Jacobs Geometry
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Contents 9 sections

Chapter 11 — Answers

Selected answers for Chapter 11: The Right Triangle, transcribed from the book’s appendix. They are meant for checking your work, not for copying — the book prints answers to selected exercises only.

Back to Chapter 11 · Teacher’s answers: Chapter 11 · Complete key: Answers to Exercises PDF · Printed answers: appendices PDF

Lesson 1 (pages 430–432)

2. The altitude to the hypotenuse.

3. AP and PC.

5. EP.

6. EP and EG.

11. Their projections on the hypotenuse.

12. AC.

15. 4.

17. 2102\sqrt{10} (or about 6.32).

18. 4.05.

21. 16.

26. 916\dfrac{9}{16}.

29. False.

33. 464\sqrt{6}.

35. 21\dfrac{\sqrt{2}}{1} (or 2\sqrt{2}).

37. 113 in².

38. OAOB=OBOC\dfrac{OA}{OB}=\dfrac{OB}{OC}.

41. r2r^2.

42. r3r^3.

48. 5.

p. 758

Lesson 2 (pages 436–440)

1. 30 ft.

2. About 95 ft.

5. About 101 ft.

8. 8 cubits.

12. 20 ft.

15. About 8.7 ft.

18. 72+242=49+576=625=2527^2+24^2=49+576=625=25^2.

21. Each number in 14-48-50 is twice the corresponding number in 7-24-25.

23. 6-8-10 and 9-12-15.

28. AB.

29. y2−y1y_2-y_1.

31. d=5xd=5x.

33. 8.4 in by 11.2 in.

35. 2714\dfrac{27}{14} or about 1.9.

38. OH = 45\sqrt{45} or 353\sqrt{5}, OP = 65\sqrt{65}, PH = 20\sqrt{20} or 252\sqrt{5}.

42. The length of AP.

43. x2+202=(40−x)2x^2+20^2=(40-x)^2; x=15x=15.

53. D=R−R2−B2D=R-\sqrt{R^2-B^2}.

Lesson 3 (pages 443–447)

1. Yes. They are similar by AA.

2. a1=c2\dfrac{a}{1}=\dfrac{c}{\sqrt{2}}.

3. c=a2c=a\sqrt{2}.

4. a=c2a=\dfrac{c}{\sqrt{2}}.

5. In an isosceles right triangle, the hypotenuse is 2\sqrt{2} times the length of one leg.

11. c=2b3c=\dfrac{2b}{\sqrt{3}}.

15. 828\sqrt{2}.

16. 525\sqrt{2}.

17. x=12x=12, y=63y=6\sqrt{3}.

19. x=53x=5\sqrt{3}, y=103y=10\sqrt{3}.

23. 3\sqrt{3}.

24. 434\sqrt{3}.

27. 902≈12790\sqrt{2}\approx127 ft.

30. 753≈13075\sqrt{3}\approx130 ft.

31. About 23 in.

34. 21 square units.

39. x2\dfrac{x}{2}.

43. OA = 12\dfrac{1}{2}, AP = 123\dfrac{1}{2}\sqrt{3}.

44. (−12,123)\left(-\dfrac{1}{2},\dfrac{1}{2}\sqrt{3}\right).

53. 12\dfrac{1}{2} or 0.5.

54. 34\dfrac{\sqrt{3}}{4}.

Lesson 4 (pages 450–453)

1. cb\dfrac{c}{b}.

3. fd\dfrac{f}{d}.

4. 0.445.

6. 57.290.

7. 5.711°.

11. tan⁡A=ab=aa=1\tan A=\dfrac{a}{b}=\dfrac{a}{a}=1.

13. ∠A > 45°.

14. tan⁡A>1\tan A>1.

18. tan⁡60∘=BCAC=b3b=3\tan 60^\circ=\dfrac{BC}{AC}=\dfrac{b\sqrt{3}}{b}=\sqrt{3}.

22. A right triangle because 32+42=523^2+4^2=5^2.

24. 43\dfrac{4}{3}.

25. 37°.

27. ∠A < ∠B; ∠A and ∠B are complementary.

29. 16.5.

32. 20°.

35. 58°.

39. 197 ft.

43. About 22 ft.

46. 3.7°.

48. 26.565051°.

51. AB = 5\sqrt{5}, BC = 10\sqrt{10}, and AC = 5\sqrt{5}.

53. 40.8°.

55. About 4,172 mi.

Lesson 5 (pages 456–460)

1. ha\dfrac{h}{a} or bx+y\dfrac{b}{x+y}.

2. hy\dfrac{h}{y} or ab\dfrac{a}{b}.

3. yb\dfrac{y}{b} or bx+y\dfrac{b}{x+y}.

7. 0.996.

10. 82.935°.

11. 45.000°.

14. 1213\dfrac{12}{13}.

