

2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

What are the coordinates of the vertices of the image after a 90° counterclockwise rotation about the origin?

y

4 2

What are the coordinates of the vertices of the image after a 270° clockwise rotation about the origin? −4

What are the coordinates of the vertices of the image after turning the figure 90° to the left about the origin?

B

A

C

−2

4 x

2 −2 −4

What are the coordinates of the vertices of the image after a 270° counterclockwise rotation about the origin?

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, trace the polygon and point P. Then draw a rotation of the polygon about point P using the given number of degrees. (See Example 1.) 3. 30°

8. 180°

y

E

4. 80° B

4x

D

P

D

−2

F

E

P

9. 180°

C

A

G

5. 150°

F

y

2

R P

F P

Q

In Exercises 7–10, graph the polygon and its image after a rotation of the given number of degrees about the origin. (See Example 2.) 7. 90°

L

M 4

6

−4

11. Translation: (x, y) → (x, y + 2)

Rotation: 90° about the origin

Reflection: in the y-axis

566

C −2

Chapter 11

2

Q

4x

14. Reflection: in the line y = x

Rotation: 180° about the origin

Transformations

T

— with endpoints X(−3, 1) In Exercises 11–14, graph XY and Y(4, −5) and its image after the composition. (See Example 3.)

13. Rotation: 270° about the origin A

x −2

Translation: (x, y) → (x − 1, y + 1)

B

S

x

12. Rotation: 180° about the origin

y 4

R −6

2

J

y

K

J 4

6. 130° G

10. 270°

In Exercises 15 and 16, graph △LMN with vertices L(1, 6), M(−2, 4), and N(3, 2) and its image after the composition. (See Example 3.)

26.

✗

C (−1, 1) → C ′ (1, −1) D (2, 3) → D ′ (3, 2)

15. Rotation: 90° about the origin

Translation: (x, y) → (x − 3, y + 2)

27. CONSTRUCTION Follow these steps to construct a

rotation of △ABC by angle D around a point O. Use a compass and straightedge.

16. Reflection: in the x-axis

Rotation: 270° about the origin

A′

In Exercises 17–20, determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. (See Example 4.) 17.

18.

A

B C

D

O

Step 1 Draw △ABC, ∠D, and O, the center of rotation.

19.

—. Use the construction for copying Step 2 Draw OA an angle to copy ∠D at O, as shown. Then use distance OA and center O to find A′.

20.

Step 3 Repeat Step 2 to find points B′ and C′. Draw △A′B′C′. 28. REASONING You enter the revolving door at a hotel. REPEATED REASONING In Exercises 21–24, select the angles of rotational symmetry for the regular polygon. Select all that apply.

A 30° ○

B 45° ○

C 60° ○

D 72° ○

E 90° ○

F 120° ○

G 144° ○

H 180° ○

21.

22.

a. You rotate the door 180°. What does this mean in the context of the situation? Explain. b. You rotate the door 360°. What does this mean in the context of the situation? Explain. 29. MATHEMATICAL CONNECTIONS Use the graph of

y = 2x − 3.

23.

24.

ERROR ANALYSIS In Exercises 25 and 26, the endpoints

— are C(−1, 1) and D(2, 3). Describe and correct of CD the error in finding the coordinates of the vertices of the image after a rotation of 270° about the origin. 25.

✗

C (−1, 1) → C ′ (−1, −1) D (2, 3) → D ′ (2, −3)

a. Rotate the line 90°, 180°, 270°, and 360° about the origin. Write the equation of the line for each image. Describe the relationship between the equation of the preimage and the equation of each image.

y −2

2

x

−2

b. Do you think that the relationships you described in part (a) are true for any line that is not vertical or horizontal? Explain your reasoning. 30. MAKING AN ARGUMENT Your friend claims that

rotating a figure by 180° is the same as reflecting a figure in the y-axis and then reflecting it in the x-axis. Is your friend correct? Explain your reasoning.

Section 11.3

Rotations

567

31. DRAWING CONCLUSIONS A figure only has point

38. HOW DO YOU SEE IT? You are finishing the puzzle.

symmetry. How many rotations that map the figure onto itself can be performed before it is back where it started?

The remaining two pieces both have rotational symmetry.

32. ANALYZING RELATIONSHIPS Is it possible for a

figure to have 90° rotational symmetry but not 180° rotational symmetry? Explain your reasoning.

1

2

33. ANALYZING RELATIONSHIPS Is it possible for a

figure to have 180° rotational symmetry but not 90° rotational symmetry? Explain your reasoning. a. Describe the rotational symmetry of Piece 1 and of Piece 2.

34. THOUGHT PROVOKING Can rotations of 90°, 180°,

270°, and 360° be written as the composition of two reflections? Justify your answer.

b. You pick up Piece 1. How many different ways can it fit in the puzzle?

35. USING AN EQUATION Inside a kaleidoscope, two

mirrors are placed next to each other to form a V. The angle between the mirrors determines the number of lines of symmetry in the mirror image. Use the formula 1 n(m∠1) = 180° to find the measure of ∠1, the angle black glass between the mirrors, for the number n of lines of symmetry. a.

c. Before putting Piece 1 into the puzzle, you connect it to Piece 2. Now how many ways can it fit in the puzzle? Explain. 39. USING STRUCTURE A polar coordinate system locates

a point in a plane by its distance from the origin O and by the measure of an angle with its vertex at the origin. For example, the point A(2, 30°) is 2 units from the origin and m∠XOA = 30°. What are the polar coordinates of the image of point A after a 90° rotation? a 180° rotation? a 270° rotation? Explain.

b.

90°

120°

60°

150°

30°

A

36. REASONING Use the coordinate rules for

180°

counterclockwise rotations about the origin to write coordinate rules for clockwise rotations of 90°, 180°, or 270° about the origin.

1

O

210°

Y(3, 1), and Z(0, 2). Rotate △XYZ 90° about the point P(−2, −1).

Maintaining Mathematical Proficiency

300°

270°

Reviewing what you learned in previous grades and lessons

The figures are congruent. Name the corresponding angles and the corresponding sides. (Skills Review Handbook) W Q

41. A

B

V

J D S

568

K

C

X

T

Chapter 11

R

Z

Y

Transformations

X 0°

3

330° 240°

37. USING STRUCTURE △XYZ has vertices X(2, 5),

40. P

2

M

L

11.4 Congruence and Transformations Essential Question

What conjectures can you make about a figure

reflected in two lines?

Reflections in Parallel Lines Work with a partner. Use dynamic geometry software to draw any scalene triangle and label it △ABC. a. Draw any line ⃖##⃗ DE. Reflect △ABC in ⃖##⃗ DE to form △A′B′C′.

Sample D

b. Draw a line parallel to ⃖##⃗ DE. Reflect △A′B′C′ in the new line to form △A″B″C″. c. Draw the line through point A that is perpendicular to ⃖##⃗ DE. What do you notice?

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to make conjectures and justify your conclusions.

A A′ A″

B

C

B′ C′

d. Find the distance between points A and A″. Find the distance between the two parallel lines. What do you notice?

C″

B″

E F

e. Hide △A′B′C′. Is there a single transformation that maps △ABC to △A″B″C″? Explain. f. Make conjectures based on your answers in parts (c)–(e). Test your conjectures by changing △ABC and the parallel lines.

Reflections in Intersecting Lines Work with a partner. Use dynamic geometry software to draw any scalene triangle and label it △ABC. a. Draw any line ⃖##⃗ DE. Reflect △ABC in ⃖##⃗ DE to form △A′B′C′. b. Draw any line ⃖##⃗ DF so that angle EDF is less than or equal to 90°. Reflect △A′B′C′ in ⃖##⃗ DF to form △A″B″C″. c. Find the measure of ∠ EDF. Rotate △ABC counterclockwise about point D using an angle twice the measure of ∠ EDF.

Sample D B A C

B″

B′

C″

C′ E

d. Make a conjecture about a figure reflected in two intersecting lines. Test your conjecture by changing △ABC and the lines.

A′

F

A″

Communicate Your Answer 3. What conjectures can you make about a figure reflected in two lines? 4. Point Q is reflected in two parallel lines, ⃖##⃗ GH and ⃖##⃗ JK , to form Q′ and Q″.

The distance from ⃖##⃗ GH to ⃖##⃗ JK is 3.2 inches. What is the distance QQ″? Section 11.4

Congruence and Transformations

569

11.4 Lesson

What You Will Learn Identify congruent figures. Describe congruence transformations.

Core Vocabul Vocabulary larry

Use theorems about congruence transformations.

congruent figures, p. 570 congruence transformation, p. 571

Identifying Congruent Figures Two geometric figures are congruent figures if and only if there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other. Congruent figures have the same size and shape. Congruent

Not congruent

same size and shape

different sizes or shapes

You can identify congruent figures in the coordinate plane by identifying the rigid motion or composition of rigid motions that maps one of the figures onto the other. Recall from the Translation Postulate, the Reflection Postulate, the Rotation Postulate, and the Composition Theorem that translations, reflections, rotations, and compositions of these transformations are rigid motions.

Identifying Congruent Figures Identify any congruent figures in the coordinate plane. Explain.

I

SOLUTION

H

5

F

J

Square NPQR is a translation of square ABCD 2 units left and 6 units down. So, square ABCD and square NPQR are congruent.

G M

A

R

N

−5

5 x

S

T

Help in English and Spanish at BigIdeasMath.com D y

coordinate plane. Explain.

E

4

I

H

J

G

B

F C

−2

L

T

Q

P

K

S

R

N

−4

M

Transformations

D

U

1. Identify any congruent figures in the

Chapter 11

B

P

L

△STU is a 180° rotation of △HIJ. So, △HIJ and △STU are congruent.

570

C

E K Q

△KLM is a reflection of △EFG in the x-axis. So, △EFG and △KLM are congruent.

Monitoring Progress

y

A 2

4

U

x

Congruence Transformations Another name for a rigid motion or a combination of rigid motions is a congruence transformation because the preimage and image are congruent. The terms “rigid motion” and “congruence transformation” are interchangeable.

READING You can read the notation ▱ABCD as “parallelogram A, B, C, D.”

Describing a Congruence Transformation Describe a congruence transformation that maps ▱ABCD to ▱EFGH.

4

y

D

C

2

A G

H

B 2

4 x

−2

F

E

SOLUTION The two vertical sides of ▱ABCD rise from left to right, and the two vertical sides of ▱EFGH fall from left to right. If you reflect ▱ABCD in the y-axis, as shown, then the image, ▱A′B′C′D′, will have the same orientation as ▱EFGH.

C′

D′

B′ G

Then you can map ▱A′B′C′D′ to ▱EFGH using a translation of 4 units down.

4

y

D

A′ A H

C

B 2

4 x

−2

F

E

So, a congruence transformation that maps ▱ABCD to ▱EFGH is a reflection in the y-axis followed by a translation of 4 units down.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

2. In Example 2, describe another congruence transformation that

maps ▱ABCD to ▱EFGH.

3. Describe a congruence transformation that maps △JKL to △MNP. K

4

L −4

J −2

2 −2 −4

Section 11.4

y

P

4 x

M

N

Congruence and Transformations

571

Using Theorems about Congruence Transformations Compositions of two reflections result in either a translation or a rotation. A composition of two reflections in parallel lines results in a translation, as described in the following theorem.

Theorem Reflections in Parallel Lines Theorem If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation.

k

B

B′

m

B″

If A″ is the image of A, then 1. 2.

— is perpendicular to k and m, and AA″

A

A′

AA″ = 2d, where d is the distance between k and m.

Proof

A″

d

Ex. 31, p. 576

Using the Reflections in Parallel Lines Theorem In the diagram, a reflection in line k — to G′H′ —. A reflection in line m maps GH — to G″H″ —. Also, HB = 9 maps G′H′ and DH″ = 4. a. Name any segments congruent to —, HB —, and GA —. each segment: GH

H

H′

B A

G

H″ D

G′

k

C G″

m

b. Does AC = BD? Explain.

—? c. What is the length of GG″ SOLUTION

— ≅ G′H′ —, and GH — ≅ G″H″ —. HB — ≅ H′B —. GA — ≅ G′A —. a. GH

— and HH″ — are perpendicular to both k and m. So, BD — b. Yes, AC = BD because GG″ — and AC are opposite sides of a rectangle. c. By the properties of reflections, H′B = 9 and H′D = 4. The Reflections in Parallel — is Lines Theorem implies that GG″ = HH″ = 2 BD, so the length of GG″ 2(9 + 4) = 26 units.

⋅

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Use the figure. The distance between line k and line m is 1.6 centimeters. k

4. The preimage is reflected in line k, then in

m

line m. Describe a single transformation that maps the blue figure to the green figure.

— and 5. What is the relationship between PP′ line k? Explain. 6. What is the distance between P and P ″?

572

Chapter 11

Transformations

P

P′

P″

A composition of two reflections in intersecting lines results in a rotation, as described in the following theorem.

Theorem Reflections in Intersecting Lines Theorem If lines k and m intersect at point P, then a reflection in line k followed by a reflection in line m is the same as a rotation about point P.

m

B″

B′ k

A′

A″

The angle of rotation is 2x°, where x° is the measure of the acute or right angle formed by lines k and m.

2x° x° A

P

B

m∠ BPB″ = 2x° Proof

Ex. 31, p. 606

Using the Reflections in Intersecting Lines Theorem In the diagram, the figure is reflected in line k. The image is then reflected in line m. Describe a single transformation that maps F to F ″. m

F′

F″

70°

k

F

P

SOLUTION By the Reflections in Intersecting Lines Theorem, a reflection in line k followed by a reflection in line m is the same as a rotation about point P. The measure of the acute angle formed between lines k and m is 70°. So, by the Reflections in Intersecting Lines Theorem, the angle of rotation is 2(70°) = 140°. A single transformation that maps F to F ″ is a 140° rotation about point P. You can check that this is correct by tracing lines k and m and point F, then rotating the point 140°.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com m

7. In the diagram, the preimage is reflected in

line k, then in line m. Describe a single transformation that maps the blue figure onto the green figure. 8. A rotation of 76° maps C to C′. To map C

to C′ using two reflections, what is the measure of the angle formed by the intersecting lines of reflection?

Section 11.4

80° P

Congruence and Transformations

k

573

11.4 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE Two geometric figures are _________ if and only if there is a rigid motion

or a composition of rigid motions that moves one of the figures onto the other. 2. VOCABULARY Why is the term congruence transformation used to refer to a rigid motion?

Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, identify any congruent figures in the coordinate plane. Explain. (See Example 1.) 3.

y

J

Q

6.

y

W

R

Z

4

H M

K N 2

L

P −4

E

−2 −2

B

−6

S 4

D

X

4

Y −4

−2

Fx

R

−4

G

U

4

2 −2

6x

S Q

P

−4

A

C T

4.

y

A F

C −3

Q

D

T J

−4

R V

H

G

x

K L

In Exercises 5 and 6, describe a congruence transformation that maps the blue preimage to the green image. (See Example 2.) 5.

4

−6

A

2

B

C

−4

G

−2

In Exercises 7–10, determine whether the polygons with the given vertices are congruent. Use transformations to explain your reasoning. 7. Q(2, 4), R(5, 4), S(4, 1) and T(6, 4), U(9, 4), V(8, 1)

U

M

−2

E

S

P

N

B

V

8. W(−3, 1), X(2, 1), Y(4, −4), Z(−5, −4) and

C(−1, −3), D(−1, 2), E(4, 4), F(4, −5)

9. J(1, 1), K(3, 2), L(4, 1) and M(6, 1), N(5, 2), P(2, 1) 10. A(0, 0), B(1, 2), C(4, 2), D(3, 0) and

E(0, −5), F(−1, −3), G(−4, −3), H(−3, −5)

In Exercises 11–14, k " m, △ABC is reflected in line k, and △A′B′C′ is reflected in line m. (See Example 3.) 11. A translation maps

k

△ABC onto which triangle?

y

B

m B′

B″

C C′

C″

12. Which lines are

—? perpendicular to AA″

2

4x

F

13. If the distance between

A

A′

A″

k and m is 2.6 inches, —″ ? what is the length of CC

E

14. Is the distance from B′ to m the same as the distance

from B ″ to m? Explain.

574

Chapter 11

Transformations

In Exercises 15 and 16, find the angle of rotation that maps A onto A″. (See Example 4.) 15.

m

In Exercises 19–22, find the measure of the acute or right angle formed by intersecting lines so that C can be mapped to C′ using two reflections. 19. A rotation of 84° maps C to C′.

A″

A′

20. A rotation of 24° maps C to C′. 21. The rotation (x, y) → (−x, −y) maps C to C′.

k

55°

22. The rotation (x, y) → (y, −x) maps C to C′. A

23. REASONING Use the Reflections in Parallel Lines

Theorem to explain how you can make a glide reflection using three reflections. How are the lines of reflection related?

16. A″ A

m

24. DRAWING CONCLUSIONS The pattern shown is

15°

called a tessellation.

k A′

17. ERROR ANALYSIS Describe and correct the error in

describing the congruence transformation.



y

B

2

A −4

C −2

A″ −2

2

4 x

C″

B″

△ABC is mapped to △A″B ″C ″ by a translation 3 units down and a reflection in the y-axis. 18. ERROR ANALYSIS Describe and correct the error in

using the Reflections in Intersecting Lines Theorem.



a. What transformations did the artist use when creating this tessellation? b. Are the individual figures in the tessellation congruent? Explain your reasoning. CRITICAL THINKING In Exercises 25–28, tell whether the statement is always, sometimes, or never true. Explain your reasoning. 25. A congruence transformation changes the size of

a figure. 26. If two figures are congruent, then there is a rigid

P 72°

motion or a composition of rigid motions that maps one figure onto the other. 27. The composition of two reflections results in the

same image as a rotation. 28. A translation results in the same image as the

composition of two reflections. A 72° rotation about point P maps the blue image to the green image.

29. REASONING During a presentation, a marketing

representative uses a projector so everyone in the auditorium can view the advertisement. Is this projection a congruence transformation? Explain your reasoning. Section 11.4

Congruence and Transformations

575

30. HOW DO YOU SEE IT? What type of congruence

transformation can be used to verify each statement about the stained glass window? 2 3

7 6

5

P(1, 3) and Q(3, 2), is reflected in the y-axis. — is then reflected in the x-axis to The image P′Q′ — produce the image P ″Q ″ . One classmate says that — — PQ is mapped to P ″Q ″ by the translation (x, y) → (x − 4, y − 5). Another classmate says that — is mapped to P— PQ ″Q ″ by a (2 90)°, or 180°, rotation about the origin. Which classmate is correct? Explain your reasoning.

⋅

8

1

—, with endpoints 33. MAKING AN ARGUMENT PQ

34. CRITICAL THINKING Does the order of reflections

4

for a composition of two reflections in parallel lines matter? For example, is reflecting △XYZ in lineℓand then its image in line m the same as reflecting △XYZ in line m and then its image in lineℓ?

a. Triangle 5 is congruent to Triangle 8. b. Triangle 1 is congruent to Triangle 4. c. Triangle 2 is congruent to Triangle 7.

m

d. Pentagon 3 is congruent to Pentagon 6. Y

31. PROVING A THEOREM Prove the Reflections in

Parallel Lines Theorem.

Z

X

K

K′

m

K″

CONSTRUCTION In Exercises 35 and 36, copy the figure. J

J′

J″

d

Given

Prove

— to J′K′ —, A reflection in lineℓmaps JK — to J— a reflection in line m maps J′K′ ″K″ , andℓ # m. —″ is perpendicular toℓand m. a. KK

b. KK ″ = 2d, where d is the distance betweenℓand m.

