Schaum’s outline Geometry Sixth Edition

Answers to Supplementary Problems

Chapter 1

1. (a) point; (b) line; (c) plane; (d) plane; (e) line; (f) point

2. (a) , (b) , , , ; (c) , , , ; (d) F

3. (a) AB = 16; (b) AE = 10

4. (a) 18; (b) 90°; (c) 50°; (d) 130°; (e) 230°

5. (a) ∠CBE; (b) ∠AEB; (c) ∠ABE; (d) ABC, ∠BCD, ∠BED; (e) ∠AED

6. (a) 130°; (b) 120°; (c) 75°; (d) 132°

7. (a) 75°; (b) 40°; (c) 10° or 10°20′; (d) 9°11′

8. (a) 90°; (b) 120°; (c) 135°; (d) 270°; (e) 180°

9. (a) 90°; (b) 60°; (c) 15°; (d) 165°

10. (a) ⊥ and ⊥ ; (b) 129°; (c) 102°; (d) 51°; (e) 129°

11. (a) ΔABC, hypotenuse , legs and

ΔACD, hypotenuse , legs and

ΔBCD, hypotenuse , legs and

(b) ΔDAB and ΔABC

ΔCED, legs and , base , vertex angle ∠CED

12. (a) ≅ and ∠PRA ≅ ∠PRB; (b) ∠ABF ≅ ∠CBF; (c) ∠CGA ≅ ∠CGD; (d) ≅

13. (a) vert. ; (b) comp. adj. ; (c) adj. ; (d) supp. adj. ; (e) comp. ; (f) vert.

14. (a) 25°, 65°; (b) 18°, 72°; (c) 60°, 120°; (d) 61°, 119°; (e) 50°, 130°; (f) 56°, 84°; (g) 90°, 90°

15. (a) a + b = 75°, a − b = 21°, a = 48° and b = 27°

(b) a + b = 90°, a = 3b − 10°, a = 65° and b = 25°

Chapter 2

1. (a) A is H; (b) P is D; (c) R is S; (d) E is K; (e) A is G; (f) triangles are geometric figures; (g) a rectangle is a quadrilateral

2. (a) a = c = f; (b) g = 15; (c) f = a; (d) a = f, a = h, c = h; (e) b = g, b = e, d = e

3. (a) 130; (b) 4; (c) yes; (d) x=8°; (e) y = 15; (f) x = 6; (g) x ±6

4. (a) AC = 12, AE = 11, AF = 15, DF = 9
(b) m∠ADC = 92°, m∠BAE = 68°, m∠FAD = 86°, m∠BAD = 128°

5. (a) AB = DF; (b) AB = AC; (c) ∠ECA ≅ ∠DCB; (d) ∠BAD ≅ ∠BCD

6. (a) If equals are divided by equals, the quotients are equal.

(b) Doubles of equals are equal.

(d) Halves of equals are equal.

7. (a) If equals are divided by equals, the quotients are equal.

(b) If equals are multiplied by equals, the products are equal.

(d) Halves of equals are equal.

8. (a) Their new rates of pay per hour will be the same. (Add. Post.)

(b) Those stocks have the same value now. (Mult. Post.)

(d) 100°C = 212°F (Trans. Post.)

(e) Their parts will be the same length. (Div. Post.)

(f) He has a total of $10,000 in Banks A, B, and C. (Part. Post.)

(g) Their values are the same. (Trans. Post.)

9. (a) Vertical angles are congruent.

(b) All straight angles are congruent.

(d) Perpendiculars form right angles and all right angles are congruent.

(e) Complements of congruent angles are congruent.

10. In each answer, (H) indicates the hypothesis and (C) indicates the conclusion.

(a) (H) Stars, (C) twinkle.

(b) (H) Jet planes, (C) are the speediest.

(d) (H) If it is the American flag, (C) its colors are red, white, and blue.

(e) (H) If you fail to do homework in the subject, (C) you cannot learn geometry.

(f) (H) If the umpire calls a fourth ball, (C) a batter goes to first base.

(g) (H) If A is B’s brother and C is B’s daughter, (C) then A is C’s uncle.

(h) (H) An angle bisector, (C) divides the angle into two equal parts.

(i) (H) If it is divided into three equal parts, (C) a segment is trisected.

(j) (H) A pentagon (C) has five sides and five angles.

(k) (H) Some rectangles (C) are squares.

(l) (H) If their sides are made longer, (C) angles do not become larger.

(m) (H) If they are congruent and supplementary, (C) angles are right angles.

(n) (H) If one of its sides is not a straight line segment, (C) the figure cannot be a polygon.

11. (a) An acute angle is half a right angle. Not necessarily true.

(b) A triangle having one obtuse angle is an obtuse triangle. True.

(d) If you are shorter than I, then I am taller than you. True.

(e) If our weights are unequal, then I am heavier than you. Not necessarily true.

Chapter 3

1. (a) ΔI ≅ ΔII ≅ ΔIII, SAS; (b) ΔI ≅ ΔIII, ASA; (c) ΔI ≅ ΔII ≅ ΔIII, SSS.

2. (a) ASA; (b) SAS; (c) SSS; (d) SAS; (e) ASA; (f) SAS; (g) SAS; (h) ASA.

