Chapter 12 — Answers

Selected answers for Chapter 12: Circles, 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 12 · Teacher’s answers: Chapter 12 · Complete key: Answers to Exercises PDF · Printed answers: appendices PDF

Lesson 1 (pages 486–490)

2. Chords.

3. A diameter.

6. Five.

9. Six.

10. Concentric.

14. They are perpendicular.

17. Two points determine a line.

18. All radii of a circle are equal.

22. If a line divides a line segment into two equal parts, it bisects the segment.

26. In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment.

27. If a line through the center of a circle bisects a chord that is not a diameter, it is also perpendicular to the chord.

28. Through a point on a line, there is exactly one line perpendicular to the line.

34. An isosceles right triangle.

35. About 46 mm.

38. 1,000 seconds.

40. $\dfrac{10}{27}$, or about 0.37.

45. They are rhombuses because all of their sides are equal.

Lesson 2 (pages 493–496)

2. In a plane, two lines perpendicular to a third line are parallel to each other.

4. OB ⊥ AC.

5. If a line through the center of a circle bisects a chord that is not a diameter, it is also perpendicular to the chord.

9. $\dfrac{4}{3}$.

10. $-\dfrac{3}{4}$.

15. AB is tangent to circle O.

16. If a line is perpendicular to a radius at its outer endpoint, it is tangent to the circle.

18. $\dfrac{AB}{1}$, or AB.

32. Because ∠FDE is a right angle and corresponding parts of congruent triangles are equal.

34. Line AB.

36. If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of contact.

41. AD = 4 cm and AF = AB = 8 cm.

42. $(8-r)$ cm.

44. A 3-4-5 right triangle.

48. A parallelogram (also, a rectangle).

Lesson 3 (pages 500–503)

1. Central angles.

2. $\widehat{PJ}$.

3. $\widehat{PCJ}$.

5. 283°.

7. 77°.

8. Two points determine a line.

10. An arc is equal in measure to its central angle.

16. The Arc Addition Postulate.

19. A semicircle.

24. No. The sum of any two sides of a triangle is greater than the third side.

27. AB = $r$, BC = $r\sqrt{3}$, AC = $2r$.

31. △ABC and △ABD are equilateral and congruent.

33. Line l is tangent to circle A. If a line is perpendicular to a radius at its outer endpoint, it is tangent to the circle.

41. All of the arcs have the same measure.

Lesson 4 (pages 506–509)

2. 147°.

3. 24.5°.

4. 33°.

7. All radii of a circle are equal.

9. An exterior angle of a triangle is equal to the sum of the remote interior angles.

12. A central angle is equal in measure to its intercepted arc.

15. Betweenness of Rays Theorem.

17. Arc Addition Postulate.

20. An inscribed angle is equal in measure to half its intercepted arc.

p. 761

22. The measure of a semicircle is 180°.

25. A 90° angle is a right angle.

28. Inscribed angles.

30. $x=40^\circ$, $y=20^\circ$, and $z=30^\circ$.

33. 9°.

36. They are complementary.

37. It is a right angle. An angle inscribed in a semicircle is a right angle.

43. $n^\circ$.

44. $2n^\circ$.

50. It is an altitude (to the hypotenuse) of the triangle.

52. BF = $\sqrt{3}$ in.

Lesson 5 (pages 512–515)

2. An exterior angle of a triangle is equal to the sum of the remote interior angles.

3. An inscribed angle is equal in measure to half its intercepted arc.

5. 36.

7. 54° and 126°.

10. Subtraction.

14. 40°.

15. 75°.

18. 55°.

19. A secant angle.

20. It is less than 55°.

23. $m\widehat{AD}+m\widehat{CB}=180^\circ$.

27. 72°.

29. 36°.

34. 20°.

40. If a line through the center of a circle bisects a chord that is not a diameter, it is also perpendicular to the chord.

42. 17 ft.

45. Each angle is 90°.

48. About 92.5 days.

51. 6°.

Lesson 6 (pages 517–521)

3. HL.

7. Inscribed angles that intercept the same arc are equal.

9. Corresponding sides of similar triangles are proportional.

12. The tangent segments to a circle from an external point are equal.

19. Right angles. An angle inscribed in a semicircle is a right angle.

24. CFOE is a square.

31. PA is tangent to circle O because it is perpendicular to radius OA at its outer endpoint.

33. PA = PB = 10.

35. 20 units.

41. (12, 10).

45. If a line through the center of a circle is perpendicular to a chord, it also bisects the chord.

46. BP.

Review (pages 523–526)

2. A central angle.

7. 60°.

9. 35°.

10. 16.5.

15. 30°.

18. 60°.

25. AB = AD. The tangent segments to a circle from an external point are equal.

27. They are similar to △ABC.

29. $\dfrac{1}{2}$.

34. PB ⊥ AC.

36. Isosceles. (In the figure, they are also obtuse.)

40. 4 cm.

42. Approximately 3.7 cm.

44. Approximately 68°.

46. 53°.

47. 74°.

51. C is the midpoint of AB.

52. Approximately 9.7 in.

53. Approximately 75.5°.

56. $\sqrt{5}$.

58. $\dfrac{\sqrt{5}+1}{2}$.

Algebra Review (page 528)

1. (8, 8).

2. (7, 14).

3. (11, 9).

4. (3, −4).

5. (0, −1).

6. (5, 14).

7. (14, 5).

8. (3.5, 10.5).

9. (45, 9).

10. (−2, 1).

11. (−19, −2).

12. (0, −5).

13. (51, −16).

14. (3, −11).

15. (18.5, 0.9).

16. (−1, −10).

17. (11, −4).

18. (7, 1).

19. (7, 3).

20. (9, 2).

21. (4, 0).

22. (−1, 5).

23. (8, −2).

24. (0, −8).