Then use a compass and straightedge to construct two lines of reflection that produce a composition of reflections resulting in the same image as the given transformation. 35. Translation: △ABC → △A″B ″C ″ B″

B

A″

C″

A

36. Rotation about P: △XYZ → △X ″Y ″Z ″ Z″

32. THOUGHT PROVOKING A tessellation is the covering

of a plane with congruent figures so that there are no gaps or overlaps (see Exercise 24). Draw a tessellation that involves two or more types of transformations. Describe the transformations that are used to create the tessellation.

Maintaining Mathematical Proficiency Solve the equation. Check your solution.

X

Z

Y″

Y

X″ P

Reviewing what you learned in previous grades and lessons

(Section 1.3)

37. 5x + 16 = −3x

38. 12 + 6m = 2m

39. 4b + 8 = 6b − 4

40. 7w − 9 = 13 − 4w

41. 7(2n + 11) = 4n

42. −2(8 − y) = −6y

43. Last year, the track team’s yard sale earned $500. This year, the yard sale earned $625. What is the

percent of increase? 576

Chapter 11

(Skills Review Handbook)

Transformations

C

11.3–11.4

What Did You Learn?

Core Vocabulary rotation, p. 562 center of rotation, p. 562 angle of rotation, p. 562 rotational symmetry, p. 565

center of symmetry, p. 565 congruent figures, p. 570 congruence transformation, p. 571

Core Concepts Section 11.3 Rotations, p. 562 Coordinate Rules for Rotations about the Origin, p. 563

Rotation Postulate, p. 564 Rotational Symmetry, p. 565

Section 11.4 Identifying Congruent Figures, p. 570 Describing a Congruence Transformation, p. 571 Reflections in Parallel Lines Theorem, p. 572 Reflections in Intersecting Lines Theorem, p. 573

Mathematical Practices 1.

Describe the steps you would take to arrive at the answer to Exercise 29 part (a) on page 567.

2.

Describe a real-life situation that can be modeled by Exercise 15 on page 575.

3.

Revisit Exercise 31 on page 576. Try to recall the process you used to reach the solution. Did you have to change course at all? If so, how did you approach the situation?

Performance Task:

Revolving Doors Can the rotational symmetry of a revolving door help with balancing the air pressure between the interior and exterior of a building? To explore the answer to this question and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.

577

11

Chapter Review 11.1

Dynamic Solutions available at BigIdeasMath.com

Translations (pp. 543–550)

— are Y(5, 3) and Z(1, −1). Translate a. The endpoints of YZ — YZ using the vector 〈−3, 2〉.

y

6

—. Use 〈−3, 2〉 to move each endpoint Graph YZ 3 units left and 2 units up. Label the image —. endpoints. Then draw Y′Z′

Y′(2, 5)

4

Z′(−2, 1) −4

Y(5, 3)

−2

4

2

Z(1, −1)

−2

6x

b. Graph quadrilateral ABCD with vertices A(1, −2), B(3, −1), C(0, 3), and D(−4, 1) and its image after the translation (x, y) → (x + 2, y − 2). Graph quadrilateral ABCD. To find the coordinates of the vertices of the image, add 2 to the x-coordinates and subtract 2 from the y-coordinates of the vertices of the preimage. Then graph the image.

4

y

C C′

D

(x, y) → (x + 2, y − 2) A(1, −2) → A′(3, −4) B(3, −1) → B′(5, −3) C(0, 3) → C′(2, 1) D(−4, 1) → D′(−2, −1)

−4

4

D′

A

x

B B′

−4

A′

— are E(−2, −3) and F(1, 4). Translate EF — using the given vector. Graph EF — The endpoints of EF and its image. 1. 〈4, 1〉

2. 〈−2, −2〉

3. 〈5, −3〉

Graph △XYZ with vertices X(2, 3), Y(−3, 2), and Z(−4, −3) and its image after the translation. 4. (x, y) → (x, y + 2)

5. (x, y) → (x − 3, y)

6. (x, y) → (x + 3, y − 1)

7. (x, y) → (x + 4, y + 1)

Graph △PQR with vertices P(0, −4), Q(1, 3), and R(2, −5) and its image after the composition. 8. Translation: (x, y) → (x + 1, y + 2)

Translation: (x, y) → (x − 4, y + 1)

11.2

Reflections

9. Translation: (x, y) → (x, y + 3)

Translation: (x, y) → (x − 1, y + 1)

(pp. 551–558)

a. Graph △ABC with vertices A(1, −1), B(3, 2), and C(4, −4) and its image after a reflection in the line y = x. Graph △ABC and the line y = x. Then use the coordinate rule for reflecting in the line y = x to find the coordinates of the vertices of the image. (a, b) → (b, a) A(1, −1) → A′(−1, 1) B(3, 2) → B′(2, 3) C(4, −4) → C′(−4, 4)

578

Chapter 11

Transformations

C′

4

y

B′

B

A′ −4

y=x

−2 −2 −4

A

4x

C

b. Graph △JKL with vertices J(2, 2), K(3, 4), and L(5, 0) and its image after the glide reflection. Translation: (x, y) → (x, y − 4) Reflection: in the y-axis

4

Begin by graphing △JKL. Then graph △J′K′L′ after a translation 4 units down. Finally, graph △J″K″L″ after a reflection in the y-axis.

K(3, 4)

y

J(2, 2)

2

K″(−3, 0) −8

−6

K′(3, 0)

−4

J″(−2, −2)

L″(−5, −4)

L(5, 0)

2

−4

6

J′(2, −2)

8x

L′(5, −4)

Graph the polygon and its image after a reflection in the given line. 10. x = 4

11. y = 3

y

B

4 2

4

y

E

F

2

A

H

C 2

4

4

G 6x

6x

Graph △TUV with vertices T(−3, 2), U(−1, 0), and V(−4, −2) and its image after the glide reflection. 12. Translation: (x, y) → (x + 5, y)

13. Translation: (x, y) → (x, y + 3)

Reflection: in the x-axis

Reflection: in the y-axis

14. How many lines of symmetry does the figure have?

15. Draw a triangle that has one line of symmetry.

11.3

Rotations (pp. 561–568)

Graph △LMN with vertices L(1, −1), M(2, 3), and N(4, 0) and its image after a 270° rotation about the origin. Use the coordinate rule for a 270° rotation to find the coordinates of the vertices of the image. Then graph △LMN and its image. (a, b) → (b, −a) L(1, −1) → L′(−1, −1) M(2, 3) → M′(3, −2) N(4, 0) → N′(0, −4)

4

y

M

2

N −4

−2 L′

L

4x

M′

N′

Chapter 11

Chapter Review

579

Graph the polygon with the given vertices and its image after a rotation of the given number of degrees about the origin. 16. A(−3, −1), B(2, 2), C(3, −3); 90° 17. W(−2, −1), X(−1, 3), Y(3, 3), Z(3, −3); 180°

—

18. Graph XY with endpoints X(5, −2) and Y(3, −3) and its image after a reflection in

the x-axis and then a rotation of 270° about the origin. Determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. 19.

11.4

20.

Congruence and Transformations (pp. 569–576)

Describe a congruence transformation that maps quadrilateral ABCD to quadrilateral WXYZ, as shown at the right.

y

4

A

D

2

B

Y X C

2

4x

Z W

— falls from left to right, and WX — rises from left to right. If you reflect AB quadrilateral ABCD in the x-axis as shown at the bottom right, then the image, quadrilateral A′B′C′D′, will have the same orientation as quadrilateral WXYZ. Then you can map quadrilateral A′B′C′D′ to quadrilateral WXYZ using a translation of 5 units left. So, a congruence transformation that maps quadrilateral ABCD to quadrilateral WXYZ is a reflection in the x-axis followed by a translation of 5 units left.

y

4

D

2

Y

A B

C X C′

Z

x

B′

D′ A′

W

Describe a congruence transformation that maps △DEF to △JKL. 21. D(2, −1), E(4, 1), F(1, 2) and J(−2, −4), K(−4, −2), L(−1, −1)

m

22. D(−3, −4), E(−5, −1), F(−1, 1) and J(1, 4), K(−1, 1), L(3, −1) 23. Which transformation is the same as reflecting an object in two

parallel lines? in two intersecting lines? 24. What is the angle of rotation that maps H onto H″?

H′

H″ 67°

H

580

Chapter 11

Transformations

k

11

Chapter Test

Graph △RST with vertices R(−4, 1), S(−2, 2), and T(3, −2) and its image after the translation. 1. (x, y) → (x − 4, y + 1)

2. (x, y) → (x + 2, y − 2)

Graph the polygon with the given vertices and its image after a rotation of the given number of degrees about the origin. 3. D(−1, −1), E(−3, 2), F(1, 4); 270°

4. J(−1, 1), K(3, 3), L(4, −3), M(0, −2); 90°

— with endpoints A(2, −3) and B(5, 1) and its image after the composition. Graph AB 5. Translation: (x, y) → (x − 3, y + 4)

6. Translation: (x, y) → (x − 2, y + 2)

Translation: (x, y) → (x + 1, y − 1)

Reflection: in the y-axis

7. Reflection: in the x-axis

8. Rotation: 180° about the origin

Translation: (x, y) → (x + 5, y)

Reflection: in the line y = −x

Determine whether the polygons with the given vertices are congruent. Use transformations to explain your reasoning. 9. Q(2, 4), R(5, 4), S(6, 2), T(1, 2) and

W(6, −12), X(15, −12), Y(18, −6), Z(3, −6)

10. A(−6, 6), B(−6, 2), C(−2, −4) and

D(9, 7), E(5, 7), F(−1, 3)

Determine whether the object has line symmetry and whether it has rotational symmetry. Identify all lines of symmetry and angles of rotation that map the figure onto itself. 11.

12.

13.

14. Sketch a pentagon that has (a) one line of symmetry, (b) five lines of symmetry, and

(c) no lines of symmetry. Describe the characteristics of each pentagon. 15. Draw a diagram using a coordinate plane, two parallel lines, and a parallelogram that

demonstrates the Reflections in Parallel Lines Theorem. 16. A rectangle with vertices W(−2, 4), X(2, 4), Y(2, 2), and Z(−2, 2) is reflected in the

y-axis. Your friend says that the image, rectangle W′X′Y′Z′, is exactly the same as the preimage. Is your friend correct? Explain your reasoning. 17. Write a composition of transformations that maps △ABC onto △CDB in the tesselation

shown. Is the composition a congruence transformation? Explain your reasoning.

y 4

0

B

A

2

C 0

2

D 4

6

8 x

18. In the diagram, the preimage is reflected in line q

p, then in line q. Describe a single transformation that maps the blue car onto the green car.

20° p

Chapter 11

Chapter Test

581

11

Cumulative Assessment

1. Which composition of transformations maps △ABC to △DEF?

A Rotation: 90° counterclockwise about the origin ○

4

Translation: (x, y) → (x + 4, y − 3)

B Translation: (x, y) → (x − 4, y − 3) ○

y

B

C

A

Rotation: 90° counterclockwise about the origin

C Translation: (x, y) → (x + 4, y − 3) ○

−4

Rotation: 90° counterclockwise about the origin

−2

E

2

4 x

F

D Rotation: 90° counterclockwise about the origin ○ Translation: (x, y) → (x − 4, y − 3)

−4

D

2. Use the diagrams to describe the steps you would take to construct a line perpendicular

to line m through point P, which is not on line m. Step 1

Step 2

P

P A

Step 3

B

m

A

P B

m

A

B

m

Q

3. Find all numbers between 0 and 100 that are in the range of the function defined below.

f (1) = 1, f (2) = 1, f (n) = f (n − 1) + f (n − 2) 4. Your friend claims that she can find the perimeter of the school crossing

4

sign without using the Distance Formula. Do you support your friend’s claim? Explain your reasoning. −2

4 x −2

5. What are the coordinates of the vertices of the image of △QRS after the

composition of transformations shown? Translation: (x, y) → (x + 2, y + 3) Rotation: 180° about the origin

A Q′(1, 2), R′(5, 4), S′(4, −1) ○ B Q′(−1, −2), R′(−5, −4), S′(−4, 1) ○

2

−2

y

R 4 x

2

Q

C Q′(3, −2), R′(−1, −4), S′(0, 1) ○ D Q′(−2, 1), R′(−4, 5), S′(1, 4) ○

582

Chapter 11

Transformations

−4

S

y

— 6. The midpoint of ST is M(1, −2). One endpoint is T(4.5, −7). Find the coordinates

of endpoint S. 7. The graph shows quadrilateral WXYZ and quadrilateral ABCD.

a. Write a composition of transformations that maps quadrilateral WXYZ to quadrilateral ABCD.

W

b. Are the quadrilaterals congruent? Explain your reasoning.

Z D

y

X A

Y C

−4

2

4 x

−2

B

−4

8. Which equation represents the line passing through the point (−6, 3) that is parallel to

the line y = − —13 x − 5?

A y = 3x + 21 ○

1 B y = −—3 x − 5 ○

C y = 3x − 15 ○

1 D y = −—3 x + 1 ○

9. You conduct a survey that asks students in your school about whether they have visited

a foreign country. Some of the results are shown in the two-way table. What percent of the students surveyed are sophomores and have visited a foreign country? Round your answer to the nearest percent. Visited a Foreign Country Yes Class

Freshman

No

Total

29

Sophomore

45

Total

96

173

10. List one possible set of coordinates of the vertices of quadrilateral ABCD for

each description. a. A reflection in the y-axis maps quadrilateral ABCD onto itself. b. A reflection in the x-axis maps quadrilateral ABCD onto itself. c. A rotation of 90° about the origin maps quadrilateral ABCD onto itself. d. A rotation of 180° about the origin maps quadrilateral ABCD onto itself.

Chapter 11

Cumulative Assessment

583

12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Congruent Triangles Angles of Triangles Congruent Polygons Proving Triangle Congruence by SAS Equilateral and Isosceles Triangles Proving Triangle Congruence by SSS Proving Triangle Congruence by ASA and AAS Using Congruent Triangles Coordinate Proofs SEE the Big Idea

Hang Glider Hang G lid li der (p. (p. 634) 634)

Lifeguard Lif Li ifeguard d Tower Tower (p. (p. 611) 611))

Barn (p. B ( 604) 604))

Home Decor Decor (p. (p. 597) 597) Home

Painting Paiinti ting (p. (p 591) 591)

Maintaining Mathematical Proficiency Using the Midpoint and Distance Formulas — are A(−2, 3) and B(4, 7). Find the coordinates of the midpoint M. Example 1 The endpoints of AB Use the Midpoint Formula.

(

) ( )

−2 + 4 3 + 7 2 10 M —, — = M —, — 2 2 2 2 = M(1, 5)

The coordinates of the midpoint M are (1, 5). Example 2

Find the distance between C (0, −5) and D(3, 2). ——

CD = √ (x2 − x1)2 + (y2 − y1)2 ——

= √ (3 − 0)2 + [2 − (−5)]2 =

—

√32

+

72

Distance Formula Substitute. Subtract.

—

= √ 9 + 49

Evaluate powers.

—

= √ 58

Add.

≈ 7.6

Use a calculator.

The distance between C(0, −5) and D(3, 2) is about 7.6.

Find the coordinates of the midpoint M of the segment with the given endpoints. Then find the distance between the two points. 2. G(3, 6) and H(9, −2)

1. P(−4, 1) and Q(0, 7)

3. U(−1, −2) and V(8, 0)

Solving Equations with Variables on Both Sides Example 3

Solve 2 − 5x = −3x. 2 − 5x = −3x +5x

Write the equation.

+5x

Add 5x to each side.

2 = 2x 2 2

Simplify.

2x 2

—=—

Divide each side by 2.

1=x

Simplify.

The solution is x = 1.

Solve the equation. 4. 7x + 12 = 3x

5. 14 − 6t = t

6. 5p + 10 = 8p + 1

7. w + 13 = 11w − 7

8. 4x + 1 = 3 − 2x

9. z − 2 = 4 + 9z

10. ABSTRACT REASONING Is it possible to find the length of a segment in a coordinate plane

without using the Distance Formula? Explain your reasoning. Dynamic Solutions available at BigIdeasMath.com

585

Mathematical Practices

Mathematically proficient students understand and use given definitions.

Definitions, Postulates, and Theorems

Core Concept Definitions and Biconditional Statements A definition is always an “if and only if” statement. Here is an example. Definition: Two geometric figures are congruent figures if and only if there is a rigid motion

or a composition of rigid motions that maps one of the figures onto the other. Because this is a definition, it is a biconditional statement. It implies the following two conditional statements. 1.

If two geometric figures are congruent figures, then there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other.

2.

If there is a rigid motion or a composition of rigid motions that maps one geometric figure onto another, then the two geometric figures are congruent figures.

Definitions, postulates, and theorems are the building blocks of geometry. In two-column proofs, the statements in the reason column are almost always definitions, postulates, or theorems.

Identifying Definitions, Postulates, and Theorems Classify each statement as a definition, a postulate, or a theorem. a.

If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

b.

If two coplanar lines have no point of intersection, then the lines are parallel.

c.

If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.

SOLUTION a.

This is a theorem. It is the Alternate Interior Angles Converse Theorem studied in Section 10.3.

b.

This is the definition of parallel lines.

c.

This is a postulate. It is the Parallel Postulate studied in Section 10.1. In Euclidean geometry, it is assumed, not proved, to be true.

Monitoring Progress Classify each statement as a definition, a postulate, or a theorem. Explain your reasoning. 1. In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their

slopes is −1.

2. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. 3. If two lines intersect to form a right angle, then the lines are perpendicular. 4. Through any two points, there exists exactly one line.

586

Chapter 12

Congruent Triangles

12.1 Angles of Triangles Essential Question

How are the angle measures of a

triangle related?

Writing a Conjecture

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to reason inductively about data and write conjectures.

Work with a partner. a. Use dynamic geometry software to draw any triangle and label it △ABC. b. Find the measures of the interior angles of the triangle. c. Find the sum of the interior angle measures. d. Repeat parts (a)–(c) with several other triangles. Then write a conjecture about the sum of the measures of the interior angles of a triangle.

Sample A C

Angles m∠A = 43.67° m∠B = 81.87° m∠C = 54.46°

B

Writing a Conjecture Work with a partner. a. Use dynamic geometry software to draw any triangle and label it △ABC. A b. Draw an exterior angle at any vertex and find its measure. c. Find the measures of the two nonadjacent interior angles of the triangle. d. Find the sum of the measures of the two nonadjacent interior angles. B Compare this sum to the measure of the exterior angle. e. Repeat parts (a)–(d) with several other triangles. Then write a conjecture that compares the measure of an exterior angle with the sum of the measures of the two nonadjacent interior angles.

D

C

Sample Angles m∠A = 43.67° m∠B = 81.87° m∠ACD = 125.54°

Communicate Your Answer 3. How are the angle measures of a triangle related? 4. An exterior angle of a triangle measures 32°. What do you know about the

measures of the interior angles? Explain your reasoning. Section 12.1

Angles of Triangles

587

12.1 Lesson

What You Will Learn Classify triangles by sides and angles. Find interior and exterior angle measures of triangles.

Core Vocabul Vocabulary larry interior angles, p. 589 exterior angles, p. 589 corollary to a theorem, p. 591 Previous triangle

Classifying Triangles by Sides and by Angles Recall that a triangle is a polygon with three sides. You can classify triangles by sides and by angles, as shown below.

Core Concept Classifying Triangles by Sides Scalene Triangle

Isosceles Triangle

Equilateral Triangle

no congruent sides

at least 2 congruent sides

3 congruent sides

READING Notice that an equilateral triangle is also isosceles. An equiangular triangle is also acute.