3. (a) ≅ ; (b) ∠ABD = ∠DBC; (c) ∠1 ≅ ∠4; (d) ≅ ; (e) ≅ ; (f) ∠BAD ≅ ∠CDA

4. (a) ∠1 ≅ ∠3, ∠2 ≅ ∠4, ≅ ; (b) ≅ , ≅ , ∠B ≅ ∠C; (c) ∠E ≅ ∠C, ∠A ≅ ∠F, ∠EDF ≅ ∠ABC

5. (a) x = 19, y = 8; (b) x = 4, y = 12; (c) x = 48, y = 12

8. (a) ∠b ≅ ∠d, ∠E ≅ ∠G; (b) ∠A ≅ ∠1 ≅ ∠4, ∠2 ≅ ∠C; (c) ∠1 ≅ ∠5, ∠4 ≅ ∠6, ∠EAD ≅ ∠EDA

9. (a) ≅ ; (b) ≅ ≅ , ≅ ; (c) ≅ , ≅ , ≅

Chapter 4

1. (a) x = 105°, y = 75°; (b) x = 60°, y = 40°; (c) x = 85°, y = 95°; (d) x = 50°, y = 50°;

(e) x = 65°, y = 65°; (f) x = 40°, y = 30°; (g) x = 60°, y = 120°; (h) x = 90°, y = 35°;

(i) x = 30°, y = 40°; (j) x = 80°, y = 10°; (k) x = 30°, y = 150°; (l) x = 85°, y = 95°

2. (a) x = 22°, y = 102°; (b) x = 40°, y = 100°; (c) x = 80°, y = 40°

3. (a) Each angle measures 105°. (b) Each angle measures 70°. (c) Angles measure 72° and 1088.

7. (a) 25; (b) 9; (c) 20; (d) 8

8. (a) 8; (b) 10; (c) 2; (d) 14

10. (a) P is equidistant from B and C. P is on ⊥ bisector of .

Q is equidistant from A and B. Q is on ⊥ bisector of .

R is equidistant from A, C, and D. R is on ⊥ bisectors of and .

(b) P is equidistant from and . P is on bisector of ∠A.

Q is equidistant from and . Q is on bisector of ∠B.

R is equidistant from , , and . R is on the bisectors of ∠C and ∠D.

11. (a) P is equidistant from , , and . Q is equidistant from and and equidistant from A and D. R is equidistant from and and equidistant from A and D.

(b) P is equidistant from and and equidistant from B and C. Q is equidistant from A, B, and C.

R is equidistant from and and equidistant from A and B.

12. (a) x = 50°, y = 110°; (b) x = 65°, y = 65°; (c) x = 30°, y = 100°; (d) x = 51°, y = 112°; (e) x = 52°, y = 40°; (f) x = 120°, y = 90°

13. (a) x = 55°, y = 125°; (b) x = 80°, y = 90°; (c) x = 56°, y = 68°; (d) x = 100°, y = 30°; (e) x = 30°, y = 120°; (f) x = 90°, y = 30°

14. (a) 18°, 54°, 108°; (b) 40°, 50°, 90°; (c) 36°, 36°, 108°; (d) 36°, 72°, 108°, 144°; (e) 50°, 75°; (f) 100°, 60°, and 20°

16. (a) Since x = 45, each angle measures 60°.

(b) Since x = 25, x + 15 = 40 and 3x − 35 = 40; that is, two angles each measure 40°.

(d) If x and 5x − 10 represent the unknown angles, x = 21 and 5x − 10 = 95; that is, one of the angles measures 95°.

17. (a) 7 st. , 30 st. ; (b) 1620°, 5400°, 180,000°; (c) 30, 12, 27, 202

18. (a) 20°, 18°, 9°; (b) 160°, 162°, 171°; (c) 3, 9, 20, 180; (d) 3, 12, 36, 72, 360

19. (a) 65°, 90°, 95°, 110°; (b) 140°, 100°, 60°, 60°

20. (a) ΔI ≅ ΔIII by hy-leg; (b) ΔI ≅ ΔIII by SAA.

Chapter 5

1. (a) x = 15, y = 25; (b) x = 20, y = 130; (c) x = 20, y = 140

4. (a) EFGH; (b) ABCD and EBFD; (c) GHKJ, HILK, GILJ; (d) ACHB, CEFH

5. (a) Two sides are congruent and ‖. (b) Opposite sides are congruent. (c) Opposite angles are congruent. (d) and are congruent and parallel ( ≅ ≅ BC).

6. (a) x = 6, y = 12; (b) x = 5, y = 9; (c) x = 120, y = 30; (d) x = 15, y = 45

7. (a) x = 14, y = 6; (b) x = 18, y=4; (c) x = 8, y = 5; (d) x = 3, y = 9

10. (a) x = 5, y = 7; (b) x = 10, y = 35; (c) x=2, y = 17; (d) x = 8, y = 4; (e) x = 25, y = 25; (f) x = 11, y = 118

13. (a) x = 6, y = 40; (b) x=3, y=5; (c) x=8, y=22

14. (a) x = 28, y = 25; (b) x = 12 (since y does not join midpoints, Pr. 3 does not apply);

15. (a) m = 19; (b) b′ = 36; (c) b = 73

16. (a) x = 11, y = 33; (b) x = 32, y = 26; (c) x = 12, y = 36

17. (a) 22; (b) 70

18. (a) 21; (b) 30; (c) 14; (d) 26

Chapter 6

5. (a) square; (b) isosceles triangle; (c) trapezoid; (d) right triangle

6. (a) 140°; (b) 60°; (c) 90°; (d) (180 − x)°; (e) x°; (f) (90 + x)°

7. (a) 100°; (b) 50°, 80°; (c) 54°, 27°; (d) 45°; (e) 35°; (f) 45°

8. (a) x = 22; (b) y = 6; (c) AB + CD = 22; (d) perimeter = 44; (e) x = 21; (f) r = 14

9. (a) 0; (b) 40; (c) 33; (d) 7

10. (a) tangent externally; (b) tangent internally; (c) the circles are 5 units apart; (d) overlapping

11. (a) concentric; (b) tangent internally; (c) tangent externally; (d) outside each other; (e) the smaller entirely inside the larger; (f) overlapping

13. (a) 40; (b) 90; (c) 170; (d) 180; (e) 2x; (f) 180 − x; (g) 2x − 2y

14. (a) 20; (b) 45; (c) 85; (d) 90; (e) 130; (f) 174; (g) x; (h) 90 − x; (i) x − y

