geo book 8.5 key

Proportions in Triangles GEOMETRY LESSON 8-5

(For help, go to Lesson 8-2.)

Find the value of x. The triangles in each are similar.

1.

2.

3.

4.

Proportions in Triangles GEOMETRY LESSON 8-5

Solutions 1. Simplify 30 + 15 in 84 = 30 + 15 : 84 = 3 ; 84(1) = (3)x; 15

x

15

x

1

divide by 3: x = 28 cm 2. Write a proportion: x = x + 8 ; 4(x + 8) = 22x; Simplify: 4x + 32 = 22x: 4

22

Subtract 4x: 32 = 18x; divide by 18; x = 32 = 16 or 1 7 mm 18

9

9

3. Write a proportion: 7 = 7 + x ; 7(12) = 5(7 + x); Simplify: 84 = 35 + 5x; 5

12

Subtract 35: 49 = 5x; divide by 5: x = 49 or 9.8 in.

5

2 7.5 4. Simplify 6 +63 in 6 +63 = 7.5 : = ; (2)x = 7.5(3); x 3 x Simplify: 2x = 22.5; divide by 2: x = 11.25 ft

Proportions in Triangles GEOMETRY LESSON 8-5

Find y.

CN CM = NA MB 12 = 12 y 6

10y = 72 y = 7.2

Proportions in Triangles GEOMETRY LESSON 8-5

The segments joining the sides of trapezoid RSTU are parallel to its bases. Find x and y.

Because RSTU is a trapezoid, RS || UT. 5 6 = x 12.5

5x = 75 x = 15 12.5 x = 9 y 12.5 15 = 9 y

15y = 112.5 y = 7.5 x = 15 and y = 7.5.

Proportions in Triangles GEOMETRY LESSON 8-5

Find the value of x.

IG IK = GH HK 24 x = 40 30 24 (30) = x 40

x = 18

Proportions in Triangles GEOMETRY LESSON 8-5

Pages 448-452 Exercises 1. 7.5

9. 3

1 3

17. KS

2. 8

10. 9.6

18. SQ

3. 5.2

11. 6

19. JP

4. d

12. 4.8

20. KP

5. c

13. 35

21. KM

6. b

14. 3.6

22. PM

7. d

15. 7

23. JP

8. 7.5

16. 12

24. LW

40

Proportions in Triangles GEOMETRY LESSON 8-5

25. 559 ft

31. 20

26. 671 ft

32. 2.5

27. 3.8 cm and 9.2 cm

33. 9

28. Answers may vary.

Sample: 9 cm and 13.5 cm

34. a. AB

29. x = 18 m; y = 12 m

c.

30. a.

b. isosceles; Bisector Thm.

36. 4.5 cm or 12.5 cm 37. 6 38. 2.5

BC WX b.

XY AB WX = BC XY

35. Measure AC, CE, and BD. Use the SideSplitter Thm. Write the proport. AC = AB and CE BD solve for AB.

39. 19.5 40. h = 10.0, h1 = 4.3, h2 = 5.7 41. h = 13.0, 2 = 11.9

42.

1

= 5.3,

= 5.0, h1 = 3.8, h2 = 9.2 1

Proportions in Triangles GEOMETRY LESSON 8-5

43.

= 7.1, h2 = 5.0, h = 10.0 2

44. h = 17.0, h1 = 11.1, h2 = 5.9

46. The ratio of the legs of a 30째-60째-90째 is 3. Let the rt. bisector divide the hyp. into segments x and y. Then by the - - Bis. Thm., 3, so x =

2. Prop. of Proportions

3 y.

28

5. SAS ~ Thm.

7. If corr. s are lines are ||.

s

s

24

49. No; 12 =/ 10 .

4. Reflexive Prop. of

6. Corr. s of ~ are .

9

48. Yes; since 10 = 15 , the segments are || by the Converse of the Side-Splitter Thm.

3. Segment Add. Post.

45. h2 = 4, 1 = 5.7, 2 = 5.7

x = y

6

47. 1. Given

,

15 20 50. Yes; since 12 = 16 ,

the segments are || by the Converse of the Side-Splitter Thm.

Proportions in Triangles GEOMETRY LESSON 8-5

51. a. A midsegment of a connects the midpts. of 2 opp. sides.

b.

51. (continued) 1.

ABCD (Given) 2. AE || BF and ED || FC (Def. of

51. (continued) 6. AE = BF, ED = FC (Subst.) )

3. AD BC (Opp. sides of are .)

Given: ABCD with EF connecting the midpts. of AD and BC Prove: AB || EF; EF || CD

4. E and F are midpts. of AD and BC. (Given) 1 2 BF = FC = 1 BC 2

5. AE = ED = AD;

(Def. of midpt.)

7. ABFE and EFCD are s (If one pair of opp. sides of a quad. is and ||, it is a .)

8. AB || EF and EF || CD (Opp. sides of a are ||.)

Proportions in Triangles GEOMETRY LESSON 8-5

51. (continued) c.

55. [4] a.

52. D 53. F

54. [2] n + 1 = 20 ; 28

35

35n + 35 = 560; Given: ABCD with midsegment EF Prove: EF bisects AC and BD. Since AB || EF || DC by part (b), and EF bisects AD, by the Side-Splitter Thm., EF bisects AC and BD.

n = 15 [1] correct proportion solved incorrectly b. x = 6 4

5

x = 4.8 cm 6 y = 4 5

y = 7.5 cm

Proportions in Triangles GEOMETRY LESSON 8-5

55. (continued) [3] correct drawings and proportions with one computational error

56. m

65. RT = SV = 48

57. m

66. RT = SV = 6.5

58. c 59. h

[2] one possibility drawn and done correctly OR two correct drawings [1] one correct drawing OR one correct proportion

60. x = 24; y = 12 61. x = 9; y = 9

3

62. x = 15; y = 30 63. RT = SV = 38 64. RT = SV = 37

3

Proportions in Triangles GEOMETRY LESSON 8-5

Solve for x in each diagram.

1.

2.

3.

11.25 8

24

Use the diagram for Exercises 4–6.

4. Find x. 9.6 5. Find y. 5 6. Explain how you know that AM is not the angle bisector of BAC.

If it were, AB would equal BM = 1. But AC > AB. AC

MC