Classifying Triangles by Angles Acute Triangle

Right Triangle

Obtuse Triangle

Equiangular Triangle

3 acute angles

1 right angle

1 obtuse angle

3 congruent angles

Classifying Triangles by Sides and by Angles Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles.

SOLUTION The triangle has a pair of congruent sides, so it is isosceles. By measuring, the angles are 55°, 55°, and 70°. So, it is an acute isosceles triangle.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1. Draw an obtuse isosceles triangle and an acute scalene triangle.

588

Chapter 12

Congruent Triangles

Classifying a Triangle in the Coordinate Plane Classify △OPQ by its sides. Then determine whether it is a right triangle.

4

y

Q(6, 3)

P(−1, 2)

O(0, 0)

−2

4

6

8 x

SOLUTION Step 1 Use the Distance Formula to find the side lengths. ——

——

——

——

——

——

—

OP = √(x2 − x1)2 + (y2 − y1)2 = √ (−1 − 0)2 + (2 − 0)2 = √5 ≈ 2.2 —

OQ = √ (x2 − x1)2 + (y2 − y1)2 = √(6 − 0)2 + (3 − 0)2 = √ 45 ≈ 6.7 —

PQ = √(x2 − x1)2 + (y2 − y1)2 = √ [6 − (−1)]2 + (3 − 2)2 = √50 ≈ 7.1 Because no sides are congruent, △OPQ is a scalene triangle. 2−0 — is — — = −2. The slope of OQ Step 2 Check for right angles. The slope of OP −1 − 0 3−0 1 1 — ⊥ OQ — and is — = —. The product of the slopes is −2 — = −1. So, OP 6−0 2 2 ∠POQ is a right angle.

()

So, △OPQ is a right scalene triangle.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

2. △ABC has vertices A(0, 0), B(3, 3), and C(−3, 3). Classify the triangle by its

sides. Then determine whether it is a right triangle.

Finding Angle Measures of Triangles When the sides of a polygon are extended, other angles are formed. The original angles are the interior angles. The angles that form linear pairs with the interior angles are the exterior angles.

B

B

A

A

C

interior angles

C

exterior angles

Theorem Triangle Sum Theorem

B

The sum of the measures of the interior angles of a triangle is 180°. A

Proof p. 590; Ex. 53, p. 594

C

m∠A + m∠B + m∠C = 180°

Section 12.1

Angles of Triangles

589

To prove certain theorems, you may need to add a line, a segment, or a ray to a given diagram. An auxiliary line is used in the proof of the Triangle Sum Theorem.

Triangle Sum Theorem B

Given △ABC

D

4 2 5

Prove m∠1 + m∠2 + m∠3 = 180° Plan a. Draw an auxiliary line through B that —. for is parallel to AC Proof

A

1

3

C

b. Show that m∠4 + m∠2 + m∠5 = 180°, ∠1 ≅ ∠4, and ∠3 ≅ ∠5. c. By substitution, m∠1 + m∠2 + m∠3 = 180°.

Plan STATEMENTS in —. BD parallel to AC Action a. 1. Draw ⃖&&⃗

REASONS 1. Parallel Postulate

b. 2. m∠4 + m∠2 + m∠5 = 180°

2. Angle Addition Postulate and

definition of straight angle

3. ∠1 ≅ ∠4, ∠3 ≅ ∠5

3. Alternate Interior Angles Theorem

4. m∠l = m∠4, m∠3 = m∠5

4. Definition of congruent angles

c. 5. m∠l + m∠2 + m∠3 = 180°

5. Substitution Property of Equality

Theorem Exterior Angle Theorem B

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

1 C

A

m∠1 = m∠A + m∠B

Proof Ex. 42, p. 593

Finding an Angle Measure Find m∠JKM.

J x°

SOLUTION Step 1 Write and solve an equation to find the value of x. (2x − 5)° = 70° + x° x = 75

70° L

Apply the Exterior Angle Theorem. Solve for x.

Step 2 Substitute 75 for x in 2x − 5 to find m∠JKM.

⋅

2x − 5 = 2 75 − 5 = 145 So, the measure of ∠JKM is 145°.

590

Chapter 12

Congruent Triangles

(2x − 5)° K M

A corollary to a theorem is a statement that can be proved easily using the theorem. The corollary below follows from the Triangle Sum Theorem.

Corollary Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary.

C

A

B

m∠A + m∠B = 90°

Proof Ex. 41, p. 593

Modeling with Mathematics In the painting, the red triangle is a right triangle. The measure of one acute angle in the triangle is twice the measure of the other. Find the measure of each acute angle.

SOLUTION

2x°

x°

1. Understand the Problem You are given a right triangle and the relationship between the two acute angles in the triangle. You need to find the measure of each acute angle. 2. Make a Plan First, sketch a diagram of the situation. You can use the Corollary to the Triangle Sum Theorem and the given relationship between the two acute angles to write and solve an equation to find the measure of each acute angle. 3. Solve the Problem Let the measure of the smaller acute angle be x°. Then the measure of the larger acute angle is 2x°. The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary. Use the corollary to set up and solve an equation. x° + 2x° = 90°

Corollary to the Triangle Sum Theorem

x = 30

Solve for x.

So, the measures of the acute angles are 30° and 2(30°) = 60°. 4. Look Back Add the two angles and check that their sum satisfies the Corollary to the Triangle Sum Theorem. 30° + 60° = 90°



Monitoring Progress 3. Find the measure of ∠1.

Help in English and Spanish at BigIdeasMath.com

4. Find the measure of each acute angle.

2x°

3x° 40°

1 (5x − 10)°

(x − 6)°

Section 12.1

Angles of Triangles

591

12.1 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Can a right triangle also be obtuse? Explain your reasoning. 2. COMPLETE THE SENTENCE The measure of an exterior angle of a triangle is equal to the sum of the

measures of the two ____________________ interior angles.

Monitoring Progress and Modeling with Mathematics In Exercises 3– 6, classify the triangle by its sides and by measuring its angles. (See Example 1.) 3.

4.

X

In Exercises 15–18, find the measure of the exterior angle. (See Example 3.) 15.

M

16. 75° 64°

Y

5.

L

Z

N

6.

K

J

(2x − 2)°

2

45°

x°

A

17. 24°

C

(2x + 18)° (3x + 6)°

B

H

In Exercises 7–10, classify △ABC by its sides. Then determine whether it is a right triangle. (See Example 2.)

18.

7. A(2, 3), B(6, 3), C(2, 7)

(7x − 16)°

8. A(3, 3), B(6, 9), C(6, −3) (x + 8)°

9. A(1, 9), B(4, 8), C(2, 5) 10. A(−2, 3), B(0, −3), C(3, −2)

In Exercises 11–14, find m∠1. Then classify the triangle by its angles. 11.

12.

1 78°

In Exercises 19–22, find the measure of each acute angle. (See Example 4.) 19.

1 38°

592

Chapter 12

14. 60°

x° 2x°

40°

31°

21. 13.

20. 3x°

30°

1

4x°

1

60°

Congruent Triangles

(6x + 7)° (11x − 2)°

(3x + 2)°

22. (19x − 1)° (13x − 5)°

In Exercises 23–26, find the measure of each acute angle in the right triangle. (See Example 4.) 23. The measure of one acute angle is 5 times the measure

of the other acute angle.

37. USING TOOLS Three people are standing on a stage.

The distances between the three people are shown in the diagram. Classify the triangle by its sides and by measuring its angles. 8 ft

24. The measure of one acute angle is 8 times the measure

of the other acute angle. 5 ft

6.5 ft

25. The measure of one acute angle is 3 times the sum of

the measure of the other acute angle and 8. 26. The measure of one acute angle is twice the difference

of the measure of the other acute angle and 12. ERROR ANALYSIS In Exercises 27 and 28, describe and

correct the error in finding m∠1. 27.



39°

115° + 39° + m∠1 = 360° 154° + m∠1 = 360° m∠1 = 206° 115°

28.

1



38. USING STRUCTURE Which of the following sets of

angle measures could form a triangle? Select all that apply.

A 100°, 50°, 40° ○

B 96°, 74°, 10° ○

C 165°, 113°, 82° ○

D 101°, 41°, 38° ○

E 90°, 45°, 45° ○

F 84°, 62°, 34° ○

39. MODELING WITH MATHEMATICS You are bending a

strip of metal into an isosceles triangle for a sculpture. The strip of metal is 20 inches long. The first bend is made 6 inches from one end. Describe two ways you could complete the triangle. 40. THOUGHT PROVOKING Find and draw an object

80° 1

50°

m∠1 + 80° + 50° = 180° m∠1 + 130° = 180° m∠1 = 50°

(or part of an object) that can be modeled by a triangle and an exterior angle. Describe the relationship between the interior angles of the triangle and the exterior angle in terms of the object. 41. PROVING A COROLLARY Prove the Corollary to the

Triangle Sum Theorem. Given △ABC is a right triangle.

In Exercises 29–36, find the measure of the numbered angle.

Prove ∠A and ∠B are complementary. A

1 40°

2 4

7 3 5

8

6

C

B

42. PROVING A THEOREM Prove the Exterior Angle

Theorem. 29. ∠1

30. ∠2

Given △ABC, exterior ∠BCD

31. ∠3

32. ∠4

Prove m∠A + m∠B = m∠BCD

33. ∠5

34. ∠6

35. ∠7

36. ∠8

B

A

Section 12.1

C

D

Angles of Triangles

593

43. CRITICAL THINKING Is it possible to draw an obtuse

48. MAKING AN ARGUMENT Your friend claims

isosceles triangle? obtuse equilateral triangle? If so, provide examples. If not, explain why it is not possible.

the measure of an exterior angle will always be greater than the sum of the nonadjacent interior angle measures. Is your friend correct? Explain your reasoning.

44. CRITICAL THINKING Is it possible to draw a right MATHEMATICAL CONNECTIONS In Exercises 49–52, find

isosceles triangle? right equilateral triangle? If so, provide an example. If not, explain why it is not possible.

the values of x and y. 49.

43°

45. MATHEMATICAL CONNECTIONS △ABC is isosceles,

AB = x, and BC = 2x − 4.

y°

75°

a. Find two possible values for x when the perimeter of △ABC is 32.

x°

50.

b. How many possible values are there for x when the perimeter of △ABC is 12?

118° x°

46. HOW DO YOU SEE IT? Classify the triangles, in

as many ways as possible, without finding any measurements. a.

22°

y°

51.

b.

52. 25° y° x°

x°

64°

y° 20°

c.

d. 53. PROVING A THEOREM Use the diagram to write

a proof of the Triangle Sum Theorem. Your proof should be different from the proof of the Triangle Sum Theorem shown in this lesson. B

47. ANALYZING RELATIONSHIPS Which of the following

could represent the measures of an exterior angle and two interior angles of a triangle? Select all that apply.

A 100°, 62°, 38° ○

B 81°, 57°, 24° ○

C 119°, 68°, 49° ○

D 95°, 85°, 28° ○

E 92°, 78°, 68° ○

F 149°, 101°, 48° ○

2

A

Maintaining Mathematical Proficiency

55. m∠ABC

594

Chapter 12

(Section 8.2 and Section 8.5) L G 3y

5y − 8 (6x + 2)° B

Congruent Triangles

3z + 6

C

4

5

C

A

56. GH 57. BC

3

Reviewing what you learned in previous grades and lessons

Use the diagram to find the measure of the segment or angle. 54. m∠KHL

1

D

K 8z − 9 (3x + 1)°

(5x − 27)° H

E

12.2 Congruent Polygons Essential Question

Given two congruent triangles, how can you use rigid motions to map one triangle to the other triangle? Describing Rigid Motions Work with a partner. Of the three transformations you studied in Chapter 11, which are rigid motions? Under a rigid motion, why is the image of a triangle always congruent to the original triangle? Explain your reasoning.

LOOKING FOR STRUCTURE

Translation

To be proficient in math, you need to look closely to discern a pattern or structure.

Reflection

Rotation

Finding a Composition of Rigid Motions Work with a partner. Describe a composition of rigid motions that maps △ABC to △DEF. Use dynamic geometry software to verify your answer.

a. △ABC ≅ △DEF A

b. △ABC ≅ △DEF A

3

2

3

C

2

1

B −4

−3

1

E

0

−2

−1

0

1

B 3

2

4

5

−4

−3

−1

D

−3

3

4

5

3

4

5

D

d. △ABC ≅ △DEF A

3

2

3

C

2

1

−1

B 0

1

3

2

E

4

5

−4

−3

D

−1

F

0

−2

−1

0

1

2

−1

E

−2

−3

C

1

0

−2

2

−2

c. △ABC ≅ △DEF

B

1

F

F

−3

−3

0 −1

−2

−4

E

0

−2

−1

A

C

−2

D

−3

F

Communicate Your Answer 3. Given two congruent triangles, how can you use rigid motions to map one triangle

to the other triangle? 4. The vertices of △ABC are A(1, 1), B(3, 2), and C(4, 4). The vertices of △DEF

are D(2, −1), E(0, 0), and F(−1, 2). Describe a composition of rigid motions that maps △ABC to △DEF. Section 12.2

Congruent Polygons

595

What You Will Learn

12.2 Lesson

Identify and use corresponding parts. Use the Third Angles Theorem.

Core Vocabul Vocabulary larry corresponding parts, p. 596

Identifying and Using Corresponding Parts

Previous congruent figures

Recall that two geometric figures are congruent if and only if a rigid motion or a composition of rigid motions maps one of the figures onto the other. A rigid motion maps each part of a figure to a corresponding part of its image. Because rigid motions preserve length and angle measure, corresponding parts of congruent figures are congruent. In congruent polygons, this means that the corresponding sides and the corresponding angles are congruent. When △DEF is the image of △ABC after a rigid motion or a composition of rigid motions, you can write congruence statements for the corresponding angles and corresponding sides.

STUDY TIP Notice that both of the following statements are true.

E

B

F C

1. If two triangles are congruent, then all their corresponding parts are congruent.

A

D

Corresponding angles ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F

Corresponding sides

— ≅ DE —, BC — ≅ EF —, AC — ≅ DF — AB

When you write a congruence statement for two polygons, always list the corresponding vertices in the same order. You can write congruence statements in more than one way. Two possible congruence statements for the triangles above are △ABC ≅ △DEF or △BCA ≅ △EFD.

2. If all the corresponding parts of two triangles are congruent, then the triangles are congruent.

When all the corresponding parts of two triangles are congruent, you can show that the triangles are congruent. Using the triangles above, first translate △ABC so that point A maps to point D. This translation maps △ABC to △DB′C′. Next, rotate △DB′C′ counterclockwise through ∠C′DF so that the image of $$$$⃗ DC′ coincides with $$$⃗ DF. — ≅ DF —, the rotation maps point C′ to point F. So, this rotation maps Because DC′ △DB′C′ to △DB″F. E

B

F C

A

D

VISUAL REASONING To help you identify corresponding parts, rotate △TSR. J

596

F

E F

F D

D

B″

Now, reflect △DB″F in the line through points D and F. This reflection maps the sides and angles of △DB″F to the corresponding sides and corresponding angles of △DEF, so △ABC ≅ △DEF. So, to show that two triangles are congruent, it is sufficient to show that their corresponding parts are congruent. In general, this is true for all polygons.

T K

L

D C′

E

E

B′

Identifying Corresponding Parts S

R

Chapter 12

Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts.

J K

SOLUTION The diagram indicates that △JKL ≅ △TSR. Corresponding angles ∠J ≅ ∠T, ∠K ≅ ∠S, ∠L ≅ ∠R — ≅ TS —, KL — ≅ SR —, LJ — ≅ RT — Corresponding sides JK Congruent Triangles

R

S L

T

Using Properties of Congruent Figures In the diagram, DEFG ≅ SPQR.

D

a. Find the value of x. b. Find the value of y. G

SOLUTION

— ≅ QR —. a. You know that FG

E

8 ft

(2x − 4) ft

Q

R

(6y + x)°

102°

84°

68°

F

12 ft

S

P

b. You know that ∠F ≅ ∠Q.

FG = QR

m∠F = m∠Q

12 = 2x − 4

68° = (6y + x)°

16 = 2x

68 = 6y + 8

8=x

10 = y

Showing That Figures Are Congruent Y divide the wall into orange and blue You —. Will the sections of the ssections along JK wall be the same size and shape? Explain. w

A

J 1 2

B

SOLUTION S F From the diagram, ∠A ≅ ∠C and ∠D ≅ ∠B 3 4 C D bbecause all right angles are congruent. Also, K bby the Lines Perpendicular to a Transversal — # DC —. Then ∠1 ≅ ∠4 and ∠2 ≅ ∠3 by the Alternate Interior Angles Theorem, AB T Theorem. So, all pairs of corresponding angles are congruent. The diagram shows T — ≅ CK —, KD — ≅ JB —, and DA — ≅ BC —. By the Reflexive Property of Congruence, AJ A — — JJK ≅ KJ . So, all pairs of corresponding sides are congruent. Because all corresponding parts p are congruent, AJKD ≅ CKJB. Yes, the two sections will be the same size and shape.

Monitoring Progress M

Help in English and Spanish at BigIdeasMath.com

In the diagram, ABGH ≅ CDEF. P

Q

R

B

F

1. Identify all pairs of congruent

corresponding parts. 2. Find the value of x.

T S

A

H

(4x + 5)°

C 105° 75°

G

D

E

3. In the diagram at the left, show that △PTS ≅ △RTQ.

Theorem Properties of Triangle Congruence

STUDY TIP

Triangle congruence is reflexive, symmetric, and transitive.

The properties of congruence that are true for segments and angles are also true for triangles.

Reflexive

For any triangle △ABC, △ABC ≅ △ABC.

Symmetric If △ABC ≅ △DEF, then △DEF ≅ △ABC. Transitive

If △ABC ≅ △DEF and △DEF ≅ △JKL, then △ABC ≅ △JKL.

Proof BigIdeasMath.com Section 12.2

Congruent Polygons

597

Using the Third Angles Theorem

Theorem Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. A Proof Ex. 19, p. 600

B

E

C

D

F

If ∠A ≅ ∠D and ∠B ≅ ∠E, then ∠C ≅ ∠F.

Using the Third Angles Theorem A

B 45°

Find m∠BDC.

SOLUTION

N

∠A ≅ ∠B and ∠ADC ≅ ∠BCD, so by the Third Angles Theorem, ∠ACD ≅ ∠BDC. By the Triangle Sum Theorem, m∠ACD = 180° − 45° − 30° = 105°.

30° C

D

So, m∠BDC = m∠ACD = 105° by the definition of congruent angles.

Proving That Triangles Are Congruent Use the information in the figure to prove that △ACD ≅ △CAB.

A D

SOLUTION

B C

—, DC — ≅ BA —, ∠ACD ≅ ∠CAB, ∠CAD ≅ ∠ACB Given — AD ≅ CB Prove △ACD ≅ △CAB

— ≅ CA —. Plan a. Use the Reflexive Property of Congruence to show that AC for Proof b. Use the Third Angles Theorem to show that ∠B ≅ ∠D. Plan STATEMENTS in — ≅ CB —, DC — ≅ BA — 1. AD Action

— ≅ CA — a. 2. AC

3. ∠ACD ≅ ∠CAB,

∠CAD ≅ ∠ACB

b. 4. ∠B ≅ ∠D 5. △ACD ≅ △CAB

Monitoring Progress

D

C

N 75° 68°

598

Chapter 12

S

R

REASONS 1. Given 2. Reflexive Property of Congruence 3. Given 4. Third Angles Theorem 5. All corresponding parts are congruent.

Help in English and Spanish at BigIdeasMath.com

Use the diagram. 4. Find m∠DCN. 5. What additional information is needed to conclude that △NDC ≅ △NSR?