15. (a) 85; (b) 170; (c) c; (d) 2i; (e) 60; (f) 30

16. (a) 60, 120, 180; (b) 80, 120, 160; (c) 100, 120, 140; (d) 36, 144, 180

17. (a) m∠x = 136°; (b) = 11°; (c) m∠x = 130°; (d) m∠y = 126°; (e) m∠x = 110°; = 77°

18. (a) 135°; (b) 90°; (c) (180 − x)°; (d) (90 + x)°; (e) 100°; (f) 80°; (g) 55°; (h) 72°

19. (a) 85°; (b) y°; (c) 110°; (d) 95°; (e) 72°; (f) 50°; (g) 145°; (h) 87°

20. (a) 50; (b) 60

21. (a) = 65°, = 65°; (b) m∠x = 90°, m∠y = 55°; (c) m∠x = 37°, m∠y = 50°

22. (a) 19; (b) 45; (c) 69; (d) 90; (e) 125; (f) 167; (g) x; (h) 180 − x; (i) x + y

23. (a) 110; (b) 135; (c) 180; (d) 270; (e) 180 − 2x; (f) 360 − 2x; (g) 2x − 2y; (h) 7x

24. (a) 45°; (b) 60°; (c) 30°; (d) 18°

25. (a) =120°, m∠y = 60°; (b) m∠x = 62°, m∠y = 28°; (c) m∠x = 46°, m∠y = 588

26. (a) 75°; (b) 75°; (c) 115°; (d) 100°; (e) 140°; (f) 230°; (g) 80°; (h) 48°

27. (a) 85°; (b) 103°; (c) 80°; (d) 72°; (e) 90°; (f) 110°; (g) 130°; (h) 110°

28. (a) =68°, m∠y = 95°; (b) m∠x = 90°, m∠y = 120°; (c) =34°, m∠ =68°

29. (a) 30°; (b) 37°; (c) 20°; (d) 36°; (e) 120°; (f) 130°; (g) 94°; (h) 25°

30. (a) 45°; (b) 75°; (c) 50°; (d) 36°; (e) 90°; (f) 140°; (g) 115°; (h) 45°; (i) 80°

31. (a) 20°; (b) 85°; (c) (180 − x)°; (d) (90 + x)°; (e) 90°; (f) 25°; (g) 42°; (h) 120°; (i) 72°; (j) 110°; (k) 145°; (l) (180 − y)°; (m) 240°; (n) (180 + x)°; (o) 270°

32. (a) =43°, m∠y = 43°, (b) =190°, m∠y = 55°; (c) =140°, m∠y = 40°

33. (a) 120°; (b) 150°; (c) 180°; (d) 50°; (e) 22; (f) 45°

34. (a) =150°, =40°; (b) =190°, = 70°; (c) = 252°, = 108°

35. (a) 25°; (b) 39°; (c) 50°; (d) 30°; (e) 40°; (f) 76°; (g) 45°; (h) 95°; (i) 75°; (j) 120°

36. (a) 74°; (b) 90°; (c) 55°; (d) 60°; (e) 40°; (f) 37°; (g) 84°; (h) 110°; (i) 66°; (j) 98°; (k) 75°; (l) 79°

37. (a) m∠x = 120°, m∠y = 60°; (b) m∠x = 45°, m∠y = 22°; (c) m∠x = 36°, m∠y = 72°

38. (a) =40°, m∠y = 80°; (b) =45°, m∠y=67°; (c) m∠x = 78°, m∠y = 103°

Chapter 7

3. (a) 2:3:10; (b) 12:6:1; (c) 5:2:1; (d) 1:4:7; (e) 4:3:1; (f) 8:2:1; (g) 50:5:1; (h) 6:2:1; (i) 8:2:1

6. (a) 5x and 4x, sum = 9x; (b) 9x and x, sum = 10x; (c) 2x, 5x, and 11x, sum = 18x; (d) x, 2x, 2x, 3x, and 7x, sum = 15x

7. (a) 5x + 4x = 45, x = 5, 25° and 20°; (b) 5x + 4x = 90, x = 10, 50° and 40°; (c) 5x + 4x = 180, x = 20, 100° and 80°; (d) 5x + 4x + x = 180, x = 18, 90° and 72°

8. (a) 7x + 6x = 91, x = 7, 49°, 42° and 35°; (b) 7x + 5x = 180, x = 15, 105°, 90° and 75°; (c) 7x + 3x = 90, x = 9, 63°, 54° and 45°; (d) 7x + 6x + 5x = 180, x = 10, 70°, 60° and 50°

10.

11. (a) 15; (b) 3; (c) 6; (d) 2; (e) 3; (f) 30; (g) 32; (h) 6a

12.

13.

14.

15. Only (b) is not a proportion since 3(12) ≠ 5(7); that is, 36 ≠ 35.

16.

17. (a) d; (b) 35; (c) = (d) 4

18. (a) 21; (b) ; (c) 5

19. (a) 16; (b) 6; (c) 10

20.

21. (a) 12; (b) 8; (c) 60

22. (a) 15; (b) 15; (c) 6

24. (a) 35°; (b) 53°

25. (a) a = 16; (b) b = 15; (c) c = 126

27.

28. (a) ∠D ≅ ∠B, ∠AED ≅ ∠FGB; (b) ∠ADB ≅ ∠ABC, ∠A ≅ ∠A; (c) ∠ABC ≅ ∠AED, ∠BAE ≅ ∠EDA

29.

30.