15. 67°.

20. 513\dfrac{5}{13}.

23. 5.3.

25. 61°.

26. 36.5.

29. BH = 2.6 cm, CI = 5.0 cm, DJ = 7.1 cm, EK = 8.7 cm, and FL = 9.7 cm.

36. The sine of an acute angle of a right triangle is the ratio of the length of the opposite leg to the length of the hypotenuse.

38. In a 30°-60° right triangle, the side opposite the 30° angle is half the hypotenuse.

40. Approximately 14,500 ft, or more than 2.7 mi.

43. About 806 ft.

46. About 4°.

48. Parallel lines form equal alternate interior angles.

49. AAS.

51. 5a5a.

57. sin⁡∠2=d−wh\sin\angle2=\dfrac{d-w}{h}.

Lesson 6 (pages 463–467)

1. hh.

3. The x-axis.

5. ee.

6. jj, 14\dfrac{1}{4}; ee, −4.

11. The slope is tan⁡30∘≈0.58\tan30^\circ\approx0.58.

13. −60 m.

16. About 3.4°.

18. 59\dfrac{5}{9} (or about 0.56).

23. 311\dfrac{3}{11}.

27. −32-\dfrac{3}{2}.

36. Its run is 550 m, and its rise is −10 m.

39. (300−x55)\left(300-\dfrac{x}{55}\right) m above the ground.

40. 16,500 m.

42. Run = x−x1x-x_1; rise = y−y1y-y_1.

43. y=m(x−x1)+y1y=m(x-x_1)+y_1.

47. From the start to about 2.5 seconds.

52. ACBC\dfrac{AC}{BC}.

56. Corresponding sides of similar triangles are proportional.

58. Division.

60. l2l_2.

62. The altitude to the hypotenuse of a right triangle is the geometric mean between the segments of the hypotenuse.

Lesson 7 (pages 471–474)

2. They are opposite each other.

4. The Law of Sines.

7. They include ∠A.

12. 40°.

13. 17.0.

15. 60°.

16. 32.0.

p. 759

18. sin⁡∠1y=sin⁡∠4x\dfrac{\sin\angle1}{y}=\dfrac{\sin\angle4}{x}.

21. 108°.

25. 56°.

26. 27.7 mi.

28. 34.6 mi.

32. 238,000 mi.

34. a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A.

35. cos⁡A=12\cos A=\dfrac{1}{2}.

40. By multiplication.

44. c2=a2+b2c^2=a^2+b^2.

Review (pages 476–480)

1. 2.

5. −43-\dfrac{4}{3}.

8. HF ⊥ BG.

11. d=s2d=s\sqrt{2}.

15. 28 ft.

16. About 24.2 ft.

19. About 195 ft.

21. About 5.7°.

26. Two.

30. 100 ft.

31. About 137.6 ft.

34. About 408 ft.

35. 0.87.

40. 7 units.

43. AF = BF.

47. About 44 ft.

53. About 380 times as far.

55. Approximately 13\dfrac{1}{3} in.

57. About 600 ft.

62. tan⁡∠ABE=h−xd\tan\angle ABE=\dfrac{h-x}{d}.

Algebra Review (page 482)

1. y=−34x+6y=-\dfrac{3}{4}x+6.

2. m=−34m=-\dfrac{3}{4}; b=6b=6.

3.

Graph of a line with slope −3/4 and y-intercept 6.

4.

Graph of a line through (−1, 3) and (3, 5).

5. y−1=12(x+5)y-1=\dfrac{1}{2}(x+5).

6. y=12x+72y=\dfrac{1}{2}x+\dfrac{7}{2}.

7. b=72b=\dfrac{7}{2} (or 3.5).

8.

Graph of a line with slope 2 through (0, 1).

9. They have the same slope, 2.

10. y=2x+1y=2x+1.

11. y=−x+5y=-x+5; y=x−3y=x-3.

12.

Graph of the lines y = −x + 5 and y = x − 3 intersecting at (4, 1).

13. −1 and 1.

14. The product of their slopes is −1.

15. −13-\dfrac{1}{3}.

16. −6.

17.

Graph of a line with slope −1/3 and y-intercept 2.

18. y=−13x+2y=-\dfrac{1}{3}x+2.

19. x-intercept, 6; y-intercept, 2.

20. x+3y=6x+3y=6.

p. 760

21.

Graph of a line through A(0, −2) and B(1, 3).

22. 5.

23. y−3=5(x−1)y-3=5(x-1).

24. y=5x−2y=5x-2.

25. 5x−y=25x-y=2.

26. y=43xy=\dfrac{4}{3}x (or y=43x+0y=\dfrac{4}{3}x+0).

27. m=43m=\dfrac{4}{3}; b=0b=0.

28.

Graph of a line through A(−3, −4) and B(3, 4).

29. 0.

30. 10.