Congruent Triangles

12.2 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING Based on this lesson, what information do you need to prove that two triangles are

congruent? Explain your reasoning. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

Is △JKL ≅ △RST?

Is △KJL ≅ △SRT?

Is △JLK ≅ △STR?

Is △LKJ ≅ △TSR?

K

S

J

L

T

R

Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, identify all pairs of congruent corresponding parts. Then write another congruence statement for the polygons. (See Example 1.) 3. △ABC ≅ △DEF

A

10. △MNP ≅ △TUS N

(2x − 50)°

(2x − y) m

E

D

T

S

13 m

142° P

M 24°

U

C F

B

4. GHJK ≅ QRST

In Exercises 11 and 12, show that the polygons are congruent. Explain your reasoning. (See Example 3.)

S

H

11.

K

Q

R

In Exercises 5– 8, △XYZ ≅ △MNL. Copy and complete the statement. 5. m∠Y = ______

X

12

L

6. m∠M = ______

33°

7. m∠Z = ______

Y

J

T

J

G

W

12.

V

X

Z

Y

N

Z

W M

In Exercises 13 and 14, find m∠1. (See Example 4.) 13.

L

M

70°

In Exercises 9 and 10, find the values of x and y. (See Example 2.) E (4y − 4)° B 135° 28°

Y

1

X B

Z Q

S

80°

(10x + 65)° F

H

N

14.

C D

M Y

X

8

8. XY = ______

A

L

N 124° Z

9. ABCD ≅ EFGH

K

A

45°

1 C

R

G

Section 12.2

Congruent Polygons

599

15. PROOF Triangular postage stamps, like the ones

19. PROVING A THEOREM Prove the Third Angles

shown, are highly valued by stamp collectors. Prove that △AEB ≅ △CED. (See Example 5.) A

Theorem by using the Triangle Sum Theorem. 20. THOUGHT PROVOKING Draw a triangle. Copy the

B

triangle multiple times to create a rug design made of congruent triangles. Which property guarantees that all the triangles are congruent?

E

21. REASONING △JKL is congruent to △XYZ. Identify D

all pairs of congruent corresponding parts.

C

— # DC —, AB — ≅ DC —, E is the midpoint of Given AB — and BD —. AC Prove

22. HOW DO YOU SEE IT? In the diagram,

ABEF ≅ CDEF.

△AEB ≅ △CED

G

16. PROOF Use the information in the figure to prove that

△ABG ≅ △DCF.

B B

C A

E A

D

E

F

C

— ≅ DE — and a. Explain how you know that BE ∠ABE ≅ ∠CDE.

D

G

F

b. Explain how you know that ∠GBE ≅ ∠GDE. ERROR ANALYSIS In Exercises 17 and 18, describe and

c. Explain how you know that ∠GEB ≅ ∠GED.

correct the error. 17.



d. Do you have enough information to prove that △BEG ≅ △DEG? Explain.

Given △QRS ≅ △XZY R

Y 42°

Q

X

S

Z

∠S ≅ ∠Z m∠S = m∠Z m∠S = 42°

MATHEMATICAL CONNECTIONS In Exercises 23 and 24,

use the given information to write and solve a system of linear equations to find the values of x and y. 23. △LMN ≅ △PQR, m∠L = 40°, m∠M = 90°,

18.



m∠P = (17x − y)°, m∠R = (2x + 4y)°

N △MNP ≅ △RSP

M R

because the corresponding angles are congruent.

S P

24. △STU ≅ △XYZ, m∠T = 28°, m∠U = (4x + y)°,

m∠X = 130°, m∠Y = (8x − 6y)°

25. PROOF Prove that the criteria for congruent triangles

in this lesson is equivalent to the definition of congruence in terms of rigid motions.

Maintaining Mathematical Proficiency What can you conclude from the diagram? 26.

X

V

27.

S

Y

600

Z

U

Chapter 12

W

(Section 8.6) 28.

P

N

R

Congruent Triangles

Reviewing what you learned in previous grades and lessons

29.

L

D

Q

F T

J

K

M

I

E G H

12.3 Proving Triangle Congruence by SAS Essential Question

What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent? Drawing Triangles Work with a partner. Use dynamic geometry software. a. Construct circles with radii of 2 units and 3 units centered at the origin. Construct a 40° angle with its vertex at the origin. Label the vertex A.

4 3 2 1

40°

0

USING TOOLS STRATEGICALLY To be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data.

b. Locate the point where one ray of the angle intersects the smaller circle and label this point B. Locate the point where the other ray of the angle intersects the larger circle and label this point C. Then draw △ABC.

−4

−3

−2

−1

A

0

1

2

3

4

5

4

5

−1 −2 −3

4 3

c. Find BC, m∠B, and m∠C.

B

2

d. Repeat parts (a)–(c) several times, redrawing the angle in different positions. Keep track of your results by copying and completing the table below. What can you conclude?

A

B

C

1

C

40°

0 −4

−3

−2

−1

A

0

1

2

3

−1 −2 −3

AB

AC

BC

m∠A

1.

(0, 0)

2

3

40°

2.

(0, 0)

2

3

40°

3.

(0, 0)

2

3

40°

4.

(0, 0)

2

3

40°

5.

(0, 0)

2

3

40°

m∠B

m∠C

Communicate Your Answer 2. What can you conclude about two triangles when you know that two pairs of

corresponding sides and the corresponding included angles are congruent? 3. How would you prove your conclusion in Exploration 1(d)?

Section 12.3

Proving Triangle Congruence by SAS

601

12.3 Lesson

What You Will Learn Use the Side-Angle-Side (SAS) Congruence Theorem. Solve real-life problems.

Core Vocabul Vocabulary larry

Using the Side-Angle-Side Congruence Theorem

Previous congruent figures rigid motion

Theorem Side-Angle-Side (SAS) Congruence Theorem If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

STUDY TIP The included angle of two sides of a triangle is the angle formed by the two sides.

— ≅ DE —, ∠A ≅ ∠D, and AC — ≅ DF —, If AB then △ ABC ≅ △DEF.

E

B

F C

Proof p. 602

A

D

Side-Angle-Side (SAS) Congruence Theorem

— ≅ DE —, ∠A ≅ ∠D, AC — ≅ DF — Given AB

E

B

Prove △ABC ≅ △DEF

F C

A

D

First, translate △ABC so that point A maps to point D, as shown below. E

B

E

B′ F

C

A

F

D C′

D

This translation maps △ABC to △DB′C′. Next, rotate △DB′C′ counterclockwise through ∠C′DF so that the image of $$$$⃗ DC′ coincides with $$$⃗ DF, as shown below. E

E

B′

F

F

D C′

D B″

— ≅ DF —, the rotation maps point C′ to point F. So, this rotation maps Because DC′

△DB′C′ to △DB″F. Now, reflect △DB″F in the line through points D and F, as shown below. E

E F

F D

B″

D

Because points D and F lie on ⃖$$⃗ DF, this reflection maps them onto themselves. Because a reflection preserves angle measure and ∠B″DF ≅ ∠EDF, the reflection maps $$$$⃗ DB″ — ≅ DE —, the reflection maps point B″ to point E. So, this reflection to $$$⃗ DE. Because DB″ maps △DB″F to △DEF. Because you can map △ABC to △DEF using a composition of rigid motions, △ABC ≅ △DEF. 602

Chapter 12

Congruent Triangles

Using the SAS Congruence Theorem B

Write a proof.

STUDY TIP Make your proof easier to read by identifying the steps where you show congruent sides (S) and angles (A).

— ≅ DA —, BC — " AD — Given BC

C

Prove △ABC ≅ △CDA A

SOLUTION STATEMENTS

D

REASONS

— ≅ DA — S 1. BC

1. Given

— — 2. BC " AD

2. Given

A 3. ∠BCA ≅ ∠DAC

3. Alternate Interior Angles Theorem

S

4. Reflexive Property of Congruence

— ≅ CA — 4. AC

5. △ABC ≅ △CDA

5. SAS Congruence Theorem

Using SAS and Properties of Shapes

— and RP — pass through the center M of the circle. What can you In the diagram, QS conclude about △MRS and △MPQ? S R

P

M Q

SOLUTION Because they are vertical angles, ∠PMQ ≅ ∠RMS. All points on a circle are the same —, MQ —, MR —, and MS — are all congruent. distance from the center, so MP So, △MRS and △MPQ are congruent by the SAS Congruence Theorem.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

In the diagram, ABCD is a square with four congruent sides and four right — ⊥ SU — angles. R, S, T, and U are the midpoints of the sides of ABCD. Also, RT — ≅ VU —. and SV S

B

R

A

V

U

C

T

D

1. Prove that △SVR ≅ △UVR. 2. Prove that △BSR ≅ △DUT.

Section 12.3

Proving Triangle Congruence by SAS

603

Copying a Triangle Using SAS C

Construct a triangle that is congruent to △ABC using the SAS Congruence Theorem. Use a compass and straightedge. A

SOLUTION Step 1

Step 2

Step 3

Step 4 F

D

E

Construct a side — so that it Construct DE —. is congruent to AB

D

E

Construct an angle Construct ∠D with vertex $$$⃗ so that it is D and side DE congruent to ∠A.

B

D

F

E

D

Construct a side — so that Construct DF —. it is congruent to AC

E

Draw a triangle Draw △DEF. By the SAS Congruence Theorem, △ABC ≅ △DEF.

Solving Real-Life Problems Solving a Real-Life Problem You are making a canvas sign to hang on the triangular portion of the barn wall shown in the picture. You think you can use two identical triangular sheets of — ⊥ QS — and canvas. You know that RP — ≅ PS —. Use the SAS Congruence PQ Theorem to show that △PQR ≅ △PSR.

R

S

Q

P

SOLUTION

— ≅ PS —. By the Reflexive Property of Congruence, RP — ≅ RP —. You are given that PQ By the definition of perpendicular lines, both ∠RPQ and ∠RPS are right angles, so they are congruent. So, two pairs of sides and their included angles are congruent. △PQR and △PSR are congruent by the SAS Congruence Theorem.

Monitoring Progress

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3. You are designing the window shown in the photo. You want to make △DRA

— ≅ DG — and congruent to △DRG. You design the window so that DA ∠ADR ≅ ∠GDR. Use the SAS Congruence Theorem to prove △DRA ≅ △DRG. D

A

604

Chapter 12

Congruent Triangles

R

G

12.3 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING What is an included angle? 2. COMPLETE THE SENTENCE If two sides and the included angle of one triangle are congruent to

two sides and the included angle of a second triangle, then ___________.

Monitoring Progress and Modeling with Mathematics In Exercises 3–8, name the included angle between the pair of sides given. J

In Exercises 15–18, write a proof. (See Example 1.)

—

— —

15. Given PQ bisects ∠SPT, SP ≅ TP

L

Prove △SPQ ≅ △TPQ P

K

P

S

— and KL — 3. JK

— and LK — 4. PK

5.

— and LK — LP

6.

— and JK — JL

7.

— and JL — KL

8.

— and PL — KP

Q

— ≅ CD —, AB — # CD — 16. Given AB Prove △ABC ≅ △CDA

In Exercises 9–14, decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem. Explain. 9. △ABD, △CDB A

A 1 2

M

L

B

Q N B

11. △YXZ, △WXZ

—

D

12. △QRV, △TSU R

Y

13. △EFH, △GHF

Q

S

V

U

A

K

E

C B

T

—≅ RT —, QT — ≅ ST — 18. Given PT

14. △KLM, △MNK E

—

Prove △ABC ≅ △EDC

P

Z

F

C

17. Given C is the midpoint of AE and BD .

C

X

D

10. △LMN, △NQP D

W

T

L

Prove △PQT ≅ △RST P

Q T

G

H

N

M S

Section 12.3

R

Proving Triangle Congruence by SAS

605

In Exercises 19–22, use the given information to name two triangles that are congruent. Explain your reasoning. (See Example 2.) 19. ∠SRT ≅ ∠URT, and

four congruent sides and four congruent angles. B

T

R

A

C

D

C

27. PROOF The Navajo rug is made of isosceles

triangles. You know ∠B ≅ ∠D. Use the SAS Congruence Theorem to show that △ABC ≅ △CDE. (See Example 3.)

U A

21. RSTUV is a regular

D

— ⊥ MN —, KL — ⊥ NL —, 22. MK

pentagon.

B

and M and L are centers of circles.

T S

A

E

L 10 m

V

28. THOUGHT PROVOKING There are six possible subsets

of three sides or angles of a triangle: SSS, SAS, SSA, AAA, ASA, and AAS. Which of these correspond to congruence theorems? For those that do not, give a counterexample.

N

CONSTRUCTION In Exercises 23 and 24, construct a

triangle that is congruent to △ABC using the SAS Congruence Theorem.

29. MATHEMATICAL CONNECTIONS Prove that

24.

△ABC ≅ △DEC. Then find the values of x and y.

B

B

A

C

10 m M

23.

D

K

U

R

B

What additional information do you need to prove that △ABC ≅ △DBC?

20. ABCD is a square with

R is the center of the circle. S

26. HOW DO YOU SEE IT?

A

C

C

A

4y − 6

B

3y + 1

4x E

25. ERROR ANALYSIS Describe and correct the error in

finding the value of x.

✗

30. MAKING AN ARGUMENT Your friend claims it is

possible to construct a triangle congruent to △ABC by first — and AC —, and constructing AB then copying ∠C. Is your friend correct? Explain your reasoning.

Y 5x − 1

4x + 6 X

5x − 5 Z

W 3x + 9

4x + 6 = 3x + 9 x+6=9 x=3

C

A

31. PROVING A THEOREM Prove the Reflections in

Intersecting Lines Theorem.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Classify the triangle by its sides and by measuring its angles. 32.

606

33.

Chapter 12

D

2x + 6

C

Congruent Triangles

34.

(Section 12.1) 35.

B

12.4 Equilateral and Isosceles Triangles Essential Question

What conjectures can you make about the side lengths and angle measures of an isosceles triangle? Writing a Conjecture about Isosceles Triangles Work with a partner. Use dynamic geometry software. a. Construct a circle with a radius of 3 units centered at the origin. b. Construct △ABC so that B and C are on the circle and A is at the origin.

Sample Points A(0, 0) B(2.64, 1.42) C(−1.42, 2.64) Segments AB = 3 AC = 3 BC = 4.24 Angles m∠A = 90° m∠B = 45° m∠C = 45°

3

C

2

B 1 0 −4

−3

−2

−1

A

0

1

2

3

4

−1

−2

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.

−3

c. Recall that a triangle is isosceles if it has at least two congruent sides. Explain why △ABC is an isosceles triangle. d. What do you observe about the angles of △ABC? e. Repeat parts (a)–(d) with several other isosceles triangles using circles of different radii. Keep track of your observations by copying and completing the table below. Then write a conjecture about the angle measures of an isosceles triangle. A

Sample

1.

(0, 0)

2.

(0, 0)

3.

(0, 0)

4.

(0, 0)

5.

(0, 0)

B

C

(2.64, 1.42) (−1.42, 2.64)

AB

AC

BC

3

3

4.24

m∠A m∠B m∠C

90°

45°

45°

f. Write the converse of the conjecture you wrote in part (e). Is the converse true?

Communicate Your Answer 2. What conjectures can you make about the side lengths and angle measures of an

isosceles triangle? 3. How would you prove your conclusion in Exploration 1(e)? in Exploration 1(f)?

Section 12.4

Equilateral and Isosceles Triangles

607

12.4 Lesson

What You Will Learn Use the Base Angles Theorem.

Core Vocabul Vocabulary larry legs, p. 608 vertex angle, p. 608 base, p. 608 base angles, p. 608

Use isosceles and equilateral triangles.

Using the Base Angles Theorem

vertex angle

A triangle is isosceles when it has at least two congruent sides. When an isosceles triangle has exactly two congruent sides, these two sides are the legs. The angle formed by the legs is the vertex angle. The third side is the base of the isosceles triangle. The two angles adjacent to the base are called base angles.

leg

leg base angles base

Theorems Base Angles Theorem

A

If two sides of a triangle are congruent, then the angles opposite them are congruent.

— ≅ AC —, then ∠B ≅ ∠C. If AB Proof p. 608

B

Converse of the Base Angles Theorem

C

A

If two angles of a triangle are congruent, then the sides opposite them are congruent.

— ≅ AC —. If ∠B ≅ ∠C, then AB Proof Ex. 27, p. 631

B

C

Base Angles Theorem

B

— ≅ AC — Given AB A

Prove ∠B ≅ ∠C

— so that it bisects ∠CAB. Plan a. Draw AD for Proof b. Use the SAS Congruence Theorem to show that △ADB ≅ △ADC.

D C

c. Use properties of congruent triangles to show that ∠B ≅ ∠C. Plan STATEMENTS in — Action a. 1. Draw AD , the angle

bisector of ∠CAB.

2. ∠CAD ≅ ∠BAD

— — 3. AB ≅ AC — ≅ DA — 4. DA

608

Chapter 12

REASONS 1. Construction of angle bisector 2. Definition of angle bisector 3. Given 4. Reflexive Property of Congruence

b. 5. △ADB ≅ △ADC

5. SAS Congruence Theorem

c. 6. ∠B ≅ ∠C

6. Corresponding parts of congruent triangles

Congruent Triangles

are congruent.

Using the Base Angles Theorem

— ≅ DF —. Name two congruent angles. In △DEF, DE F

E

D

SOLUTION

— ≅ DF —, so by the Base Angles Theorem, ∠E ≅ ∠F. DE

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Copy and complete the statement.

— ≅ HK —, then ∠ 1. If HG

H

≅∠

2. If ∠KHJ ≅ ∠KJH, then

.

≅

.

G

K

J

Recall that an equilateral triangle has three congruent sides.

Corollaries Corollary to the Base Angles Theorem

READING The corollaries state that a triangle is equilateral if and only if it is equiangular.

If a triangle is equilateral, then it is equiangular. Proof Ex. 37, p. 614

A

Corollary to the Converse of the Base Angles Theorem If a triangle is equiangular, then it is equilateral. Proof Ex. 39, p. 614

B

C

Finding Measures in a Triangle Find the measures of ∠P, ∠Q, and ∠R.

P

SOLUTION The diagram shows that △PQR is equilateral. So, by the Corollary to the Base Angles Theorem, △PQR is equiangular. So, m∠P = m∠Q = m∠R. 3(m∠P) = 180° m∠P = 60°

Triangle Sum Theorem

R

Q

Divide each side by 3.

The measures of ∠P, ∠Q, and ∠R are all 60°. S

T 5 U

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

— for the triangle at the left. 3. Find the length of ST Section 12.4

Equilateral and Isosceles Triangles

609

Using Isosceles and Equilateral Triangles Constructing an Equilateral Triangle

—. Use a Construct an equilateral triangle that has side lengths congruent to AB compass and straightedge. A

B

SOLUTION Step 1

Step 2

Step 3

Step 4 C

A

B

A

—. Copy a segment Copy AB

B

C

A

Draw an arc Draw an arc with center A and radius AB.

B

A

Draw an arc Draw an arc with center B and radius AB. Label the intersection of the arcs from Steps 2 and 3 as C.

B

Draw a triangle Draw — and △ABC. Because AB — AC are radii of the same — ≅ AC —. Because circle, AB — — AB and BC are radii of the — ≅ BC —. By same circle, AB the Transitive Property of — ≅ BC —. So, Congruence, AC △ABC is equilateral.

Using Isosceles and Equilateral Triangles Find the values of x and y in the diagram. K 4 y

COMMON ERROR You cannot use N to refer to ∠LNM because three angles have N as their vertex.

