Similar Polygons GEOMETRY LESSON 8-2

(For help, go to Lessons 4-1 and 8-1.)

Solve each proportion.

1.

ABC

2. 3 = x 4

8

HIJ. Name three pairs of congruent sides.

3. 2 = 8 x

24

4. x = 1 9

3

5. 10 = 2 25

x

Similar Polygons GEOMETRY LESSON 8-2

Solutions 1.Choose pairs of letters in the same position on each part of the congruence statement

ABC

HIJ: AB

2. Multiply both sides of 3 = x by 8: x = 3 4

8

4

HI, AC 8 = 3 1 1

HJ, BC 2 =6 1

3. Simplify 8 in 2 = 8 : 2 = 1 ; use the Cross Product Property: 24

x

24

x

3

(1)x = (2)(3); simplify: x = 6

4. Multiply both sides of x = 1 by 9: x = 1 9

3

3

9 = 9 =3 1 3

5. Simplify 10 in 10 = 2 : 2 = 2 ; use the Cross Product Property: 25

25

x

5

x

(2)x = (5)(2); simplify: 2x = 10; divide by 2: x = 5

IJ

Similar Polygons GEOMETRY LESSON 8-2

ABC ~

XYZ Complete each statement. a. m B = ? b. BC = ? YZ

XZ

Two polygons are similar if (1) corresponding angles are congruent and (2) corresponding sides are proportional. a.

B Y and m Y = 78, so m B = 78 because congruent angles have the same measure.

b. Because AC corresponds to XZ, BC = AC. YZ

XZ

Similar Polygons GEOMETRY LESSON 8-2

Determine whether the parallelograms are similar. Explain.

Check that the corresponding sides are proportional. AB 2 = JK 4

BC 1 = KL 2

CD 2 = LM 4

DA 1 = MJ 2

Corresponding sides of the two parallelograms are proportional. Check that corresponding angles are congruent.

B corresponds to congruent.

K, but m B =/ m K, so corresponding angles are not

Although corresponding sides are proportional, the parallelograms are not similar because the corresponding angles are not congruent.

Similar Polygons GEOMETRY LESSON 8-2

If

Because AC BC = YZ XZ x = 12 40 30

x = 12 30

x = 16

40

ABC ~

ABC ~

YXZ, find the value of x.

YXZ, you can write and solve a proportion.

Similar Polygons GEOMETRY LESSON 8-2

A painting is 24 in. wide by 36 in. long. The length of a postcard reduction of the painting is 6 in. How wide is the postcard? The postcard and the painting are similar rectangles, so you can write a proportion. Let x represent the width of the postcard. Postcard width = postcard length Painting width painting length x 6 = 24 36 6

x = 36

24

x=4 The postcard is 4 in. wide.

Similar Polygons GEOMETRY LESSON 8-2

The dimensions of a rectangular tabletop are in the Golden Ratio. The shorter side is 40 in. Find the longer side. Let 40

represent the longer side of the tabletop.

= 1.1618 1

= 64.72

The table is about 65 in. long.

Similar Polygons GEOMETRY LESSON 8-2

Pages 425-429 Exercises 1.

JHY

2. 3.

9. yes; KLMJ ~ PQNO; 3

17. 6.6 in. by 11 in.

R

5 10. yes; ABCD ~ FGHE; 4 5

18. 3.6 in. by 6 in.

JXY

11. No; corr.

19. 70 mm

s

are not

7 5

4. HY

12. yes;

5. JT

13. x = 4; y = 3

21. 2 : 3

6. HY

14. x = 20; y = 17.5; z = 7.5

22. 3 : 2

7. no;

20 36 = / 30 52

8. yes; QRST ~ XWZY; 3

4

ABC ~ FED;

.

15. x = 16; y = 4.5; z = 7.5 16. x = 6; y = 8; z = 10

20. 54 in. by 87.37 in.

23. 50 24. 50

Similar Polygons GEOMETRY LESSON 8-2

25. 70

31. 20 ft

2

26. 3

s

of

41. sides of 2 cm; 60° and 120°

s

of

32. x = 60, y = 25

27. 7.5 m

33. 2.6 cm

28. 5.6 m

34. 3 : 4

29. Yes; corr. sides are

40. sides of 2 cm; 60° and 120°

s

and

s

of

43. sides of 0.8 cm; 60° and 120°

s

of

35. 3 : 1

. 36. 2 : 1

30. equal sign, similarity symbol; Answers may vary. Sample: figures are similar with = areas.

42. sides of 3.2 cm; 60° and 120°

37. 1 : 2

44. sides of 1 cm; 60° and 120°

s

of

45. sides of 3 cm; 60° and 120°

s

of

38. 4 : 3 39. 2 : 3

Similar Polygons GEOMETRY LESSON 8-2

46. 16.2 in.

50. Answers may vary. Sample:

47. 6.2 in. 48. No; corr. sides are not in proportion. 49. Yes; explanations may vary. Sample: The ratios of radii, diameters, and circumferences of 2 circles are =.

51. a. (1) Corr. sides of ~ polygons are proport. (2) Subst. (3) Cross-Product Prop. (4) Subtr. Prop.

52. a. 21, 34, 55, 89, 144, 233, 377 b. 1.6; 1.625; 1.6154; 1.6190; 1.6176; 1.6182; 1.6180; 1.6181; 1.6180 c. The ratios get closer to the golden ratio. 53. D

b. Length cannot be negative.

54. C

c. 1.6180

55. A

Similar Polygons GEOMETRY LESSON 8-2

56. [2] a. 12 cm

61. no; only one pair of || sides

b. m KLM = 38 [1] incorrect length OR incorrect angle measure

62. Yes; both pairs of sides are ||. 63.

CEA, BCD

57. 7y 64. BD, FA

58. 7 9

65. 8

59. y + 9 9

60. Yes; the diagonals bisect each other.

66. 69

FED,

Similar Polygons GEOMETRY LESSON 8-2

Use the trapezoids below for Exercises 1â€“3. DFHN ~ BMLP. Complete each statement. 1. m H = 2. x =

?

3. m D =

74

? 49 ?

99

4. A 4-in. by 6-in. drawing is enlarged to fit on a poster that measures 20 in. by 24 in. What are the dimensions of the largest drawing possible? 16 in. by 24 in. 5. A rectangle with a perimeter 20 cm has a side 4 cm long. A rectangle with perimeter 40 cm has a side 8 cm long. Determine whether the rectangles are similar. If they are, give the similarity ratio. If they are not, explain. yes; 1 : 2 6. The longer side of the golden rectangle is 20 ft. Find the length of the shorter side, rounded to the nearest tenth. 12.4 ft