32. (a) q = 20; (b) p = 8; (c) b = 7; (d) a = 12; (e) AB = 35; (f) d=2

33. (a) 8; (b) 6; (c) 26

34. (a) 42 ft; (b) 66ft

37. (a) 8:5; (b) 3:5; (c) halved (in each case)

38. (a) 15; (b) 60; (c) 25, 35, 40; (d) 4; (e) 6, 3

39. (a) 3:7; (b) 7:2; (c) quadrupled; (d) 7

43. (a) 5; (b) 14; (c) 6; (d) 5; (e) 12; (f) 13; (g) 48; (h) 2

44. 30, 18

45. (a) 8; (b) 6; (c) 12; (d) 5; (e) 7; (f) 12; (g) 30; (h) 7; (i) 5; (j) 8

46. (a) 8; (b) 13; (c) 21; (d) 6; (e) 9; (f) 14; (g) 3; (h) 8

47.

48.

49.

50. (a) 9, 12; (b) 10, 24; (c) 80, 150; (d) 2, 4

51. (a) 41; (b) 5

52. (a) 12; (b) 10; (c) 5

53. All except (h)

54. (a) yes; (b) no, since (2x)² + (3x)² ≠ (4x)²

55. (a) 8; (b) 6; (c) ; (d) 5

56. (a) 15; (b) 2; (c) 6

57. (a) 16; (b) 30; (c) 4; (d) 10

58. (a) 10; (b) 12; (c) 28; (d) 15

59. (a) 5; (b) 20; (c) 15; (d) 25

60. (a) 12; (b) 24

61. 12

62. 30

63. (a) 10 and 10; (b) 7 and 14; (c) = and 10

64. (a) 11; (b) a; (c) 48; (d) 16

65. (a) 25 and 25; (b) 35 and 35

66. (a) 28, 8; (b) 17, 14

67. (a) 17; (b) a; (c) 34; (d) 30

68. (a) 20; (b) 40

69. (a) 45, 13; (b) 11, 27; (c) 15, 55

70. 6, 5

Chapter 8

1. (a) 0.4226, 0.7431, 0.8572, 0.9998; (b) 0.9659, 0.6157, 0.2756, 0.0349; (c) 0.0699, 0.6745, 1.4281, 19.0811; (d) sine and tangent; (e) cosine; (f) tangent

2. (a) x = 20°; (b) A = 29°; (c) B = 71°; (d) A′ = 21°; (e) y = 45°; (f) Q = 69°; (g) W = 19°; (h) B′ = 67°

3. (a) 26°; (b) 47°; (c) 69°; (d) 8°; (e) 40°; (f) 74°; (g) 7°; (h) 27°; (i) 80°; (j) 13° since sin x = 0.2200; (k) 45° since sin x = 0.707; (l) 59° since cos x = 0.5200; (m) 68° since cos x = 0.3750; (n) 30° since cos x = 0.866; (o) 16° since tan x = 0.2857; (p) 10° since tan x = 0.1732

5. (a) m∠A = 27° since cos A = 0.8900; (b) m∠A = 58° since sin A = 0.8500; (c) m∠A = 52° since tan A = 1.2800

6. (a) m∠A = 42° since sin B = 0.6700; (b) m∠B = 74° since cos B = 0.2800; (c) m∠B = 68° since tan B = 2.500; (d) m∠B = 30° since tan B = 0.577

8. (a) 23°, 67°; (b) 28°, 62°; (c) 16°, 74°; (d) 10°, 80°

9. (a) x = 188, y = 313; (b) x = 174, y = 250; (c) x = 123, y = 182

10. (a) 82 ft; (b) 88 ft

11. 156 ft

12. (a) 2530 ft; (b) 2560 ft

13. (a) 21 in; (b) 79 in

14. 14

15. 16 and 18 in

16. 31 ft

17. 15 yd

18. (a) 1050 ft; (b) 9950 ft

19. 7°

20. 282 ft

21. (a) 81°; (b) 45°

22. (a) 22 ft; (b) 104 ft

23. 754 ft

24. 404 ft

25. (a) 295 ft; (b) 245 ft; (c) 960 ft

26. (a) 234 ft; (b) 343 ft

27. (a) 96 ft; (b) 166 ft

28. 9.1

Chapter 9

1. (a) 99 in²; (b) 3 ft² or 432 in²; (c) 500; (d) 120; (e) 36; (f) 100; (g) 300; (h) 150

2. (a) 48; (b) 432; (c) 25; (d) 240

3. (a) 7 and 4; (b) 12 and 6; (c) 9 and 6; (d) 6 and 2; (e) 10 and 7; (f) 20 and 8

4. (a) 1296 in²; (b) 100 square decimeters (100 dm²)

5. (a) 225; (b) 12; (c) 3.24; (d) 64a²; (e) 121; (f) 6; (g) 9b²; (h) 32; (i) 40; (j) 64

6. (a) 128; (b) 72; (c) 100; (d) 49; (e) 400

7. (a) 1600; (b) 400; (c) 100

8. (a) 9; (b) 36; (c) 9; (d) 4; (e)

9. (a) 2; (b) 52; (c) 10; (d) 5; (e) 6; (f) 4

10. (a) 16 ft²; (b) 6 ft² or 864 in²; (c) 70; (d) 1.62 m²

11. (a) 3x²; (b) x² + 3x; (c) x² − 25; (d) 12x² + 11x + 2

12. (a) 36; (b) 15; (c) 16

13. (a) 2; (b) 20; (c) 9; (d) 3; (e) 15; (f) 12; (g) 8; (h) 7

14. (a) 11 in²; (b) 3 ft² (c) 4x − 28; (d) 10x²; (e) 2x² + 18x; (f) (x² −16); (g) x² − 9