N

Chapter 12

x+1 M

SOLUTION Step 1 Find the value of y. Because △KLN is equiangular, it is also equilateral and — ≅ KL —. So, y = 4. KN

— ≅ LM —, and △LMN is Step 2 Find the value of x. Because ∠LNM ≅ ∠LMN, LN isosceles. You also know that LN = 4 because △KLN is equilateral. LN = LM

610

L

Congruent Triangles

Definition of congruent segments

4=x+1

Substitute 4 for LN and x + 1 for LM.

3=x

Subtract 1 from each side.

Solving a Multi-Step Problem

— ≅ QR — and ∠QPS ≅ ∠PQR. In the lifeguard tower, PS

P

2

1

3

T

S

Q

4 R

a. Explain how to prove that △QPS ≅ △PQR. b. Explain why △PQT is isosceles.

COMMON ERROR When you redraw the triangles so that they do not overlap, be careful to copy all given information and labels correctly.

SOLUTION a. Draw and label △QPS and △PQR so that they do not overlap. You can see that — ≅ QP —, PS — ≅ QR —, and ∠QPS ≅ ∠PQR. So, by the SAS Congruence Theorem, PQ △QPS ≅ △PQR. P

3

2

Q

P

T

Q 1 T

S

4 R

b. From part (a), you know that ∠1 ≅ ∠2 because corresponding parts of congruent — ≅ QT —, triangles are congruent. By the Converse of the Base Angles Theorem, PT and △PQT is isosceles.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

4. Find the values of x and y in the diagram. y° x°

5. In Example 4, show that △PTS ≅ △QTR.

Section 12.4

Equilateral and Isosceles Triangles

611

12.4 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY Describe how to identify the vertex angle of an isosceles triangle. 2. WRITING What is the relationship between the base angles of an isosceles triangle? Explain.

Monitoring Progress and Modeling with Mathematics In Exercises 3– 6, copy and complete the statement. State which theorem you used. (See Example 1.)

12. MODELING WITH MATHEMATICS A logo in an

advertisement is an equilateral triangle with a side length of 7 centimeters. Sketch the logo and give the measure of each side.

E

A

B

C

In Exercises 13–16, find the values of x and y. (See Example 3.)

D

— ≅ DE —, then ∠___ ≅ ∠___. 3. If AE

13.

— ≅ EB —, then ∠___ ≅ ∠___. 4. If AB

14. x°

y°

5. If ∠D ≅ ∠CED, then ___ ≅ ___.

40 8y

40 x°

In Exercises 7–10, find the value of x. (See Example 2.) 7.

M

8.

A

16.

B

9.

x

12

x

L

C

10.

S

60° 60° N 16

T

3x°

CONSTRUCTION In Exercises 17 and 18, construct an F

5 D

11. MODELING WITH MATHEMATICS The dimensions

of a sports pennant are given in the diagram. Find the values of x and y. 79°

WC

y°

x°

612

Chapter 12

y + 12

E 5

x°

3x − 5 5y − 4

5

R

40° x°

15.

6. If ∠EBC ≅ ∠ECB, then ___ ≅ ___.

y°

equilateral triangle whose sides are the given length. 17. 3 inches 18. 1.25 inches 19. ERROR ANALYSIS Describe and correct the error in

—. finding the length of BC



B

A

Congruent Triangles

Because ∠A ≅ ∠C,

— ≅ BC —. AC

5 6

C

So, BC = 6.

MODELING WITH MATHEMATICS In Exercises 25–28, use the diagram based on the color wheel. The 12 triangles in the diagram are isosceles triangles with congruent vertex angles.

20. PROBLEM SOLVING

The diagram represents part of the exterior of the Bow Tower in Calgary, Alberta, Canada. In the diagram, △ABD and △CBD are congruent equilateral triangles. (See Example 4.)

A B E

yellowgreen

D

green C

a. Explain why △ABC is isosceles.

bluepurple

d. Find the measure of ∠BAE.

Triangle

26. The measure of the vertex angle of the yellow triangle

is 30°. Find the measures of the base angles.

Area

1 square unit

27. Trace the color wheel. Then form a triangle whose

vertices are the midpoints of the bases of the red, yellow, and blue triangles. (These colors are the primary colors.) What type of triangle is this?

b. Find the areas of the first four triangles in the pattern.

28. Other triangles can be formed on the color wheel

that are congruent to the triangle in Exercise 27. The colors on the vertices of these triangles are called triads. What are the possible triads?

c. Describe any patterns in the areas. Predict the area of the seventh triangle in the pattern. Explain your reasoning. can you prove? Select all that apply.

B ∠X ≅ ∠Y ○

— ≅ ZX — D YZ ○

In Exercises 23 and 24, find the perimeter of the triangle.

7 in.

29. CRITICAL THINKING Are isosceles triangles always

acute triangles? Explain your reasoning. 30. CRITICAL THINKING Is it possible for an equilateral

—. What The base of isosceles △XYZ is YZ

23.

redpurple purple

on the color wheel. Explain how you know that the yellow triangle is congruent to the purple triangle.

small triangle is an equilateral triangle with an area of 1 square unit.

C ∠Y ≅ ∠Z ○

red

25. Complementary colors lie directly opposite each other

21. FINDING A PATTERN In the pattern shown, each

— ≅ XZ — A XY ○

orange redorange

blue

c. Show that △ABE and △CBE are congruent.

22. REASONING

yelloworange

bluegreen

b. Explain why ∠BAE ≅ ∠BCE.

a. Explain how you know that any triangle made out of equilateral triangles is equilateral.

yellow

24.

(21 − x) in.

(x + 4) in.

triangle to have an angle measure other than 60°? Explain your reasoning. 31. MATHEMATICAL CONNECTIONS The lengths of the

sides of a triangle are 3t, 5t − 12, and t + 20. Find the values of t that make the triangle isosceles. Explain your reasoning.

32. MATHEMATICAL CONNECTIONS The measure of

an exterior angle of an isosceles triangle is x°. Write expressions representing the possible angle measures of the triangle in terms of x. 33. WRITING Explain why the measure of the vertex

(4x + 1) in.

(2x − 3) in.

(x + 5) in.

angle of an isosceles triangle must be an even number of degrees when the measures of all the angles of the triangle are whole numbers.

Section 12.4

Equilateral and Isosceles Triangles

613

34. PROBLEM SOLVING The triangular faces of the peaks

37. PROVING A COROLLARY Prove that the Corollary

on a roof are congruent isosceles triangles with vertex angles U and V.

U

to the Base Angles Theorem follows from the Base Angles Theorem. 38. HOW DO YOU SEE IT? You are designing fabric

V

purses to sell at the school fair.

6.5 m

B W

X

8m

Y

E

a. Name two angles congruent to ∠WUX. Explain your reasoning.

A

b. Name the isosceles triangles in the purse.

35. PROBLEM SOLVING A boat is traveling parallel to

the shore along !!!⃗ RT . When the boat is at point R, the captain measures the angle to the lighthouse as 35°. After the boat has traveled 2.1 miles, the captain measures the angle to the lighthouse to be 70°. 2.1 mi 35°

S

D

a. Explain why △ABE ≅ △DCE.

b. Find the distance between points U and V.

R

C 100°

c. Name three angles that are congruent to ∠EAD. 39. PROVING A COROLLARY Prove that the Corollary

to the Converse of the Base Angles Theorem follows from the Converse of the Base Angles Theorem.

T

40. MAKING AN ARGUMENT The coordinates of two

70°

points are T(0, 6) and U(6, 0). Your friend claims that points T, U, and V will always be the vertices of an isosceles triangle when V is any point on the line y = x. Is your friend correct? Explain your reasoning. L

41. PROOF Use the diagram to prove that △DEF

is equilateral.

a. Find SL. Explain your reasoning.

A

b. Explain how to find the distance between the boat and the shoreline.

D

36. THOUGHT PROVOKING The postulates and theorems

in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, do all equiangular triangles have the same angle measures? Justify your answer.

Maintaining Mathematical Proficiency Use the given property to complete the statement.

— 42. Reflexive Property of Congruence: ____ ≅ SE

B

Congruent Triangles

C

Prove △DEF is equilateral.

Reviewing what you learned in previous grades and lessons

(Section 9.4)

— ≅ PQ —, and PQ — ≅ UV —, then ____ ≅ ____. 44. Transitive Property of Congruence: If EF Chapter 12

F

Given △ABC is equilateral. ∠CAD ≅ ∠ABE ≅ ∠BCF

— ≅ JK. — 43. Symmetric Property of Congruence: If ____ ≅ ____, then RS

614

E

12.1–12.4

What Did You Learn?

Core Vocabulary interior angles, p. 589 exterior angles, p. 589 corollary to a theorem, p. 591 corresponding parts, p. 596

legs (of an isosceles triangle), p. 608 vertex angle (of an isosceles triangle), p. 608 base (of an isosceles triangle), p. 608 base angles (of an isosceles triangle), p. 608

Core Concepts Section 12.1 Classifying Triangles by Sides, p. 588 Classifying Triangles by Angles, p. 588 Triangle Sum Theorem, p. 589

Exterior Angle Theorem, p. 590 Corollary to the Triangle Sum Theorem, p. 591

Section 12.2 Identifying and Using Corresponding Parts, p. 596 Properties of Triangle Congruence, p. 597

Third Angles Theorem, p. 598

Section 12.3 Side-Angle-Side (SAS) Congruence Theorem, p. 602

Section 12.4 Base Angles Theorem, p. 608 Converse of the Base Angles Theorem, p. 608 Corollary to the Base Angles Theorem, p. 609

Corollary to the Converse of the Base Angles Theorem, p. 609

Mathematical Practices 1. 2. 3.

In Exercise 37 on page 593, what are you given? What relationships are present? What is your goal? Explain the relationships present in Exercise 23 on page 600. Describe at least three different patterns created using triangles for the picture in Exercise 20 on page 613.

Studying for Finals • Form a study group of three or four students several veral weeks before the final exam. • Find out what material you must know for the final exam, even if your teacher has not yet covered it. • Ask for a practice final exam or create one yourself self and have your teacher look at it. • Have each group member take the practice final exam. • Decide when the group is going to meet and what you will cover during each session. • During the sessions, make sure you stay on track.

615

12.1–12.4

Quiz

Find the measure of the exterior angle. (Section 12.1) 1.

2.

80°

(5x + 2)°

3.

y°

29°

30°

6x°

x°

(15x + 34)°

(12x + 26)°

Identify all pairs of congruent corresponding parts. Then write another congruence statement for the polygons. (Section 12.2) 4. △ABC ≅ △DEF

5. QRST ≅ WXYZ F

B

D

Q

Y

R

Z

C T

A

W

X

S

E

Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem. If so, write a proof. If not, explain why. (Section 12.3) 6. △CAD, △CBD

7. △GHF, △KHJ

A

8. △LMP, △NMP

G

C

D

F

M J H P

B

K

L

Copy and complete the statement. State which theorem you used. (Section 12.4) 9. If VW ≅ WX, then ∠___ ≅ ∠___. 11. If ∠ZVX ≅ ∠ZXV, then ___ ≅ ___.

W

(5y − 7) ft

E 123°

12. If ∠XYZ ≅ ∠ZXY, then ___ ≅ ___.

F

Q

38 ft

29° S

D

X

10. If XZ ≅ XY, then ∠___ ≅ ∠___.

V

Find the values of x and y. (Section 12.2 and Section 12.4) 13. △DEF ≅ △QRS

N

14.

Z

Y

5x − 1 24

6y°

(2x + 2)° R

15. In a right triangle, the measure of one acute angle is 4 times the difference of the 2

4

3

measure of the other acute angle and 5. Find the measure of each acute angle in the triangle. (Section 12.1)

1

16. The figure shows a stained glass window. (Section 12.1 and Section 12.3) 7 6 5

616

Chapter 12

8

a. Classify triangles 1–4 by their angles. b. Classify triangles 4–6 by their sides. c. Is there enough information given to prove that △7 ≅ △8? If so, label the vertices and write a proof. If not, determine what additional information is needed. Congruent Triangles

12.5 Proving Triangle Congruence by SSS Essential Question

What can you conclude about two triangles when you know the corresponding sides are congruent? Drawing Triangles Work with a partner. Use dynamic geometry software. a. Construct circles with radii of 2 units and 3 units centered at the origin. Label the origin A. — of length 4 units. Then draw BC b.

USING TOOLS STRATEGICALLY To be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data.

4

C

B

3 2

— so that B is on the Move BC

1 0

smaller circle and C is on the larger circle. Then draw △ABC.

−4

−3

−2

−1

A

0

1

2

3

4

5

1

2

3

4

5

−1 −2

c. Explain why the side lengths of △ABC are 2, 3, and 4 units.

−3

4

d. Find m∠A, m∠B, and m∠C.

3 2

e. Repeat parts (b) and (d) several — to different times, moving BC locations. Keep track of your results by copying and completing the table below. What can you conclude?

1 0 −4

−3

−2

−1

A 0

C

−1

B

−2 −3

A

B

C

AB

AC

BC

1.

(0, 0)

2

3

4

2.

(0, 0)

2

3

4

3.

(0, 0)

2

3

4

4.

(0, 0)

2

3

4

5.

(0, 0)

2

3

4

m∠A

m∠B

m∠C

Communicate Your Answer 2. What can you conclude about two triangles when you know the corresponding

sides are congruent? 3. How would you prove your conclusion in Exploration 1(e)?

Section 12.5

Proving Triangle Congruence by SSS

617

What You Will Learn

12.5 Lesson

Use the Side-Side-Side (SSS) Congruence Theorem. Use the Hypotenuse-Leg (HL) Congruence Theorem.

Core Vocabul Vocabulary larry

Using the Side-Side-Side Congruence Theorem

legs, p. 620 hypotenuse, p. 620

Theorem

Previous congruent figures rigid motion

Side-Side-Side (SSS) Congruence Theorem If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. B E

— ≅ DE —, BC — ≅ EF —, and AC — ≅ DF —, If AB then △ABC ≅ △DEF.

C

A

D

F

Side-Side-Side (SSS) Congruence Theorem

— ≅ DE —, BC — ≅ EF —, AC — ≅ DF — Given AB Prove

△ABC ≅ △DEF

B C

E A

D

F

First, translate △ABC so that point A maps to point D, as shown below. B C

B′

E A

D

F

C′

E

D

F

This translation maps △ABC to △DB′C′. Next, rotate △DB′C′ counterclockwise through ∠C′DF so that the image of $$$$⃗ DC′ coincides with $$$⃗ DF, as shown below. B′ C′

D

E

E F

F

D B″

— ≅ DF —, the rotation maps point C′ to point F. So, this rotation maps Because DC′

1

2

D 3

△DB′C′ to △DB″F. Draw an auxiliary line through points E and B″. This line creates ∠1, ∠2, ∠3, and ∠4, as shown at the left.

E F 4 B″

— ≅ DB″ —, △DEB″ is an isosceles triangle. Because FE — ≅ FB″ —, △FEB″ Because DE is an isosceles triangle. By the Base Angles Theorem, ∠1 ≅ ∠3 and ∠2 ≅ ∠4. By the definition of congruence, m∠1 = m∠3 and m∠2 = m∠4. By construction, m∠DEF = m∠1 + m∠2 and m∠DB″F = m∠3 + m∠4. You can now use the Substitution Property of Equality to show m∠DEF = m∠DB″F. m∠DEF = m∠1 + m∠2

Angle Addition Postulate

= m∠3 + m∠4

Substitute m∠3 for m∠1 and m∠4 for m∠2.

= m∠DB″F

Angle Addition Postulate

By the definition of congruence, ∠DEF ≅ ∠DB″F. So, two pairs of sides and their included angles are congruent. By the SAS Congruence Theorem, △DB″F ≅ △DEF. So, a composition of rigid motions maps △DB″F to △DEF. Because a composition of rigid motions maps △ABC to △DB″F and a composition of rigid motions maps △DB″F to △DEF, a composition of rigid motions maps △ABC to △DEF. So, △ABC ≅ △DEF. 618

Chapter 12

Congruent Triangles

Using the SSS Congruence Theorem L

Write a proof.

— ≅ NL —, KM — ≅ NM — KL

Given

K

N

Prove △KLM ≅ △NLM M

SOLUTION STATEMENTS

REASONS

— ≅ NL — S 1. KL S S

1. Given

— — 2. KM ≅ NM — — 3. LM ≅ LM

2. Given 3. Reflexive Property of Congruence

4. △KLM ≅ △NLM

4. SSS Congruence Theorem

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Decide whether the congruence statement is true. Explain your reasoning. 1. △DFG ≅ △HJK F

2. △ACB ≅ △CAD

J

3

B

A D

G

H

P

7

S

C

9

4

7

K

3. △QPT ≅ △RST

Q

D

T

R

Solving a Real-Life Problem Explain why the bench with the diagonal support is stable, while the one without the support can collapse.

SOLUTION The bench with the diagonal support forms triangles with fixed side lengths. By the SSS Congruence Theorem, these triangles cannot change shape, so the bench is stable. The bench without the diagonal support is not stable because there are many possible quadrilaterals with the given side lengths.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Determine whether the figure is stable. Explain your reasoning. 4.

5.

Section 12.5

6.

Proving Triangle Congruence by SSS

619

Copying a Triangle Using SSS Construct a triangle that is congruent to △ABC using the SSS Congruence Theorem. Use a compass and straightedge.

C

A

SOLUTION Step 1

Step 2

Step 3

Step 4 F

D

E

D

Construct a side — so that it is Construct DE —. congruent to AB

E

Draw an arc Open your compass to the length AC. Use this length to draw an arc with center D.

B

F

D

E

D

Draw an arc Draw an arc with radius BC and center E that intersects the arc from Step 2. Label the intersection point F.

E

Draw a triangle Draw △DEF. By the SSS Congruence Theorem, △ABC ≅ △DEF.

Using the Hypotenuse-Leg Congruence Theorem You know that SAS and SSS are valid methods for proving that triangles are congruent. What about SSA? In general, SSA is not a valid method for proving that triangles are congruent. In the triangles below, two pairs of sides and a pair of angles not included between them are congruent, but the triangles are not congruent. B

A hypotenuse leg leg

C

E

D

F

While SSA is not valid in general, there is a special case for right triangles. In a right triangle, the sides adjacent to the right angle are called the legs. The side opposite the right angle is called the hypotenuse of the right triangle.

Theorem Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

A

D

— ≅ DE —, AC — ≅ DF —, and If AB m∠C = m∠F = 90°, then △ABC ≅ △DEF. Proof BigIdeasMath.com

620

Chapter 12

Congruent Triangles

C

B

F

E

Using the Hypotenuse-Leg Congruence Theorem Write a proof. Given

If you have trouble matching vertices to letters when you separate the overlapping triangles, leave the triangles in their original orientations. W

Z

SOLUTION

W

Redraw the triangles so they are side by side with corresponding parts in the same position. Mark the given information in the diagram.

Z

STATEMENTS

Y Z

L

Y X

Y

Y

Z

1. Given

— —— — 2. WZ ⊥ ZY , XY ⊥ ZY

Y

Z

REASONS

— ≅ XZ — H 1. WY

X

X

— ≅ XZ —, WZ — ⊥ ZY —, XY — ⊥ Z—Y WY

Prove △WYZ ≅ △XZY

STUDY TIP

W

2. Given

3. ∠Z and ∠Y are right angles.

3. Definition of ⊥ lines

4. △WYZ and △XZY are right triangles.

