4.4 Exercises - web.lincoln.k12.mi.us

4.4 Exercises 1. Sketch a right ... Guided Practice Extra Practice ... Example 4: Exs. 27–34 Homework Help 4.4 The Pythagorean Theorem and the Distanc...

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4.4 Exercises Guided Practice Vocabulary Check

1. Sketch a right triangle and label its vertices. Then use your

triangle to state the Pythagorean Theorem.

Skill Check

Find the unknown side length. Round your answer to the nearest tenth, if necessary. x

2.

3.

4.

x

x

4

1

8

10

8

2

Find the distance between the points. Round your answer to the nearest tenth, if necessary. 5.

6.

y

y

7.

D (4, 6)

y

B (5, 3)

G (3, 3)

1 1

x

1

A(0, 0)

4

1

C(2, 1)

x 1

F (1, 23)

x

Practice and Applications Extra Practice See p. 681.

Finding a Hypotenuse Find the length of the hypotenuse. 8.

9.

10.

c c

9

c

65

40 9

72

12 11.

12.

24 10

12 8

35

c

15

13.

c

c

Homework Help Example 1: Exs. 8–13, 26 Example 2: Exs. 14–22 Example 3: Exs. 27–34 Example 4: Exs. 27–34

Finding a Leg Find the unknown side length. 14.

25 24

15.

a

16.

b 89

39

5

Ï·· 61 b

4.4 The Pythagorean Theorem and the Distance Formula

195

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Pythagorean Triples

EXAMPLE

A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c 2 5 a 2 1 b 2. For example, the integers 3, 4, and 5 form a Pythagorean triple because 52 5 32 1 42. Find the length of the hypotenuse of the right triangle. Tell whether the side lengths form a Pythagorean triple.

c 8 15

Solution (hypotenuse) 2 5 (leg)2 1 (leg)2 2

2

Pythagorean Theorem

2

c 5 8 1 15

Substitute 8 and 15 for the legs.

2

Multiply.

2

Add.

c 5 17

Find the positive square root.

c 5 64 1 225 c 5 289

ANSWER

© Because the side lengths 8, 15, and 17 are integers, they form a Pythagorean triple.

Pythagorean Triples Find the unknown side length. Tell whether the side lengths form a Pythagorean triple. 17.

3

Architecture

18.

b

19.

6

c

7 4

5

c 11

20.

30

16

21.

22.

24

50

9

48 b

c

a

Visualize It! Tell whether the side lengths form a Pythagorean triple. If so, draw a right triangle with the side lengths. 23. 21, 29, 20 PETRONAS TOWERS

These 1483 foot buildings tower over the city of Kuala Lumpur, Malaysia. When the Petronas Towers were designed by Cesar Pelli in 1991, they were the tallest buildings in the world.

shown on page 170 are connected by a skywalk with support beams. Use the diagram to find the approximate length of each support beam.

Application Links

Chapter 4

25. 5, 12, 14

26. Support Beam The skyscrapers

CLASSZONE.COM

196

24. 25, 7, 24

Triangle Relationships

x

x

support beams

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Distance Formula Find the distance between the points. Round your answer to the nearest tenth. 27.

28.

y

29.

y

y

F (5, 5)

B (5, 2) D (6, 8)

1 1

x

1 2

A(2, 22)

C(0, 2)

E(4, 21)

2

ICLASSZONE.COM

HOMEWORK HELP Extra help with problem solving in Exs. 30–32 is at classzone.com

Congruence Graph P, Q, and R. Then use the Distance Formula to &* c QR &*. decide whether PQ 30. P(4, 24)

31. P(21, 26)

Q(1, 26) R(21, 23)

32. P(5, 1)

Q(28, 5) R(3, 22)

Q(25, 27) R(23, 6)

Sum of Distances In Exercises 33 and 34, use the map below. Sidewalks around the edge of a campus quadrangle connect the buildings. Students sometimes take shortcuts by walking across the grass along the pathways shown. The coordinate system shown is measured in yards. dorm

dorm

F (0, 30)

B (65, 30)

E (100, 30) dorm

library

IStudent Help

5 x

x

C (0, 15)

A(50, 0)

G (0, 0)

D (100, 0) dorm

classroom

dorm

33. Find the distances from A to B, from B to C, and from C to A if you

have to walk around the quadrangle along the sidewalks. 34. Find the distances from A to B, from B to C, and from C to A if you

are able to walk across the grass along the pathways. Challenge Find the value of x. Use a calculator, and round your answer to the nearest tenth. 35.

36.

37.

x

x

10

6

5

x

3 11

8

8

12

4.4 The Pythagorean Theorem and the Distance Formula

197

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Standardized Test Practice

38. Multiple Choice What is the distance from (3, 5) to (21, 24)? A Ï5 w X

B X

Ï1 w7 w

C 2Ï1 w3 w X

D Ï9 w7 w X

39. Multiple Choice Which of the following is the length of the

hypotenuse of a right triangle with legs of lengths 33 and 56? F 65 X

Mixed Review

G X

72.9

H 85.8 X

J 89 X

Finding Absolute Values Evaluate. (Skills Review, p. 662) 40.

27

41.

1.05

42.

0

43.

20.02

Solving Inequalities Solve the inequality. (Algebra Review, p. 167) 44. x 1 5 < 8

Algebra Skills

45. 10 1 x ≥ 12

46. 4x ≥ 28

47. 6x 1 11 ≤ 11

Fractions and Decimals Write the decimal as a fraction in simplest form. (Skills Review, p. 657) 48. 0.4

49. 0.08

50. 0.54

51. 0.12

52. 0.250

53. 0.173

54. 0.3 xx

55. 0.1 xx

Quiz 2 Find the value of x. (Lesson 4.3) 1.

2.

13

3x 2 5

3.

x15

558 (2x 1 1)8

4x 2 16

In Exercises 4–6, find the distance between the points. (Lesson 4.4) 4.

5.

y

6.

y

y

1

A(22, 3)

A(23, 21) 1

B (1, 2) 1

x

1

B (1, 22) 1 B (3, 0) x

1

A(21, 21)

7. A device used to measure windspeed is

attached to the top of a pole. Support wires are attached to the pole 5 feet above the ground. Each support wire is 6 feet long. About how far from the base of the pole is each wire attached to the ground? (Lesson 4.4)

198

Chapter 4

Triangle Relationships

6 ft

5 ft

x