15. (a) 84; (b) 48; (c) 30; (d) 120; (e) 148; (f) 423; (g) 8; (h) 9

16. (a) 24; (b) 2; (c) 4

17. (a) 8; (b) 10; (c) 8; (d) 18; (e) 9; (f) 12; (g) 12; (h) 18

18. (a) 25; (b) 36; (c) 12; (d) 25; (e) b² ; (f) 4x² ; (g) 3r²

19. (a) 2; (b) ; (c) 24 (d) 18

20. (a) 24; (b) 54; (c) 150

21. (a) 15; (b) 8; (c) 12; (d) 5

22. (a) 140; (b) 69; (c) 225; (d) 60; (e) 94

23. (a) 150; (b) 204; (c) 39; (d) 64; (e) 160

24. (a) 4; (b) 7; (c) 18 and 9; (d) 9 and 6; (e) 10 and 5

25. (a) 17 and 9; (b) 23 and 13; (c) 17 and 11; (d) 5; (e) 13

26. (a) 36; (b) 38; (c) 12; (d) 12x²; (e) 120; (f) 96; (g) 18; (h) ; (i) 32; 98

27. (a) 737; (b) 14; (c) 77

28. (a) 10; (b) 12 and 9; (c) 20 and 10; (d) 5; (e)

29. 12

34. (a) 1:49; (b) 49:4; (c) 1:3; (d) 1:25; (e) 81: x²; (f) 9:x; (g) 1:2

35. (a) 49:100; (b) 4:9; (c) 25:36; (d) 1:9; (e) 9:4; (f) 1:2

36. (a) 10:1; (b) 1:7; (c) 20:9; (d) 5:11; (e) 2:y; (f) 3x:1; (g) :2; (h) 1:; (i) x:; (j) :4

37. (a) 6:5; (b) 3:7; (c) :1; (d) :2; (e) :3 or 1:

38. (a) 100; (b) 12; (c) 12; (d) 100; (e) 105; (f) 18; (g) 20

39. (a) 12; (b) 63; (c) 48; (d) 2; (e) 45

Chapter 10

1. (a) 200; (b) 24.5; (c) 112; (d) 13; (e) 9; (f) 3; (g) 4.5

2. (a) 12; (b) 23.47; (c) 7; (d) 18.5; (e) 3

3. (a) 24°; (b) 24°; (c) 156°

4. (a) 40°; (b) 9; (c) 140°

5. (a) 15°; (b) 15°; (c) 24

6. (a) 5°; (b) 72°; (c) 175°

7. (a) regular octagon; (b) regular hexagon; (c) equilateral triangle; (d) regular decagon; (e) square; (f) regular dodecagon (12 sides)

9. (a) 9; (b) 30; (c) 6; (d) 6; (e) 13; (f) 6; (g) 20; (h) 60

10. (a) 18; (b) 7; (c) 40; (d) 8; (e) 3.4; (f) 28; (g) 5; (h) 2

11. (a) 30; (b) 14; (c) 27; (d) 18; (e) 8; (f) 4; (g) 48; (h) 42; (i) 6; (j) 10; (k) ; (l) 3

12. (a) 817; (b) 3078

13. (a) 54; (b) 96; (c) 600

14. (a) 576; (b) 324; (c) 100

15. (a) 36; (b) 27; (c) ; (d) 144; (e) 3; (f) 48

16. (a) 10; (b) 10; (c) 5

17. (a) 18; (b) 9; (c) 6; (d) 3

18. (a) 1:8; (b) 4:9; (c) 9:10; (d) 8:11; (e) 3:1; (f) 2:5; (g) 4:3; (h) 5:2

19. (a) 5:2; (b) 1:5; (c) 1:3; (d) 3:4; (e) 5:1

20. (a) 5:1; (b) 4:7; (c) x:2; (d) :1; (e) :y; (f) :3 or :6

21. (a) 1:4; (b) 1:25; (c) 36:1; (d) 9:100; (e) 49:25

22. (a)12π; (b) 14π; (c) 10π; (d) 2π

23. (a) 9π; (b) 25π; (c) 64π; (d) π, (e) 18π

24. (a) C = 10π, A = 25π; (b) r = 8, A = 64π; (c) r = 4, C = 8π

25. (a) 12π; (b) 4π; (c) 7π; (d) 26π; (e) 8π (f) 3π

26. (a) 98π; (b) 18π; (c) 32π; (d) 25π; (e) 72π; (f) 100π

27. (a) (1) C = 8π, A = 16π; (2) C = 4π, A = 12π

(b) (1) C = 16π, A = 64π; (2) C = 8π, A = 48π

(d) (1) C = 16π, A = 64π; (2) C = 8π, A = 16π

(e) (1) C = 20π, A = 200π; (2) C = 20π, A = 100π

(f) (1) C = 6π, A = 18π; (2) C = 6π, A = 9π

28. (a) 10 ft; (b) 17 ft; (c) 3 ft or 6.7 ft

29. (a) 2π; (b) 10π; (c) 8; (d) 11π; (e) 6π; (f) 10π

30. (a) 3π; (b) 12; (c) 5π; (d) 2π; (e) π; (f) 4π

31. (a) 6π; (b) π/6; (c) 25π/6; (d) 25π; (e) 4; (f) 13; (g) 24π; (h) 8π/3

32. (a) 6π; (b) 20; (c) 3π; (d) 16π

33. (a) 120°; (b) 240°; (c) 36°; (d) 180°; (e) 135°; (f) (180/π)° or 57.3° to nearest tenth

34. (a) 72°; (b) 270°; (c) 40°; (d) 150°; (e) 320°

35. (a) 90°; (b) 270°; (c) 45°; (d) 36°

36. (a) 12; (b) 9; (c) 10; (d) 6; (e) 5; (f) 3

37. (a) 4; (b) 10; (c) 10 cm; (d) 9

38.

39. (a) 4π − 8; (b) 150π − 225; (c) 24π − 36; (d) 16π − 32; (e) 50π − 100

40.

41.

42.