4. Definition of a right triangle

— ≅ YZ — 5. ZY

5. Reflexive Property of Congruence

6. △WYZ ≅ △XZY

6. HL Congruence Theorem

Using the Hypotenuse-Leg Congruence Theorem The television antenna is perpendicular to the plane containing points B, C, D, and E. Each of the cables running from the top of the antenna to B, C, and D has the same length. Prove that △AEB, △AEC, and △AED are congruent.

A

— ⊥ EB —, AE — ⊥ EC —, AE — ⊥ ED —, AB — ≅ AC — ≅ AD — Given AE

B

Prove △AEB ≅ △AEC ≅ △AED E

D

SOLUTION

C

— ⊥ EB — and AE — ⊥ EC —. So, ∠AEB and ∠AEC are right angles You are given that AE by the definition of perpendicular lines. By definition, △AEB and △AEC are right — and AC —, triangles. You are given that the hypotenuses of these two triangles, AB — — — are congruent. Also, AE is a leg for both triangles, and AE ≅ AE by the Reflexive Property of Congruence. So, by the Hypotenuse-Leg Congruence Theorem, △AEB ≅ △AEC. You can use similar reasoning to prove that △AEC ≅ △AED. So, by the Transitive Property of Triangle Congruence, △AEB ≅ △AEC ≅ △AED.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com B

A

Use the diagram. 7. Redraw △ABC and △DCB side by side with

corresponding parts in the same position. 8. Use the information in the diagram to prove

that △ABC ≅ △DCB.

Section 12.5

C

Proving Triangle Congruence by SSS

D

621

12.5 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE The side opposite the right angle is called the ___________ of the right triangle. 2. WHICH ONE DOESN’T BELONG? Which triangle’s legs do not belong with the other three? Explain

your reasoning.

Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, decide whether enough information is given to prove that the triangles are congruent using the SSS Congruence Theorem. Explain. 3. △ABC, △DBE

9. △DEF ≅ △DGF E

4. △PQS, △RQS

C

F

Q

E

10. △JKL ≅ △LJM

B

D

P

S

A

D

P

E

C

Q

R

F

7. △RST ≅ △TQP

R

622

Chapter 12

12.

— ≅ BD —, 13. Given AC — ⊥ AD —, AB — — CD ⊥ AD

S

Prove

8. △ABD ≅ △CDB

Q

T

11.

In Exercises 13 and 14, redraw the triangles so they are side by side with corresponding parts in the same position. Then write a proof. (See Example 3.)

In Exercises 7–10, decide whether the congruence statement is true. Explain your reasoning. (See Example 1.)

S

M

6. △PQT, △SRT

T B

J

In Exercises 11 and 12, determine whether the figure is stable. Explain your reasoning. (See Example 2.)

R

In Exercises 5 and 6, decide whether enough information is given to prove that the triangles are congruent using the HL Congruence Theorem. Explain. 5. △ABC, △FED

L

D G

A

K

B

P

A

14. Given C

D

Congruent Triangles

Prove

A

B

D

C

△BAD ≅ △CDA

G is the midpoint —, FG — ≅ GI —, of EH ∠E and ∠H are right angles.

E

△EFG ≅ △HIG

G

I

F

H

In Exercises 15 and 16, write a proof.

22. MODELING WITH MATHEMATICS The distances

— ≅ JK —, MJ — ≅ KL — 15. Given LM

between consecutive bases on a softball field are the same. The distance from home plate to second base is the same as the distance from first base to third base. The angles created at each base are 90°. Prove △HFS ≅ △FST ≅ △STH. (See Example 4.)

Prove △LMJ ≅ △JKL K

L

J

M

second base (S)

— —— —— —

16. Given WX ≅ VZ , WY ≅ VY , YZ ≅ YX

third base (T)

Prove △VWX ≅ △WVZ W

first base (F)

X Y

V

home plate (H)

Z

23. REASONING To support a tree, you attach wires from

CONSTRUCTION In Exercises 17 and 18, construct

the trunk of the tree to stakes in the ground, as shown in the diagram.

a triangle that is congruent to △QRS using the SSS Congruence Theorem. 17.

18.

R

Q

S

R

Q

S

19. ERROR ANALYSIS Describe and correct the error in

L

identifying congruent triangles.



U

X

Z J

T

V

a. What additional information do you need to use the HL Congruence Theorem to prove that △JKL ≅ △MKL?

Y

△TUV ≅ △XYZ by the SSS Congruence Theorem.

b. Suppose K is the midpoint of JM. Name a theorem you could use to prove that △JKL ≅ △MKL. Explain your reasoning.

20. ERROR ANALYSIS Describe and correct the error in

determining the value of x that makes the triangles congruent.



M

K

24. REASONING Use the photo of the Navajo rug, where

— ≅ DE — and AC — ≅ CE —. BC

K 2x + 1

4x + 4

J

L 3x − 1

6x

6x = 2x + 1 4x = 1 x = —14

B A

D C

E

M

21. MAKING AN ARGUMENT Your friend claims that

in order to use the SSS Congruence Theorem to prove that two triangles are congruent, both triangles must be equilateral triangles. Is your friend correct? Explain your reasoning. Section 12.5

a. What additional information do you need to use the SSS Congruence Theorem to prove that △ABC ≅ △CDE? b. What additional information do you need to use the HL Congruence Theorem to prove that △ABC ≅ △CDE? Proving Triangle Congruence by SSS

623

In Exercises 25–28, use the given coordinates to determine whether △ABC ≅ △DEF.

32. THOUGHT PROVOKING The postulates and

theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, do you think that two triangles are congruent if their corresponding sides are congruent? Justify your answer.

25. A(−2, −2), B(4, −2), C(4, 6), D(5, 7), E(5, 1), F(13, 1) 26. A(−2, 1), B(3, −3), C(7, 5), D(3, 6), E(8, 2), F(10, 11) 27. A(0, 0), B(6, 5), C(9, 0), D(0, −1), E(6, −6), F(9, −1) 28. A(−5, 7), B(−5, 2), C(0, 2), D(0, 6), E(0, 1), F(4, 1) 29. CRITICAL THINKING You notice two triangles in

the tile floor of a hotel lobby. You want to determine whether the triangles are congruent, but you only have a piece of string. Can you determine whether the triangles are congruent? Explain. 30. HOW DO YOU SEE IT? There are several theorems

you can use to show that the triangles in the “square” pattern are congruent. Name two of them.

USING TOOLS In Exercises 33 and 34, use the given

information to sketch △LMN and △STU. Mark the triangles with the given information.

— ⊥ MN —, ST — ⊥ TU —, LM — ≅ NM — ≅ UT — ≅ ST — 33. LM

— ⊥ MN —, ST — ⊥ TU —, LM — ≅ ST —, LN — ≅ SU — 34. LM 35. CRITICAL THINKING The diagram shows the light

created by two spotlights. Both spotlights are the same distance from the stage.

E

B

A

D

C

G

F

a. Show that △ABD ≅ △CBD. State which theorem or postulate you used and explain your reasoning. b. Are all four right triangles shown in the diagram congruent? Explain your reasoning.

31. MAKING AN ARGUMENT Your cousin says that

△JKL is congruent to △LMJ by the SSS Congruence Theorem. Your friend says that △JKL is congruent to △LMJ by the HL Congruence Theorem. Who is correct? Explain your reasoning.

36. MATHEMATICAL CONNECTIONS Find all values of x

that make the triangles congruent. Explain.

K

J

5x

A

B 4x + 3

5x − 2 C

D

3x + 10

L

M

Maintaining Mathematical Proficiency Use the congruent triangles.

Reviewing what you learned in previous grades and lessons

(Section 12.2)

—. 37. Name the segment in △DEF that is congruent to AC

A

D

—. 38. Name the segment in △ABC that is congruent to EF 39. Name the angle in △DEF that is congruent to ∠B. 40. Name the angle in △ABC that is congruent to ∠F.

624

Chapter 12

Congruent Triangles

B

C

F

E

12.6 Proving Triangle Congruence by ASA and AAS Essential Question

What information is sufficient to determine whether two triangles are congruent? Determining Whether SSA Is Sufficient Work with a partner. a. Use dynamic geometry software to construct △ABC. Construct the triangle so that — has a length of 3 units, and BC — has a length of 2 units. vertex B is at the origin, AB b. Construct a circle with a radius of 2 units centered at the origin. Locate point D —. Draw BD —. where the circle intersects AC

Sample 3

A D

2

1

C

0 −3

−2

B

−1

0

1

2

3

−1

−2

CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to recognize and use counterexamples.

Points A(0, 3) B(0, 0) C(2, 0) D(0.77, 1.85) Segments AB = 3 AC = 3.61 BC = 2 AD = 1.38 Angle m∠A = 33.69°

c. △ABC and △ABD have two congruent sides and a nonincluded congruent angle. Name them. d. Is △ABC ≅ △ABD? Explain your reasoning. e. Is SSA sufficient to determine whether two triangles are congruent? Explain your reasoning.

Determining Valid Congruence Theorems Work with a partner. Use dynamic geometry software to determine which of the following are valid triangle congruence theorems. For those that are not valid, write a counterexample. Explain your reasoning. Possible Congruence Theorem

Valid or not valid?

SSS SSA SAS AAS ASA AAA

Communicate Your Answer 3. What information is sufficient to determine whether two triangles are congruent? 4. Is it possible to show that two triangles are congruent using more than one

congruence theorem? If so, give an example. Section 12.6

Proving Triangle Congruence by ASA and AAS

625

12.6 Lesson

What You Will Learn Use the ASA and AAS Congruence Theorems.

Core Vocabul Vocabulary larry Previous congruent figures rigid motion

Using the ASA and AAS Congruence Theorems

Theorem Angle-Side-Angle (ASA) Congruence Theorem If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

— ≅ DF —, and ∠C ≅ ∠F, If ∠A ≅ ∠D, AC then △ABC ≅ △DEF.

B

E

C

Proof p. 626

A

D

F

Angle-Side-Angle (ASA) Congruence Theorem

— ≅ DF —, ∠C ≅ ∠F Given ∠A ≅ ∠D, AC Prove △ABC ≅ △DEF

B

E

C

A

D

F

First, translate △ABC so that point A maps to point D, as shown below. B C

A

E

B′

E D

F

D

C′

F

This translation maps △ABC to △DB′C′. Next, rotate △DB′C′ counterclockwise through ∠C′DF so that the image of $$$$⃗ DC′ coincides with $$$⃗ DF, as shown below. E E

B′

D

D

C′

F

F B″

— ≅ DF —, the rotation maps point C′ to point F. So, this rotation maps Because DC′ △DB′C′ to △DB″F. Now, reflect △DB″F in the line through points D and F, as shown below. E E D F

D

F

B″

Because points D and F lie on ⃖$$⃗ DF, this reflection maps them onto themselves. Because a reflection preserves angle measure and ∠B″DF ≅ ∠EDF, the reflection maps $$$$⃗ DB″ to $$$⃗ DE. Similarly, because ∠B″FD ≅ ∠EFD, the reflection maps $$$$⃗ FB″ to $$$⃗ FE. The image of B″ lies on $$$⃗ DE and $$$⃗ FE. Because $$$⃗ DE and $$$⃗ FE only have point E in common, the image of B″ must be E. So, this reflection maps △DB″F to △DEF. Because you can map △ABC to △DEF using a composition of rigid motions, △ABC ≅ △DEF. 626

Chapter 12

Congruent Triangles

Theorem Angle-Angle-Side (AAS) Congruence Theorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. E

If ∠A ≅ ∠D, ∠C ≅ ∠F, — ≅ EF —, then and BC △ABC ≅ △DEF.

B D A

F

C

Proof p. 627

Angle-Angle-Side (AAS) Congruence Theorem Given ∠A ≅ ∠D, ∠C ≅ ∠F, — ≅ EF — BC Prove

B

△ABC ≅ △DEF

E

A

C

F

D

You are given ∠A ≅ ∠D and ∠C ≅ ∠F. By the Third Angles Theorem, ∠B ≅ ∠E. — ≅ EF —. So, two pairs of angles and their included sides are You are given BC congruent. By the ASA Congruence Theorem, △ABC ≅ △DEF.

Identifying Congruent Triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use. a.

b.

c.

COMMON ERROR You need at least one pair of congruent corresponding sides to prove two triangles are congruent.

SOLUTION a. The vertical angles are congruent, so two pairs of angles and a pair of non-included sides are congruent. The triangles are congruent by the AAS Congruence Theorem. b. There is not enough information to prove the triangles are congruent, because no sides are known to be congruent. c. Two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Theorem.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1. Can the triangles be proven congruent with

X

the information given in the diagram? If so, state the theorem you would use.

4 W

Section 12.6

3

1 2

Y

Z

Proving Triangle Congruence by ASA and AAS

627

Copying a Triangle Using ASA Construct a triangle that is congruent to △ABC using the ASA Congruence Theorem. Use a compass and straightedge.

C

A

SOLUTION Step 1

Step 2

Step 3

B

Step 4 F

D

E

Construct a side — so that it is Construct DE —. congruent to AB

D

E

D

Construct an angle Construct ∠D with %%%⃗ so vertex D and side DE that it is congruent to ∠A.

E

D

Construct an angle Construct ∠E with %%%⃗ so vertex E and side ED that it is congruent to ∠B.

E

Label a point Label the intersection of the sides of ∠D and ∠E that you constructed in Steps 2 and 3 as F. By the ASA Congruence Theorem, △ABC ≅ △DEF.

Using the ASA Congruence Theorem A

Write a proof.

C

— ! EC —, BD — ≅ BC — Given AD Prove △ABD ≅ △EBC

B D

SOLUTION STATEMENTS

E

REASONS

— ! EC — 1. AD

1. Given 2. Alternate Interior Angles Theorem

A 2. ∠D ≅ ∠C

— ≅ BC — S 3. BD

3. Given

A 4. ∠ABD ≅ ∠EBC

4. Vertical Angles Congruence Theorem

5. △ABD ≅ △EBC

5. ASA Congruence Theorem

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

— ⊥ AD —, DE — ⊥ AD —, and AC — ≅ DC —. Prove △ABC ≅ △DEC. 2. In the diagram, AB E A C B

628

Chapter 12

Congruent Triangles

D

Using the AAS Congruence Theorem Write a proof. Given

— ! GK —, ∠ F and ∠ K are right angles. HF

F

G

H

K

Prove △HFG ≅ △GKH

SOLUTION STATEMENTS 1.

REASONS

— ! GK — HF

1. Given

A 2. ∠GHF ≅ ∠HGK

2. Alternate Interior Angles Theorem

3. ∠ F and ∠ K are right angles.

3. Given

A 4. ∠ F ≅ ∠ K

4. Right Angles Congruence Theorem

S

5. Reflexive Property of Congruence

— ≅ GH — 5. HG

6. △HFG ≅ △GKH

6. AAS Congruence Theorem

Monitoring Progress 3.

Help in English and Spanish at BigIdeasMath.com

— ≅ VU —. Prove △RST ≅ △VUT. In the diagram, ∠S ≅ ∠U and RS R

U

T S

V

Concept Summary Triangle Congruence Theorems You have learned five methods for proving that triangles are congruent. SAS E B A

D

s only) HL (right △

SSS E B

F

C

Two sides and the included angle are congruent.

A

D

ASA

E F

B A

C

All three sides are congruent.

D

AAS E

B

F

C

The hypotenuse and one of the legs are congruent.

A

D

E B

F

C

Two angles and the included side are congruent.

A

D

F

C

Two angles and a non-included side are congruent.

In the Exercises, you will prove three additional theorems about the congruence of right triangles: Hypotenuse-Angle, Leg-Leg, and Angle-Leg. Section 12.6

Proving Triangle Congruence by ASA and AAS

629

12.6 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. WRITING How are the AAS Congruence Theorem and the ASA Congruence Theorem similar?

How are they different? 2. WRITING You know that a pair of triangles has two pairs of congruent corresponding angles.

What other information do you need to show that the triangles are congruent?

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, decide whether enough information is given to prove that the triangles are congruent. If so, state the theorem you would use. (See Example 1.) 3. △ABC, △QRS

4. △ABC, △DBC

— —

C

A Q

11. ∠B ≅ ∠E,∠C ≅ ∠F, AC ≅ DE

S

— —

12. ∠A ≅ ∠D, ∠B ≅ ∠E, BC ≅ EF A

R

5. △XYZ, △JKL Y

— ≅ DF — 9. ∠A ≅ ∠D, ∠C ≅ ∠F, AC

— ≅ DE —, BC — ≅ EF — 10. ∠C ≅ ∠F, AB

B

B

In Exercises 9–12, decide whether you can use the given information to prove that △ABC ≅ △DEF. Explain your reasoning.

C

D

6. △RSV, △UTV R

K

S

CONSTRUCTION In Exercises 13 and 14, construct a

triangle that is congruent to the given triangle using the ASA Congruence Theorem. Use a compass and straightedge. 13.

Z L J

X

U

T

In Exercises 7 and 8, state the third congruence statement that is needed to prove that △FGH ≅ △LMN using the given theorem. F

L G

14.

E

V D

F

L

correct the error. 15.



K

H L G

F

J N

— ≅ MN —, ∠G ≅ ∠M, ___ ≅ ____ 7. Given GH

16.



Q

X

W

Use the AAS Congruence Theorem.

— ≅ LM —, ∠G ≅ ∠M, ___ ≅ ____ 8. Given FG Use the ASA Congruence Theorem.

630

Chapter 12

Congruent Triangles

K

ERROR ANALYSIS In Exercises 15 and 16, describe and

M

H

J

R

S

V

△JKL ≅ △FHG by the ASA Congruence Theorem.

△QRS ≅ △VWX by the AAS Congruence Theorem.

PROOF In Exercises 17 and 18, prove that the triangles

are congruent using the ASA Congruence Theorem. (See Example 2.) 17. Given

—. M is the midpoint of NL — — — — — " PL — NL ⊥ NQ , NL ⊥ MP , QM

and a leg of a right triangle are congruent to an angle and a leg of a second right triangle, then the triangles are congruent. 24. REASONING What additional information do

you need to prove △JKL ≅ △MNL by the ASA Congruence Theorem?

Prove △NQM ≅ △MPL Q

23. Angle-Leg (AL) Congruence Theorem If an angle

— ≅ KJ — A KM ○

P

— ≅ NH — B KH ○ N

L

M

— —

M

K L

H

C ∠M ≅ ∠J ○

N

D ∠LKJ ≅ ∠LNM ○

J

18. Given AJ ≅ KC , ∠BJK ≅ ∠BKJ, ∠A ≅ ∠C 25. MATHEMATICAL CONNECTIONS This toy

Prove △ABK ≅ △CBJ

contains △ABC and △DBC. Can you conclude that △ABC ≅ △DBC from the given angle measures? Explain.

B

A

J

K

C

C A

D B

PROOF In Exercises 19 and 20, prove that the triangles

are congruent using the AAS Congruence Theorem. (See Example 3.)

m∠ABC = (8x − 32)°

— ≅ UW —, ∠X ≅ ∠Z 19. Given VW

m∠DBC = (4y − 24)° m∠BCA = (5x + 10)°

Prove △XWV ≅ △ZWU Z V

m∠BCD = (3y + 2)°

X

Y

m∠CAB = (2x − 8)°

U

m∠CDB = (y − 6)°

W

26. REASONING Which of the following congruence 20. Given

statements are true? Select all that apply.

