Unless otherwise noted, all content on this page is Cengage Learning. Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scammed, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review h s deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Exercises 6.3 6.3 Line and Segment Relationships in the Circle 295 1. Given: OO with OECD Find: CD = OC MCF 2. Given: OC 8 and OE = 6 15. Given: Find: 16. Given: Find: AE 9 and EB = 8; DE:EC = 2:1 DE and EC AE 6 and EB = 4; DE:EC = 3:1 DE and EC OECD in OO Find: CD For Exercises 17-20, see Theorem 6.3.6. 3. Given: OV 1 RS in OO Exercises 1, 2 17. Given: Find: AB = 6, BC = 8, AE = 15 DE OV 9 and OT = 6 Find: RS R D Exercises 17-20 18. Given: Exercises 3, 4 Find: AC 12, AB = 6, AE = 14 AD 4. Given: V is the midpoint of RS in O Find: mZS OR 15° and OT = 6 19. Given: Find: AB = 4, BC = 5, AD = 3 DE 20. Given: Find: AB 5, BC = 6, AD = 6 AE 5. Sketch two circles that have: a) No common tangents b) Exactly one common tangent c) Exactly two common tangents d) Exactly three common tangents e) Exactly four common tangents 6. Two congruent intersecting circles B and D (not shown) have a line (segment) of centers BD and a common chord AC that are congruent. Explain why quadrilateral ABCD is a square. In the figure for Exercises 7 to 16, O is the center of the circle. See Theorem 6.3.5. In the figure for Exercises 21 to 24, RS is tangent to the circle at S. See Theorem 6.3.7. 21. Given: Find: 22. Given: Find: RS 8 and RV = 12 RT R RT 4 and TV = 6 RS 23. Given: RS TV and RT = 6 Find: RS (HINT: Use the Quadratic Formula.) Exercises 21-24 7. Given: Find: AE = 6, EB = 4, DE = 8 EC 8. Given: Find: DE 12, EC = 5, AE = 8 EB о 9. Given: Find: AE = 8, EB = 6, DC = 16 DE and EC 110 D 10. Given: Find: AE = 7, EB = 5, DC = 12 DE and EC Exercises 7-16 11. Given: Find: 12. Given: Find: AE = 6, EC = 3, AD = 8 CB AD10, BC = 4, AE = 7 EC 13. Given: AE, EB = 12, DE = * + 6, a , and EC = 9 Find: x and AE 14. Given: AE = EB =, DE = 5, and EC = 6 Find: x and DE 24. Given: Find: RT = RS and TV = 9 RT 25. For the two circles in Figures (a), (b), and (c), find the total number of common tangents (internal and external). (a) (b) (c) 26. For the two circles in Figures (a), (b), and (c), find the total number of common tangents (internal and external). (a) (b) Unless otherwise noted, all content on this page is Cengage Learning. Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 296 CHAPTER 6 CIRCLES In Exercises 27 to 30, provide a paragraph proof. 27. Given: *35. Given: 00 and OQ are tangent at point F Secant AC to 00 Secant AE to g Prove: Common internal tangent AF AC ABAE. AD B F 28. Given: OO with OM LAB and ON 1 BC OM=ON Prove: AABC is isosceles I/N C 29. Given: Quadrilateral ABCD is Prove: circumscribed about 0 ABCD = DA + BC 30. Given: Prove: AB = DC in OP AABD ACDB 31. Does it follow from Exercise 30 that AADE is also congruent to ACBE? What can you conclude regarding AE and CE in the drawing? What can you conclude regarding DE and EB? D Exercises 30, 31 RT 32. In 0 (not shown), RS is a diameter and T is the midpoint of semicircle RTS. What is the value of the ratio ? The ratio ? 33. The cylindrical brush on a vacuum cleaner is powered by an electric motor. In the figure, the drive shaft is at point D. If mAC 160°, find Find: A M *36. Given: Tangents AB, BC, and AC to OO at points M, N, and P, respectively AB 14, BC = 16, AC = 12 AM, PC, and BN N OQ is inscribed in isosceles right ARST The perimeter Find: of ARST is 8+ 4√2 TM M •Q N B *37. Given: Find: AB is an external tangent to 0 and OQ at points A and B; radii lengths for 00 and OQ are 4 and 9, respectively AB (HINT: The line of centers OQ contains point C, the point at which OO and OQ are tangent.) 38. The center of a circle of radius 3 inches is at a distance of 20 inches from the center of a circle of radius 9 inches. What is the exact length of common internal tangent AB? (HINT: Use similar triangles to find OD and DP. Then apply the Pythagorean Theorem twice.) Exercises 38, 39 39. The center of a circle of radius 2 inches is at a distance of 10 inches from the center of a circle of radius length 3 inches. To the nearest tenth of an inch, what is the approximate length of a common internal tangent? Use the hint provided in Exercise 38. the measure of the angle formed by the drive belt D at point D; that is, find mZD. Exercises 33, 34 34. The drive mechanism on a treadmill is powered by an elec- tric motor. In the figure, find mZD if mABC is 36° larger than mAC. 40. Circles O, P, and Q are tangent (as shown) at points X, Y, and Z. Being as specific as possible, explain what type of triangle APQO is if: a) OX2, PY = 3, QZ = 1 b) OX = 2, PY = 3, QZ = 2 Q Exercises 40, 41 (c) 322 Unless otherwise noted, all content on this page is Cengage Learning. Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 41. Circles O, P, and Q are tangent (as shown for Exercise 40 on page 296) at points X, Y, and Z. Being as specific as possible, explain what type of triangle APQO is if: a) OX 3, PY = 4, QZ = 1 b) OX = 2, PY = 2, QZ = 2 *42. If the larger gear has 30 teeth and the smaller gear has 18, then the gear ratio (larger to smaller) is 5:3. When the larger gear rotates through an angle of 60°, through what angle measure does the smaller gear rotate? Exercises 42, 43 *43. For the drawing in Exercise 42, suppose that the larger gear has 20 teeth and the smaller gear has 10 (the gear ratio is 2:1). If the smaller gear rotates through an angle of 90°, through what angle measure does the larger gear rotate? 6.4 ■Some Constructions and Inequalities for the Circle 297 In Exercises 44 to 47, prove the stated theorem. 44. If a line is drawn through the center of a circle perpendicu- lar to a chord, then it bisects the chord and its minor arc. See Figure 6.37 on page 288. (NOTE: The major arc is also bisected by the line.) 45. If a line through the center of a circle bisects a chord other than a diameter, then it is perpendicular to the chord. See Figure 6.38 on page 289. 46. If two secant segments are drawn to a circle from an exter- nal point, then the products of the length of each secant with the length of its external segment are equal. See Figure 6.51 on page 294. 47. If a tangent segment and a secant segment are drawn to a circle from an external point, then the square of the length of the tangent equals the product of the length of the secant with the length of its external segment. See Figure 6.52 on page 294. 48. The sides of ZABC are tangent to OD at A and C, respec- tively. Explain why quadrilateral ABCD must be a kite.