43. (a) 200 − 25π/2; (b) 48 + 26π; (c) 25 − 25π/2; (d) 100π − 96; (e) 128 − 32π; (f) 300π + 400; (g) 39π; (h) 100

44. (a) 36π; (b) 36 + 18π, (c) 14π

Chapter 11

1. The description of each locus is left for the reader.

2. The diagrams are left for the reader.

(a) The line parallel to the banks and midway between them

(b) The perpendicular bisector of the segment joining the two floats

(d) The pair of bisectors of the angles between the roads

3. The diagrams are left for the reader.

(a) A circle having the sun as its center and the fixed distance as its radius

(b) A circle concentric to the coast, outside it, and at the fixed distance from it

(d) A circle having the center of the clock as its center and the length of the clock hand as its radius.

4. (a) ; (b) ; (c) ; (d) ; (e) ; (f) ; (g) ; (h) a 90° arc from A to G with B as center

5. (a) ; (b) ; (c) ; (d) ; (e) E

6. In each case, the letter refers to the circumference of the circle. (a) A; (b) C; (c) B; (d) A; (e) C; (f) A and C; (g) B

7. The description of each locus is left for the reader.

8. (a) ; (b) ; (c) line parallel to and midway between them; (d) ; (e) ; (f)

9. The explanation is left for the reader.

10. (a) The intersection of two of the angle bisectors

(b) The intersection of two of the ⊥ bisectors of the sides

(d) The intersection of the bisector of ∠C and a circle with C as center and = as radius

(e) The intersections of two circles, one with B as center and 5 as radius and the other with A as center and 10 as radius

11. (a) 1; (b) 1; (c) 4; (d) 2; (e) 2; (f)1

Chapter 12

1. A(3, 0); B(4, 3); C(3, 4); D(0, 2); E(–2, 4); F(–4, 2); G(–1, 0); H(-3, −2); I(–2, –3); J(0, –4); K(1, −2); L(4,−2)

3. Perimeter of square formed is 20 units; its area is 25 square units.

4. Area of parallelogram = 30 square units.

Area of ΔBCD = 15 square units.

5. (a) (4, 3); (b) (2, 3); (c) (–4, 6); (d) (7, –5); (e) (−10,−2); (f) (0, 10); (g) (4, –1);

(h) (−5,−2); (i) (5, 5); (j) (–3, –10); (k) (5, 6); (l) (0, –3)

6. (a) (4, 0), (0, 3), (4, 3); (b) (–3, 0), (0, 5), (–3, 5); (c) (6, –2), (0, –2), (6, 0);

(d) (4, 6), (4, 9), (3, 8); (e) (2, –3), (–2, 2), (0, 5); (f) (−, 0), (, ), (0, −1)

7. (a) (0, 2), (1, 7), (4, 5), (3, 0); (b) (–2, 7), (3, 6), (6, 1), (1, 2);

8. (a) (2, 6), (4, 3); (b) (1, 0), (0,−2); (c) common midpoint, (2, 2)

9. (a) (–2, 3); (b) (–3, –6); (c) (−1,−2); (d) (a, b); (e) (2a, 3b); (f) (a, b + c)

10. (a) M(4, 8); (b) A(–1, 0); (c) B(6, –3)

11. (a) B(2, 3); (b) D(3, 3); (c) A(–2, 9)

12. (a) Prove that ABCD is a parallelogram (since opposite sides are congruent) and has a rt. ∠.

(b) The point (3, 2) is the midpoint of each diagonal.

13. (a) D(3, 2), (1,1); (b) E(0, 2), (3, 1); (c) no, since the midpoint of each median is not a common point

14. (a) 5; (b) 6; (c) 10; (d) 12; (e) 5.4; (f) 7.5; (g) 9; (h) a

15. (a) 3, 3, 6; (b) 4, 14, 18; (c) 1, 3, 4; (d) a, 2a, 3a

16. (a) 13; (b) 5; (c) 15; (d) 5; (e) 10; (f) 15; (g) 3; (h) 5; (i) ; (j) 2; (k) 4; (l) a

18. (a) ΔABC; (b) ΔDEF; (c) ΔGHJ; (d) ΔKLM is not a rt. Δ

19. (a) 5; (b) ; (c)

21. (a) 10; (b) 5; (c) 5; (d) 13; (e) 4; (f) 3

22. (a) on; (b) on; (c) outside; (d) on; (e) inside; (f) inside; (g) on

23.

24. (a) 3; (b) 4; (c) − ; (d) –7; (e) 5; (f) 0; (g) 3; (h) 5; (i) –4; (j) − ; (k) –1; (l) –2; (m) 5; (n) 6; (o) –4; (p) –8

25. (a) 72°; (b) 18°; (c) 68°; (d) 22°; (e) 45°; (f) 0°

26. (a) 0.0875; (b) 0.3057; (c) 0.3640; (d) 0.7002; (e) 1; (f) 3.2709; (g) 11.430

27. (a) 0°; (b) 25°; (c) 45°; (d) 55°; (e) 7°; (f) 27°; (g) 37°; (h) 53°; (i) 66°

28. (a) , , , ; (b) , , ; (c) , ; (d) ,

29. (a) 0; (b) no slope; (c) 5; (d) –5; (e) 0.5; (f) –0.0005

30. (a) 0; (b) no slope; (c) no slope; (d) 0; (e) 5; (f) –1; (g) 2

31. (a) ; (b) ; (c) –1; (d) 6

32. (a) –2; (b) –1; (c) − ; (d) −; (e) –10; (f) 1; (g) ; (h) ; (i) no slope; (j) 0

33. (a) 0; (b) –2; (c) 3; (d) –1

34. (a) −; (b) ; (c) −

35. (a) −; (b) 1; (c) 2; (d) –1

36. (a) −; (b) ; (c)

37. (a) and (b)