∠NKM ≅ ∠LMK, ∠L ≅ ∠N

— ≅ UV — A TU ○

Prove △NMK ≅ △LKM L

B △STV ≅ △XVW ○

N

W S X

C △TVS ≅ △VWU ○ K

M

PROOF In Exercises 21–23, write a paragraph proof for

the theorem about right triangles. 21. Hypotenuse-Angle (HA) Congruence Theorem

If an angle and the hypotenuse of a right triangle are congruent to an angle and the hypotenuse of a second right triangle, then the triangles are congruent. 22. Leg-Leg (LL) Congruence Theorem If the legs of

a right triangle are congruent to the legs of a second right triangle, then the triangles are congruent. Section 12.6

D △VST ≅ △VUW ○

T

U

V

27. PROVING A THEOREM Prove the Converse of the

Base Angles Theorem. (Hint: Draw an auxiliary line inside the triangle.) 28. MAKING AN ARGUMENT Your friend claims to

be able to rewrite any proof that uses the AAS Congruence Theorem as a proof that uses the ASA Congruence Theorem. Is this possible? Explain your reasoning. Proving Triangle Congruence by ASA and AAS

631

29. MODELING WITH MATHEMATICS When a light ray

31. CONSTRUCTION Construct a triangle. Show that there

from an object meets a mirror, it is reflected back to your eye. For example, in the diagram, a light ray from point C is reflected at point D and travels back to point A. The law of reflection states that the angle of incidence, ∠CDB, is congruent to the angle of reflection, ∠ADB. a. Prove that △ABD is congruent to △CBD.

is no AAA congruence rule by constructing a second triangle that has the same angle measures but is not congruent. 32. THOUGHT PROVOKING Graph theory is a branch of

mathematics that studies vertices and the way they are connected. In graph theory, two polygons are isomorphic if there is a one-to-one mapping from one polygon’s vertices to the other polygon’s vertices that preserves adjacent vertices. In graph theory, are any two triangles isomorphic? Explain your reasoning.

A

Given

∠CDB ≅ ∠ADB, — ⊥ AC — DB

Prove

△ABD ≅ △CBD

b. Verify that △ACD is isosceles.

B

c. Does moving away from the mirror have any effect on the amount of his or her reflection a person sees? Explain.

33. MATHEMATICAL CONNECTIONS Six statements are

given about △TUV and △XYZ.

D

— ≅ XY — TU

— ≅ YZ — UV

— ≅ XZ — TV

∠T ≅ ∠X

∠U ≅ ∠Y

∠V ≅ ∠Z

U

C T

30. HOW DO YOU SEE IT? Name as many pairs of

congruent triangles as you can from the diagram. Explain how you know that each pair of triangles is congruent. P

b. You choose three statements at random. What is the probability that the statements you choose provide enough information to prove that the triangles are congruent?

R

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Find the coordinates of the midpoint of the line segment with the given endpoints. 35. J(−2, 3) and K(4, −1)

Copy the angle using a compass and straightedge. 37.

632

(Section 8.5) 38.

A

Chapter 12

B

Congruent Triangles

X

a. List all combinations of three given statements that would provide enough information to prove that △TUV is congruent to △XYZ.

T

34. C(1, 0) and D(5, 4)

Z Y

Q

S

V

(Section 8.3)

36. R(−5, −7) and S(2, −4)

12.7 Using Congruent Triangles Essential Question

How can you use congruent triangles to make

an indirect measurement?

Measuring the Width of a River

CRITIQUING THE REASONING OF OTHERS To be proficient in math, you need to listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Work with a partner. The figure shows how a surveyor can measure the width of a river by making measurements on only one side of the river.

a. Study the figure. Then explain how the surveyor can find the width of the river.

B

C

A

b. Write a proof to verify that the method you described in part (a) is valid. Given

— ≅ CD — ∠A is a right angle, ∠D is a right angle, AC

D

E

c. Exchange proofs with your partner and discuss the reasoning used.

Measuring the Width of a River Work with a partner. It was reported that one of Napoleon’s officers estimated the width of a river as follows. The officer stood on the bank of the river and F lowered the visor on his cap until the farthest thing visible was the edge of the D bank on the other side. He then turned and noted the point on his side that was E in line with the tip of his visor and his eye. The officer then paced the distance to this point and concluded that distance was the width of the river. a. Study the figure. Then explain how the officer concluded that the width of the river is EG.

G

b. Write a proof to verify that the conclusion the officer made is correct. Given

∠DEG is a right angle, ∠DEF is a right angle, ∠EDG ≅ ∠EDF

c. Exchange proofs with your partner and discuss the reasoning used.

Communicate Your Answer 3. How can you use congruent triangles to make an indirect measurement? 4. Why do you think the types of measurements described in Explorations 1 and 2

are called indirect measurements?

Section 12.7

Using Congruent Triangles

633

What You Will Learn

12.7 Lesson

Use congruent triangles to solve problems. Use congruent triangles to write proofs.

Core Vocabul Vocabulary larry

Using Congruent Triangles to Solve Problems

Previous congruent figures corresponding parts construction perpendicular lines

Using Congruent Triangles Explain how you can use the given information to prove that the hang glider parts are  congruent. Given Prove

∠1 ≅ ∠2, ∠RTQ ≅ ∠RTS

R Q T

1

— ≅ ST — QT

2

S

SOLUTION

Mark given and deduced information. R

T

Q

S

1

2

— ≅ ST —. First, copy If you can show that △QRT ≅ △SRT, then you will know that QT the diagram and mark the given information. Then mark the information that you can deduce. In this case, ∠RQT and ∠RST are supplementary to congruent angles, so — ≅ RT — by the Reflexive Property of Congruence. Two angle ∠RQT ≅ ∠RST. Also, RT pairs and a non-included side are congruent, so by the AAS Congruence Theorem, △QRT ≅ △SRT. — ≅ ST —. Because corresponding parts of congruent triangles are congruent, QT Using Congruent Triangles for Measurement

MAKING SENSE OF PROBLEMS When you cannot easily measure a length directly, you can make conclusions about the length indirectly, usually by calculations based on known lengths.

To find the distance across the river, from point N to point P, first place a stake — ⊥ NP —. at K on the near side so that NK — Then find M, the midpoint of NK . Finally, — ⊥ KL — and locate the point L so that NK L, P, and M are collinear. Explain how this plan allows you to find the distance.

N

P M

L

K

SOLUTION

— ⊥ NP — and NK — ⊥ KL —, ∠N and ∠K are congruent right angles. Because Because NK — — —. The vertical angles ∠KML and ∠NMP are M is the midpoint of NK , NM ≅ KM congruent. So, △MLK ≅ △MPN by the ASA Congruence Theorem. Then because — ≅ NP —. So, you can find corresponding parts of congruent triangles are congruent, KL — the distance NP across the river by measuring KL .

Monitoring Progress

B A

C D

634

Chapter 12

Help in English and Spanish at BigIdeasMath.com

1. Explain how you can prove that ∠A ≅ ∠C. 2. In Example 2, does it matter how far from point N you place a stake

at point K? Explain. Congruent Triangles

Using Congruent Triangles to Write Proofs Planning a Proof Involving Pairs of Triangles D

Use the given information to write a plan for proof. Given

∠1 ≅ ∠2, ∠3 ≅ ∠4

Prove

△BCE ≅ △DCE

C

E

2 1

4 3

A

B

SOLUTION

— ≅ CE —. If you can show that In △BCE and △DCE, you know that ∠1 ≅ ∠2 and CE — — CB ≅ CD , then you can use the SAS Congruence Theorem. — ≅ CD —, you can first prove that △CBA ≅ △CDA. You are given To prove that CB — ≅ CA — by the Reflexive Property of Congruence. You ∠1 ≅ ∠2 and ∠3 ≅ ∠4. CA can use the ASA Congruence Theorem to prove that △CBA ≅ △CDA. Plan for Proof Use the ASA Congruence Theorem to prove that — ≅ CD —. Use the SAS Congruence △CBA ≅ △CDA. Then state that CB Theorem to prove that △BCE ≅ △DCE.

Proving a Construction Write a proof to verify that the construction for copying an angle on page 416 is valid.

SOLUTION

— and EF — to the diagram on page 416. In the Add BC —, DE —, construction, one compass setting determines AB — — AC , and DF , and another compass setting determines — and EF —. So, you can assume the following as BC given statements.

C A

F

— ≅ DE —, AC — ≅ DF —, BC — ≅ EF — Given AB

E

D

Prove

∠D ≅ ∠A

Plan for Proof

Show that △DEF ≅ △ABC, so you can conclude that the corresponding parts ∠D and ∠A are congruent.

Plan STATEMENTS in — ≅ DE —, AC — ≅ DF —, BC — ≅ EF — Action 1. AB

B

REASONS 1. Given

2. △DEF ≅ △ABC

2. SSS Congruence Theorem

3. ∠D ≅ ∠A

3. Corresponding parts of congruent

triangles are congruent.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com P

3. Write a plan to prove that △PTU ≅ △UQP. 4. Use the construction of an angle bisector

R

on page 418. What segments can you assume are congruent? T

Section 12.7

Q S U

Using Congruent Triangles

635

You can use congruent triangles to prove the Slopes of Perpendicular Lines Theorem. Because the theorem is a biconditional statement, you must prove both parts.

Slopes of Perpendicular Lines Theorem

READING The notation mℓ is read as “the slope of line ℓ.”

STUDY TIP The case where line ℓ has a negative slope is proved similarly.

1. If two nonvertical lines are perpendicular, then the product of their slopes is −1.

Given ℓ ⊥ n, ℓ and n are nonvertical. Prove mℓ mn = −1 Paragraph Proof Draw two nonvertical, perpendicular lines, ℓ and n, so that line ℓ has a positive slope and the lines intersect at point A. Draw right △ABC as shown, — lies on ℓ, BC — is vertical with where AB — length a, and AC is horizontal with length b. a By definition, the slope of line ℓ is mℓ = —. b

n B′

a C′

b

B a

A

Rotate △ABC 90° counterclockwise about point A. The image is △AB′C′. Because ℓ ⊥ n, B′ lies on line n. Because rotations preserve angle measure, ∠C′ is a right angle. Because rotations preserve length,

b

C

B′C′ = BC = a and AC′ = AC = b. By definition, the slope of

( )

b a b line n is mn = – —. So, mℓ mn = — – — = −1. a b a 2. If the product of the slopes of two nonvertical lines is −1, then the lines

are perpendicular.

STUDY TIP The case where line ℓ has a negative slope is proved similarly.

Given mℓ mn = −1 Prove ℓ ⊥ n a Paragraph Proof Let mℓ = —, where a and b are both positive. You know that b a b mℓ mn = −1. By substitution, — • mn = −1. So, mn = – —. b a Draw lines ℓ and n and label the point n where the lines intersect as point A. Draw a E right △ABC and right △ADE, as shown. D Because DE = BC = a and AE = AC = b, — ≅ BC — and AE — ≅ AC —. By the Right Angles DE Congruence Theorem, ∠E ≅ ∠C. So, △ABC ≅ △ADE by the SAS Congruence Theorem.

— is vertical and AC — is horizontal, Because AE

∠EAC is a right angle. By definition, ∠EAB and ∠BAC are complementary angles. So, m∠EAB + m∠BAC = 90°.

b

B a

A

b

C

∠DAE ≅ ∠BAC because corresponding parts of congruent triangles are congruent. By substitution, m∠EAB + m∠DAE = 90°. By the Commutative Property of Addition, m∠DAE + m∠EAB = 90°. By the Angle Addition Postulate, m∠DAE + m∠EAB = m∠DAB. So, by the Transitive Property of Equality, m∠DAB = 90° and line ℓ is perpendicular to line n.

636

Chapter 12

Congruent Triangles

12.7 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE ______________ parts of congruent triangles are congruent. 2. WRITING Describe a situation in which you might choose to use indirect measurement with

congruent triangles to find a measure rather than measuring directly.

Monitoring Progress and Modeling with Mathematics In Exercises 3–8, explain how to prove that the statement is true. (See Example 1.) 3. ∠A ≅ ∠D

4. ∠Q ≅ ∠T

B

A

D

S

A

L

D

— ≅ VT — 8. QW

G

H

K

N

S

10.

1

E

2 J

K

H

11.

A

B

C

D

15. Prove

12. P 1

Q

A

B

— ≅ HN — FL

F

H G

J

C

T

K M

2 S

Plan for Proof Show that △APQ ≅ △BPQ by the SSS Congruence Theorem. Use corresponding parts of congruent triangles to show that ∠QPA and ∠QPB are right angles. In Exercises 15 and 16, use the information given in the diagram to write a proof.

2

1

B Q

In Exercises 9–12, write a plan to prove that ∠1 ≅ ∠2. (See Example 3.) G

P A

W

V

9.

to a line through a point on the line

U

L

F

Plan for Proof Show that △APQ ≅ △BPQ by the SSS Congruence Theorem. Then show that △APM ≅ △BPM using the SAS Congruence Theorem. Use corresponding parts of congruent triangles to show that ∠AMP and ∠BMP are right angles. 14. Line perpendicular

T

R

Q

B Q

B

C

— ≅ HJ — 7. GK

M

T

— ≅ DB — 6. AC

M

P M

R

K

to a line through a point not on the line A

P

— ≅ LM — 5. JM

J

13. Line perpendicular

Q

C

J

In Exercises 13 and 14, write a proof to verify that the construction is valid. (See Example 4.)

R

1 F

2 E

L

D

Section 12.7

Using Congruent Triangles

N

637

16. Prove

△PUX ≅ △QSY

20. THOUGHT PROVOKING The Bermuda Triangle is a

region in the Atlantic Ocean in which many ships and planes have mysteriously disappeared. The vertices are Miami, San Juan, and Bermuda. Use the Internet or some other resource to find the side lengths, the perimeter, and the area of this triangle (in miles). Then create a congruent triangle on land using cities ass vvertices. erti er tice ces. s..

Q

P T S

R

U

V

W X

Y

Bermuda

17. MODELING WITH MATHEMATICS Explain how to find

the distance across the canyon. (See Example 2.) Miami, FL D

E San Juan, Puerto Rico

B C

A

21. MAKING AN ARGUMENT Your friend claims that

△WZY can be proven congruent to △YXW using the HL Congruence W Theorem. Is your friend correct? Explain your reasoning.

18. HOW DO YOU SEE IT?

Use the tangram puzzle. a. Which triangle(s) have an area that is twice the area of the purple triangle?

Z

conditional statement is true or false. If the statement is false, rewrite it as a true statement using the converse, inverse, or contrapositive. a. If two triangles have the same perimeter, then they are congruent.

19. PROOF Prove that the green triangles in the Jamaican

b. If two triangles are congruent, then they have the same area.

— # BC — and E is the midpoint flag are congruent if AD —. of AC A

23. ATTENDING TO PRECISION Which triangles are

B

congruent to △ABC? Select all that apply. F

E

D

A

C

B

Maintaining Mathematical Proficiency Find the perimeter of the polygon with the given vertices.

Congruent Triangles

E

J

G

24. A(−1, 1), B(4, 1), C(4, −2), D(−1, −2)

M

C D

Chapter 12

Y

22. CRITICAL THINKING Determine whether each

b. How many times greater is the area of the orange triangle than the area of the purple triangle?

638

X

K Q

H

N

Reviewing what you learned in previous grades and lessons

(Section 8.4)

25. J(−5, 3), K(−2, 1), L(3, 4)

L

P

12.8 Coordinate Proofs Essential Question

How can you use a coordinate plane to write

a proof?

Writing a Coordinate Proof Work with a partner. a. Use dynamic geometry — software to draw AB with endpoints A(0, 0) and B(6, 0). b. Draw the vertical line x = 3. c. Draw △ABC so that C lies on the line x = 3. d. Use your drawing to prove that △ABC is an isosceles triangle.

Sample

4

3

C

2

1

B

0

A

0

1

3

2

4

5

Points A(0, 0) B(6, 0) C(3, y) Segments AB = 6 Line x=3

6

−1

Writing a Coordinate Proof Work with a partner. — with endpoints A(0, 0) and B(6, 0). a. Use dynamic geometry software to draw AB b. Draw the vertical line x = 3. c. Plot the point C(3, 3) and draw △ABC. Then use your drawing to prove that △ABC is an isosceles right triangle.

Sample

4

Points A(0, 0) B(6, 0) C(3, 3) Segments AB = 6 BC = 4.24 AC = 4.24 Line x=3

C

3

2

CRITIQUING THE REASONING OF OTHERS To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.

1

B

0

A

0

1

2

3

4

5

6

−1

d. Change the coordinates of C so that C lies below the x-axis and △ABC is an isosceles right triangle. e. Write a coordinate proof to show that if C lies on the line x = 3 and △ABC is an isosceles right triangle, then C must be the point (3, 3) or the point found in part (d).

Communicate Your Answer 3. How can you use a coordinate plane to write a proof? 4. Write a — coordinate proof to prove that △ABC with vertices A(0, 0), B(6, 0), and

C(3, 3√ 3 ) is an equilateral triangle.

Section 12.8

Coordinate Proofs

639

12.8 Lesson

What You Will Learn Place figures in a coordinate plane.

Core Vocabul Vocabulary larry coordinate proof, p. 640

Write coordinate proofs.

Placing Figures in a Coordinate Plane A coordinate proof involves placing geometric figures in a coordinate plane. When you use variables to represent the coordinates of a figure in a coordinate proof, the results are true for all figures of that type.

Placing a Figure in a Coordinate Plane Place each figure in a coordinate plane in a way that is convenient for finding side lengths. Assign coordinates to each vertex. a. a rectangle

b. a scalene triangle

SOLUTION It is easy to find lengths of horizontal and vertical segments and distances from (0, 0), so place one vertex at the origin and one or more sides on an axis. a. Let h represent the length and k represent the width. (0, k)

y

b. Notice that you need to use three different variables. y

(h, k)

(f, g)

k (0, 0)

(h, 0) x

h

Monitoring Progress

(0, 0)

(d, 0) x

Help in English and Spanish at BigIdeasMath.com

1. Show another way to place the rectangle in Example 1 part (a) that is convenient

for finding side lengths. Assign new coordinates. 2. A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex.

Once a figure is placed in a coordinate plane, you may be able to prove statements about the figure.

Writing a Plan for a Coordinate Proof Write a plan to prove that !!!⃗ SO bisects ∠PSR. Given

Coordinates of vertices of △POS and △ROS

Prove

!!!⃗ SO bisects ∠PSR.

y

S(0, 4) 2

P(−3, 0)

R(3, 0) O(0, 0) 4 x

SOLUTION Plan for Proof Use the Distance Formula to find the side lengths of △POS and △ROS. Then use the SSS Congruence Theorem to show that △POS ≅ △ROS. Finally, use the fact that corresponding parts of congruent triangles are congruent to conclude that SO bisects ∠PSR. ∠PSO ≅ ∠RSO, which implies that !!!⃗ 640

Chapter 12

Congruent Triangles

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

3. Write a plan for the proof.

Given

!!!⃗ GJ bisects ∠OGH.

Prove

△GJO ≅ △GJH

6

y

G

4 2

J

O

2

H x

4

The coordinate proof in Example 2 applies to a specific triangle. When you want to prove a statement about a more general set of figures, it is helpful to use variables as coordinates.

y

S(0, k)

For instance, you can use variable coordinates to duplicate the proof in Example 2. Once this is done, you can conclude that !!!⃗ SO bisects ∠PSR for any triangle whose coordinates fit the given pattern.

R(h, 0) P(−h, 0)

O(0, 0)

x

Applying Variable Coordinates Place an isosceles right triangle in a coordinate plane. Then find the length of the hypotenuse and the coordinates of its midpoint M.

SOLUTION Place △PQO with the right angle at the origin. Let the length of the legs be k. Then the vertices are located at P(0, k), Q(k, 0), and O(0, 0).

FINDING AN ENTRY POINT Another way to solve Example 3 is to place a triangle with point C at (0, h) on the y-axis and — on the hypotenuse AB x-axis. To make ∠ACB a right angle, position A and — and CB — B so that legs CA have slopes of 1 and −1, respectively. Slope is 1.

Slope is −1. C(0, h) y

P(0, k)

y

M O(0, 0)

Q(k, 0) x

Use the Distance Formula to find PQ, the length of the hypotenuse. ——

—

—

—

PQ = √(k − 0)2 + (0 − k)2 = √k2 + (−k)2 = √ k2 + k2 = √2k2 Use the Midpoint Formula to find the midpoint M of the hypotenuse.

(

) ( )

0+k k+0 k k M —, — = M —, — 2 2 2 2

—

So, the length of the hypotenuse is √2k2 and the midpoint of the hypotenuse is

( , ). k k 2 2

— —

A(−h, 0)

B(h, 0) x

Length of hypotenuse = 2h

(

)

−h + h 0 + 0 M —, — = M(0, 0) 2 2

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

4. Graph the points O(0, 0), H(m, n), and J(m, 0). Is △OHJ a right triangle? Find the

side lengths and the coordinates of the midpoint of each side. Section 12.8

Coordinate Proofs

641

Writing Coordinate Proofs Writing a Coordinate Proof Write a coordinate proof.

y

Given

Coordinates of vertices of quadrilateral OTUV

T(m, k)

Prove

△OTU ≅ △UVO

U(m + h, k)

SOLUTION

— and UT — have the same length. Segments OV

O(0, 0)

V(h, 0)

x

OV = ∣ h − 0 ∣ = h UT = ∣ (m + h) − m ∣ = h

— and OV — each have a slope of 0, which implies that they are Horizontal segments UT — — and OV — to form congruent alternate interior angles, parallel. Segment OU intersects UT — ≅ OU —. ∠TUO and ∠VOU. By the Reflexive Property of Congruence, OU So, you can apply the SAS Congruence Theorem to conclude that △OTU ≅ △UVO.

Writing a Coordinate Proof You buy a tall, three-legged plant stand. When you place a plant on the stand, the stand appears to be unstable under the weight of the plant. The diagram at the right shows a coordinate plane superimposed on one pair of the plant stand’s legs. The legs are extended to form △OBC. Prove that △OBC is a scalene triangle. Explain why the plant stand may be unstable.

SOLUTION

y

B(12, 48)

36 24 12

O(0, 0)

First, find the side lengths of △OBC. ——

48

12

C(18, 0)

x

—

OB = √(48 − 0)2 + (12 − 0)2 = √ 2448 ≈ 49.5 ——

—

BC = √(18 − 12)2 + (0 − 48)2 = √ 2340 ≈ 48.4 OC = ∣ 18 − 0 ∣ = 18 Because △OBC has no congruent sides, △OBC is a scalene triangle by definition. — is longer than BC —, so the plant stand The plant stand may be unstable because OB is leaning to the right.

Monitoring Progress 5. Write a coordinate proof.

Given

Coordinates of vertices of △NPO and △NMO

Prove

△NPO ≅ △NMO

Help in English and Spanish at BigIdeasMath.com y

P(0, 2h) N(h, h)

O(0, 0)

642

Chapter 12

Congruent Triangles

M(2h, 0) x

12.8 Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY How is a coordinate proof different from other types of proofs you have studied?

How is it the same? 2. WRITING Explain why it is convenient to place a right triangle on

y

the grid as shown when writing a coordinate proof.

(0, b)

(a, 0) x

(0, 0)

Monitoring Progress and Modeling with Mathematics In Exercises 3–6, place the figure in a coordinate plane in a convenient way. Assign coordinates to each vertex. Explain the advantages of your placement. (See Example 1.)

In Exercises 9–12, place the figure in a coordinate plane and find the indicated length. 9. a right triangle with leg lengths of 7 and 9 units;

Find the length of the hypotenuse.

3. a right triangle with leg lengths of 3 units and 2 units 10. an isosceles triangle with a base length of 60 units and 4. a square with a side length of 3 units 5. an isosceles right triangle with leg length p

a height of 50 units; Find the length of one of the legs. 11. a rectangle with a length of 5 units and a width of

4 units; Find the length of the diagonal. 6. a scalene triangle with one side length of 2m 12. a square with side length n; Find the length of

In Exercises 7 and 8, write a plan for the proof. (See Example 2.) Coordinates of vertices of △OPM and △ONM Prove △OPM and △ONM are isosceles triangles.

7. Given

y

P(3, 4)

4

M(8, 4)

In Exercises 13 and 14, graph the triangle with the given vertices. Find the length and the slope of each side of the triangle. Then find the coordinates of the midpoint of each side. Is the triangle a right triangle? isosceles? Explain. (Assume all variables are positive and m ≠ n.) (See Example 3.) 13. A(0, 0), B(h, h), C(2h, 0)

2

O(0, 0)

8. Given

the diagonal.

N(5, 0) 8 x

In Exercises 15 and 16, find the coordinates of any unlabeled vertices. Then find the indicated length(s).

—. G is the midpoint of HF

Prove △GHJ ≅ △GFO

15. Find ON and MN. y

y 4

14. D(0, n), E(m, n), F(m, 0)

H(1, 4)

J(6, 4)

y

N

O(0, 0) D(h, 0) M(2h, 0) x O(0, 0)

2

4

S

T

R

k units

G

2

16. Find OT.

O

2k units U

x

F(5, 0) x

Section 12.8

Coordinate Proofs

643

PROOF In Exercises 17 and 18, write a coordinate proof. (See Example 4.)

Coordinates of vertices of △DEC and △BOC Prove △DEC ≅ △BOC

17. Given

y

D(h, 2k)

B(h, 0)

D ○

— —

( ) ( w2 , −2v )

A (h, k) ○

B (−h, 0) ○

C (2h, 0) ○

D (2h, k) ○

24. THOUGHT PROVOKING Choose one of the theorems

you have encountered up to this point that you think would be easier to prove with a coordinate proof than with another type of proof. Explain your reasoning. Then write a coordinate proof.

y

A(0, 2k) G O E(2h, 0) x

25. CRITICAL THINKING The coordinates of a triangle

are (5d, −5d ), (0, −5d ), and (5d, 0). How should the coordinates be changed to make a coordinate proof easier to complete?

19. MODELING WITH MATHEMATICS You and your

cousin are camping in the woods. You hike to a point that is 500 meters east and 1200 meters north of the campsite. Your cousin hikes to a point that is 1000 meters east of the campsite. Use a coordinate proof to prove that the triangle formed by your position, your cousin’s position, and the campsite is isosceles. (See Example 5.)

26. HOW DO YOU SEE IT? Without performing any

calculations, how do you know that the diagonals of square TUVW are perpendicular to each other? How can you use a similar diagram to show that the diagonals of any square are perpendicular to each other? 3

y

T(0, 2)

W(−2, 0)

U(2, 0)

−4

20. MAKING AN ARGUMENT Two friends see a drawing

of quadrilateral PQRS with vertices P(0, 2), Q(3, −4), R(1, −5), and S(−2, 1). One friend says the quadrilateral is a parallelogram but not a rectangle. The other friend says the quadrilateral is a rectangle. Which friend is correct? Use a coordinate proof to support your answer.

21. MATHEMATICAL CONNECTIONS Write an algebraic

Maintaining Mathematical Proficiency

−3

Chapter 12

Congruent Triangles

x

V(0, −2)

27. PROOF Write a coordinate proof for each statement.

a. The midpoint of the hypotenuse of a right triangle is the same distance from each vertex of the triangle.

Reviewing what you learned in previous grades and lessons

$$$$⃗ YW bisects ∠XYZ such that m∠XYW = (3x − 7)° and m∠WYZ = (2x + 1)°. 28. Find the value of x.

4

b. Any two congruent right isosceles triangles can be combined to form a single isosceles triangle.

expression for the coordinates of each endpoint of a line segment whose midpoint is the origin.

644

—

—

width of k has a vertex at (−h, k). Which point cannot be a vertex of the rectangle?

x

Coordinates of △DEA, H is the midpoint —, G is the midpoint of EA —. of DA — ≅ EH — Prove DG

D(−2h, 0)

( ) ( −w2 , 2v )

23. REASONING A rectangle with a length of 3h and a

18. Given

H

(w, 0), (0, v), (−w, 0), and (0, −v). What is the midpoint of the side in Quadrant III? w v w v —, — A B −—, −— ○ ○ 2 2 2 2

C ○

E(2h, 2k)

C(h, k) O(0, 0)

22. REASONING The vertices of a parallelogram are

29. Find m∠XYZ.

(Section 8.5)

12.5–12.8

What Did You Learn?

Core Vocabulary legs (of a right triangle), p. 620 hypotenuse (of a right triangle), p. 620 coordinate proof, p. 640

Core Concepts Section 12.5 Side-Side-Side (SSS) Congruence Theorem, p. 618 Hypotenuse-Leg (HL) Congruence Theorem, p. 620

Section 12.6 Angle-Side-Angle (ASA) Congruence Theorem, p. 626 Angle-Angle-Side (AAS) Congruence Theorem, p. 627

Section 12.7 Using Congruent Triangles, p. 634

Section 12.8 Placing Figures in a Coordinate Plane, p. 640 Writing Coordinate Proofs, p. 642

Mathematical Practices 1.

Write a simpler problem that is similar to Exercise 22 on page 623. Describe how to use the simpler problem to gain insight into the solution of the more complicated problem in Exercise 22.

2.

Make a conjecture about the meaning of your solutions to Exercises 21–23 on page 631.

3.

Identify at least two external resources that you could use to help you solve Exercise 20 on page 638.

Performance Task:

Congruence in Design Why are the wings of a hang glider congruent? Is it necessary for performance or is it only for aesthetics? How can a designer prove that triangles are congruent? To explore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.

645

12

Chapter Review 12.1

Angles of Triangles

Dynamic Solutions available at BigIdeasMath.com

(pp. 587–594)

Classify the triangle by its sides and by measuring its angles. A

The triangle does not have any congruent sides, so it is scalene. The measure of ∠B is 117°, so the triangle is obtuse.

B

The triangle is an obtuse scalene triangle.

C

1. Classify the triangle at the right by its sides and

by measuring its angles. Find the measure of the exterior angle. 2.

3.

86°

(9x + 9)°

46°

x°

45°

5x°

Find the measure of each acute angle. 4.

5.

(7x + 6)°

8x°

(6x − 7)° 7x°

12.2

Congruent Polygons (pp. 595–600)

Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts.

C

B

The diagram indicates that △ABC ≅ △FED.

F

D

Corresponding angles ∠A ≅ ∠F, ∠B ≅ ∠E, ∠C ≅ ∠D Corresponding sides

E

— ≅ FE —, BC — ≅ ED —, AC — ≅ FD — AB

A

M G

6. In the diagram, GHJK ≅ LMNP. Identify all pairs of

H

congruent corresponding parts. Then write another congruence statement for the quadrilaterals.

J

7. Find m∠V. U

T

646

Chapter 12

L

K

Congruent Triangles

S

Q

74° V

R

P

N

12.3

Proving Triangle Congruence by SAS (pp. 601–606) B

Write a proof.

— ≅ EC —, BC — ≅ DC — Given AC △ABC ≅ △EDC

Prove

C

A

E

D

STATEMENTS

REASONS

— ≅ EC — 1. AC

1. Given

— — 2. BC ≅ DC

2. Given

3. ∠ACB ≅ ∠ECD

3. Vertical Angles Congruence Theorem

4. △ABC ≅ △EDC

4. SAS Congruence Theorem

Decide whether enough information is given to prove that △WXZ ≅ △YZX using the SAS Congruence Theorem. If so, write a proof. If not, explain why. 8.

W

Z

12.4

9.

X

W

X

Z

Y

Y

Equilateral and Isosceles Triangles (pp. 607–614)

— ≅ LN —. Name two congruent angles. In △LMN, LM

L

— ≅ LN —, so by the Base Angles Theorem, LM ∠M ≅ ∠N.

M

Copy and complete the statement.

— ≅ QR —, then ∠__ ≅ ∠__. 10. If QP

P

N R

T

11. If ∠TRV ≅ ∠TVR, then ___ ≅ ___.

— ≅ RS —, then ∠__ ≅ ∠__. 12. If RQ

13. If ∠SRV ≅ ∠SVR, then ___ ≅ ___.

Q

S

V

14. Find the values of x and y in the diagram. 26

8x°

Chapter 12

5y + 1

Chapter Review

647

12.5

Proving Triangle Congruence by SSS (pp. 617–624)

Write a proof. Given

— ≅ CB —, AB — ≅ CD — AD

Prove

△ABD ≅ △CDB

STATEMENTS

— ≅ CB — 1. AD

— — — — 3. BD ≅ DB 2. AB ≅ CD

4. △ABD ≅ △CDB

A

B

D

REASONS

C

1. Given 2. Given 3. Reflexive Property of Congruence 4. SSS Congruence Theorem

15. Decide whether enough information is given to prove that △LMP ≅ △NPM using the

SSS Congruence Theorem. If so, write a proof. If not, explain why. M L

N P

16. Decide whether enough information is given to prove that △WXZ ≅ △YZX using the

HL Congruence Theorem. If so, write a proof. If not, explain why.

12.6

W

X

Z

Y

Proving Triangle Congruence by ASA and AAS (pp. 625–632) B

Write a proof.

— ≅ DE —, ∠ABC ≅ ∠DEC Given AB Prove △ABC ≅ △DEC

648

STATEMENTS

REASONS

1.

1. Given

— ≅ DE — AB

C A

2. ∠ABC ≅ ∠DEC

2. Given

3. ∠ACB ≅ ∠DCE

3. Vertical Angles Congruence Theorem

4. △ABC ≅ △DEC

4. AAS Congruence Theorem

Chapter 12

Congruent Triangles

D

E

Decide whether enough information is given to prove that the triangles are congruent using the AAS Congruence Theorem. If so, write a proof. If not, explain why. 17. △EFG, △HJK F

18. △TUV, △QRS K

E

H

G

J

U

V

T

Q

S

R

Decide whether enough information is given to prove that the triangles are congruent using the ASA Congruence Theorem. If so, write a proof. If not, explain why. 19. △LPN, △LMN P

W

X

M

Z

Y

N

L

12.7

20. △WXZ, △YZX

Using Congruent Triangles (pp. 633–638)

D

Explain how you can prove that ∠A ≅ ∠D.

B

If you can show that △ABC ≅ △DCB, then you will know that — ≅ DB — and ∠ACB ≅ ∠DBC. You ∠A ≅ ∠D. You are given AC — — know that BC ≅ CB by the Reflexive Property of Congruence. Two pairs of sides and their included angles are congruent, so by the SAS Congruence Theorem, △ABC ≅ △DCB. A

C

Because corresponding parts of congruent triangles are congruent, ∠A ≅ ∠D. 21. Explain how to prove that ∠K ≅ ∠N. M

K H

L J

N

22. Write a plan to prove that ∠1 ≅ ∠2.

Q P

R 1 U

S

V

T

2 W

Chapter 12

Chapter Review

649

12.8

Coordinate Proofs (pp. 639–644)

Write a coordinate proof. Given

y

Coordinates of vertices of △ODB and △BDC

Prove △ODB ≅ △BDC

C(2j, 2j) D(j, j)

O(0, 0)

B(2j, 0) x

— and BD — have the same length. Segments OD ——

—

—

OD = √ ( j − 0)2 + ( j − 0)2 = √ j2 + j2 = √ 2j2 ——

—

—

BD = √ ( j − 2j)2 + ( j − 0)2 = √ (−j)2 + j 2 = √ 2j2

— and DC — have the same length. Segments DB —

DB = BD = √2j2

——

—

—

DC = √ (2j − j)2 + (2j − j)2 = √ j2 + j2 = √2j2

— and BC — have the same length. Segments OB OB = ∣ 2j − 0 ∣ = 2j BC = ∣ 2j − 0 ∣ = 2j So, you can apply the SSS Congruence Theorem to conclude that △ODB ≅ △BDC. 23. Write a coordinate proof.

Given

Coordinates of vertices of quadrilateral OPQR

Prove △OPQ ≅ △QRO y

Q(h, k + j)

R(0, j)

P(h, k) O(0, 0)

x

24. Place an isosceles triangle in a coordinate plane in a way that is convenient

for finding side lengths. Assign coordinates to each vertex. 25. A rectangle has vertices (0, 0), (2k, 0), and (0, k). Find the fourth vertex.

650

Chapter 12

Congruent Triangles

12

Chapter Test

Write a proof.

— ≅ CB — ≅ CD — ≅ CE — 1. Given CA

— " ML —, MJ — " KL — 2. Given JK

Prove △ABC ≅ △EDC A

— ≅ RS —, ∠P ≅ ∠T 3. Given QR

△MJK ≅ △KLM

Prove

△SRP ≅ △QRT

Prove

R

B

J

K

Q

C

S N

D

E

M

P

L

4. Find the measure of each acute angle in the figure at the right.

T

(4x − 2)°

5. Is it possible to draw an equilateral triangle that is not equiangular?

If so, provide an example. If not, explain why.

(3x + 8)°

6. Can you use the Third Angles Theorem to prove that two triangles

are congruent? Explain your reasoning. Write a plan to prove that ∠1 ≅ ∠2. 7.

A

1

2

8.

B

R

T 1

S

2

V

Z

D C

X

E

W

9. Is there more than one theorem that could be used to prove

that △ABD ≅ △CDB? If so, list all possible theorems.

Y A B

D

y

P

Q

C

24

10. Write a coordinate proof to show that the triangles

18

created by the keyboard stand are congruent.

R

12 6

T

S 6

12

18

24

3

x

11. The picture shows the Pyramid of Cestius, which is located in Rome, Italy.

The measure of the base for the triangle shown is 100 Roman feet. The measures of the other two sides of the triangle are both 144 Roman feet. a. Classify the triangle shown by its sides. b. The measure of ∠3 is 40°. What are the measures of ∠1 and ∠2? Explain your reasoning.

1

Chapter 12

Chapter Test

2

651

12

Cumulative Assessment

1. The graph of which inequality is shown? 6

A x + 2y < 4 ○

y

4

B x + 2y ≤ 4 ○ C x + 2y > 4 ○

x −4

D x + 2y ≥ 4 ○

−2

4

2 −2

2. Use the steps in the construction to explain how you know that the line through point P

is parallel to line m.

P m

Step 2

Step 1

P

P Q

Step 3 C

P

A

m

Step 4

A

m

QB

C

P

D

QB

m

A QB

C D m

3. The coordinate plane shows △JKL and △XYZ. K

4

J −4

y

L −2

2 −2

4 x

X

−4

Z

Y

a. Write a composition of transformations that maps △JKL to △XYZ. b. Is the composition a congruence transformation? If so, identify all congruent corresponding parts. 4. Which two equations form a system of linear equations that has infinitely many solutions?

x + 2y = 6

652

Chapter 12

2x + y = 6

2x + 4y = 12

Congruent Triangles

x − 2y = −6

5. The coordinate plane shows △ABC and △DEF.

8

a. Prove △ABC ≅ △DEF using the given information.

y

A C

6

b. Describe the composition of rigid motions that maps △ABC to △DEF.

F B

4 2

D 2

4

E 6

8 x

6. Which figure(s) have rotational symmetry? Select all that apply.

A ○

B ○

C ○

D ○

7. Write a coordinate proof.

Given Coordinates of vertices of quadrilateral ABCD Prove Quadrilateral ABCD is a rectangle. 8

y

B

6

A

4

C

2

D 2

4

6

8 x

8. Write a proof to verify that the construction of the equilateral triangle shown below

is valid. Step 2

Step 1

Step 3

Step 4 C

A

B

A

B

A

Chapter 12

C

B

A

B

Cumulative Assessment

653