38. (a) 19; (b) 9; (c) 2

39. (a) x = –5; (b) y=3; (c) y = 3 and y = –3; (d) y = –5; (e) x = 4 and x = –4; (f) x = 5 and x = –1;(g) y = 4; (h) x = 1; (i) x = 9

40. (a) x = 6; (b) y = 5; (c) x = 6; (d) x = 5; (e) x = 6; (f) y = 3

41. (a) x = y; (b) y = x + 5; (c) x = y – 4; (d) y – x = 10; (e) x + y = 12; (f) x – y = − or y – x = 2; (g) x = y and x = –y; (h) x + y = 5

42. (a) line having y-intercept 5, slope 2; (b) line passing through (2, 3), slope 4; (c) line passing through (–2, –3), slope ; (d) line passing through origin, slope ; (e) line having y-intercept 7, slope –1; (f) line passing through origin, slope

43. (a) y = 4x; (b) y = –2x; (c) y = x or 2y = 3x; (d) y = −x or 5y = –2x; (e) y = 0

44. (a) y = 4x + 5; (b) y = –3x + 2; (c) y = x − 1 or 3y = x – 3; (d) y = 3x + 8; (e) y = –4x – 3; (f) y = 2x or y – 2x = 0

45.

46.

47. (a) circle with center at origin and radius 7; (b) x² + y² = 16; (c) x² + y² = 64 and x² + y² = 4

48. (a) x² + y² = 25; (b) x² + y² = 81; (c) x² + y² = 4 or x² + y² = 144

49. (a) 3; (b) ; (c) 2; (d)

50. (a) x² + y² = 16; (b) x² + y² = 121; (c) x² +y² = or 9x² + 9y² = 4; (d) x² +y² = or 4x² + 4y² =9; (e) x² + y² = 5; (f) x² +y² = or 4x² + 4y² = 3

51. (a) 10; (b) 10; (c) 20; (d) 20; (e) 7; (f) 25

52. (a) 16; (b) 12; (c) 20; (d) 24

53. (a) 10; (b) 12; (c) 22

54. (a) 5; (b) 13; (c) 7

55. (a) 6; (b) 10; (c) 1.2

56. (a) 15; (b) 49; (c) 53

57. (a) 30; (b) 49; (c) 88; (d) 24; (e) 16; (f) 18

Chapter 13

1. (a) <; (b) >; (c) >; (d) >; (e) >; (f) <

2. (a) >; (b) >; (c) <; (d) >

3. (a) >; (b) <; (c) <; (d) >

4. (a) more; (b) less

5. (a) >; (b) >; (c) <; (d) >; (e) <; (f) <

6. (c), (d), and (e)

7. (a) 5 to 7; (b) 6 to 10; (c) 4 to 10; (d) 3 to 9; (e) 2 to 8; (f) 1 to 13

8. (a) ∠B, ∠A, ∠C; (b) , , ; (c) ∠3, ∠2, ∠1

9. (a) m∠BAC > m∠ACD; (b) AB > BC

10. (a) , , ; (b) ∠BOC, ∠AOB, ∠AOC; (c) , ≅ , ; (d)

Chapter 14

1. (a) Ornament, jewelry, ring, wedding ring; (b) vehicle, automobile, commercial automobile, taxi; (c) polygon, quadrilateral, parallelogram, rhombus; (d) angle, obtuse angle, obtuse triangle, isosceles obtuse triangle

2. (a) A regular polygon is an equilateral and an equiangular polygon.

(b) An isosceles triangle is a triangle having at least two congruent sides.

(d) A rectangle is a parallelogram having one right angle.

(e) An inscribed angle is an angle formed by two chords and having its vertex on the circumference of the circle.

(f) A parallelogram is a quadrilateral whose opposite sides are parallel.

(g) An obtuse angle is an angle larger than a right angle and less than a straight angle.

3. (a) x + 2 ≠ 4; (b) 3y = 15; (c) she does not love you; (d) his mark was not more than 65; (e) Joe is not heavier than Dick; (f) a + b = c

4. (a) A nonsquare does not have congruent diagonals. False (for example, when applied to a rectangle or a regular pentagon).

(b) A non-equiangular triangle is not equilateral. True.

(d) A number that is not zero is a positive number. This inverse is false when applied to negative numbers.

5. (a) Converse true, inverse true, contrapositive true

(b) Converse false, inverse false, contrapositive true

(d) Converse false, inverse false, contrapositive true

6. (a) Partial converses: interchange (2) and (3) or (1) and (3)
Partial inverses: negate (1) and (3) or (2) and (3)

(b) Partial converses: interchange (1) and (4) or (2) and (4) or (3) and (4)
Partial inverses: negate (1) and (4) or (2) and (4) or (3) and (4)

7. (a) Necessary and sufficient; (b) necessary but not sufficient; (c) neither necessary nor sufficient; (d) sufficient but not necessary; (e) necessary and sufficient; (f) sufficient but not necessary; (g) necessary but not sufficient

Chapter 17

1. (a) 6(7²) or 294 yd²; (b) 2(8)(6)+2(8)(14) + 2(6)(14) or 510 ft²; (c) 4(3.14)30² or 11,304 m²; (d) (3.14)(10)(10 + 4) or 911 yd²

2. (a) 36³ or 46,656 in³; (b) 100³ or 1,000,000 cm³

3. (a) 27 in³; (b) 91 in³; (c) 422 in³; (d) 47 in³; (e) 2744 in³

4. (a) 3(8)(8) or 204 in³; (b) 2(9)(9) or 162 ft³; (c) (6)(6.4) or 13 ft³

5. (a) 904 m³; (b) 1130 ft³; (c) 18 ft³

6. (a) V = πr²h; (b) V = s²h; (c) V = lwh; (d) V = πr³

Chapter 18

1. (a) A′(9, 2), B′(2, 4), and C′(5, 7); (b) A′(24, 2), B′(3, 4), and C′(0, 7); (c) A′(25, 1), B′(2, −1), and C′(−1, −4); (d) A′(−2, 14), B′(−4, 7), and C′(−7, 10); (e) A′(12, 4), B′(−2, 8), and C′(4, 14); (f) A′(−14, 6), B′(7, 12), and C′(−2, 21); (g) A′(0, −1), B′(−2, 6), and C′(−5, 3)

2. (a) A′(6,−1), B′(7, 2), and C′(8,−1); (b) A′(1,−5), B′(2,−2), and C′(3,−5); (c) A′(−2, 1), B′(−1, 4), and C′(0, 1)

3. (a) translation to the right 5 spaces, P(x, y) Images P′(x + 5, y); (b) translation down 5 spaces and to the right 3, P(x, y) Images P′(x + 3, y−5); (c) translation down 3 spaces and to the left 5, P(x, y) Images P′(x−5, y−3)

4. (a) P(x, y) P′(x, y−5); (b) P(x, y) P′(x + 6, y); (c) P(x, y) P′(x−7, y + 3); (d) P(x, y) P′(x + 8, y−2); (e) P(x, y) P′(x−1, y + 4)

5. (a) A′(−1, 3), B′(−5, 3), C′(−4, 1), and D′(−2, 1); (b) A″(1,−5), B″ (5,−5), C″(4,−3), and D″(2,−3); (c) A″(15, 3), B′″(11, 3), C″(12, 1), and D″′(14, 1)

6. (a) reflection across the line x = 2, P(x, y) P′(4−x, y); (b) reflection across y = 3, P(x, y) P″(x, 6−y); (c) reflection across x = 25, P(x, y) P″′(−10−x, y)

7. (a) P(x, y) P′(x, 10−y); (b) P(x, y) P′(−4−x, y); (c) P(x, y) P′(x,−2−y); (d) P(x, y) P′(5−x, y)

8. (a), (b), (e), and (f)

9. (a) A′(2, −1), B′(2, −4), C′(1, −5), and D′(1, −2); (b) A″(−1, −2), B″(−4, −2), C″(−5, −1), and D−(−2,−1); (c) A″′(−2, 1), B″′(−2, 4), C″′(−1, 5), and D″′(−1, 2)

10. (a) 180° rotation about the origin, P(x, y) P′(−x,−y); (b) 90° clockwise rotation about the origin, P(x, y) P″(y,−x); (c) 270° clockwise rotation about the origin (or 90° counter-clockwise), P(x, y) P″′(−x, y)

11. (a) P(x, y) P′(x cos 40° + y sin 40°, y cos 40°−x sin 40°) = P′(0.766x + 0.6428y, 0.766y−0.6428x);

(b) P(x, y) P′(x cos 50° + y sin 50°, y cos 50°−x sin 50°) = P′(0.6428x + 0.766y, 0.6428y−0.766x);

12. (a) 120°; (b) 60°; (c) 72°; (d) 360° (no rotational symmetry); (e) 90°; (f) 180°

13. (a) A′(4, 5), B′(5, 4), and C′(5, 7); (b) A″(2,−5), B″(3,−6), and C″(0,−6); (c) A″′(−1,−2), B″′(−2,−3), and C″′(1,−3); (d) A″″(−4, 3), B″″(−5, 4), and C″″(−5, 1)

14. (a) R(x, y) R″(x + 6, y−1); (b) R(x, y) R″(1−x, y + 2); (c) R(x, y) R″(y + 3,−x−6); (d) R(x, y) R″(−y, 4−x); (e) R(x, y) R″(6−x,−3−y)

15. (a) reflect across the y axis and then move down 2 and to the right 3 spaces, P(x, y) P′(−x + 3, y−2); (b) rotate about the origin counterclockwise 90°, then move to the left 1 space, P(x, y) P″(−y−1, x); (c) rotate around the origin 90° clockwise, then reflect across the x axis, then move down and to the right 1 space, P(x, y) P′″(y + 1, x−1)

16. (a) P(x, y) P′(x,−y−3); (b) P(x, y) P′(y + 2,−x); (c) P(x, y) P′(−x, y−4); (d) P(x, y) P′(2x, 4 + y); (e) P(x, y) P′(−7−x, y + 3)

17. A′(−3, 6), B′(3, 6), C′(3, 3), and D′(23, 3)

18. (a) P(x, y) P′(2x, 2y); (b) P(x, y) P′(8x, 8y); (c) P(x, y) P′(x, x)

Chapter 19

1. (a) Center: (5, −3). Major axis: y = −3. Minor axis: x = 5. Vertices: (−5, −3), (15, −3), (5, −1), and (5, −5). Foci: (5±, −3).

(b) Center: (−2, 0). Major axis: y = 0 (the x axis). Minor axis: x = −2. Vertices: (−5, 0), (1, 0), (−2, 1), and (−2, −1). Foci: (−2±, 0).

(c) Center: (1, 4). Major axis: x = 1. Minor axis: y = 4. Vertices: (−5, 4), (7, 4), (1, −3), and (1, 11). Foci: (1, 4±).

2. (a) Vertex: (4, −2). Axis: y = −2. Focus: (4+,−2).

(b) Vertex: (0, −2). Axis: x = 0 (the y axis). Focus: (0, −4.5).

3. (a) Center: (−7, −5). Transverse axis: x = −7. Conjugate axis: y = −5. Vertices: (−7, −3) and (−7, −7). Foci: (−7, −5±).

(b) Center: (3, 0). Transverse axis: x = 3. Conjugate axis: y = 0 (x axis). Vertices: (3, ±5). Foci: (3, ±).

4. See Fig. 19-24.

Fig. 19-24

6. See Fig. 19-25.

Fig. 19-25

7. See Fig. 19-26.

Fig. 19-26

8. See Fig. 19-27.

Fig. 19-27