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CHAPTER 7 Similarity
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(7.28)
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7.42. Prove each of the following: (a) If two chords intersect in a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other.
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(b) In a right triangle, the product of the lengths of the hypotenuse and the altitude upon it equals the product of the lengths of the legs. (c) If in inscribed nABC the bisector of /A intersects BC in D and the circle in E, then BD AC AD EC. 7.43. In Fig. 7-60 (a) Let AE (b) Let AB (c) Let AE (d) Let ED 10, EB 15, EB 6, ED 5, EB 6, CE 8, ED 4, CD 12. Find ED. 4. Find CE. 13. Find EB. 15. Find AE. In Fig. 7-61, diameter CD ' chord AB (e) Let OD (f) Let AB (g) Let OD (h) Let AB 10, OE 24, OE 25, EC 8, OD 8. Find AB. 5. Find OD. 18. Find AB. 5. Find EC. (7.29)
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2(AE), CD
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Fig. 7-60
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Fig. 7-61
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7.44. A point is 12 in from the center of a circle whose radius is 15 in. Find the lengths of the longest and shortest chords that can be drawn through this point. (Hint: The longest chord is a diameter, and the shortest chord is perpendicular to this diameter.) (7.29) 7.45. In Fig. 7-62, AB is a tangent (a) Let AC (b) Let CD (c) Let AB (d) Let AC (e) Let AB 16, AD 5, AD 6, AD 20, AB 12, AD 4. Find AB. 4. Find AB. 3. Find AC. 10. Find AD. 9. Find CD. In Fig. 7-63, CD is a diameter; AB is a tangent (f) Let AD (g) Let AD (h) Let AD (i) Let AB (j) Let OD 6, OD 2, AB 5, AB 12, AC 5, AB 9. Find AB. 8. Find CD. 10. Find OD. 18. Find OD. 12. Find AD. (7.29)
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Fig. 7-62
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Fig. 7-63
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7.46. In Fig. 7-64 (a) Let AB (b) Let AC (c) Let BD (d) Let AD 14, AD 8, AE 5, AD DB, EC 4, AE 6, AD 7, AE 7. Find AC. 3. Find BD. 4. Find AC. 4. Find AD.
In Fig. 7-65, CE is a diameter (e) Let OC (f) Let BD (g) Let OC (h) Let OC 3, AE 7, AD 11, AB 5, AE 6, AD 5, AE 15, AD 6, BD 8. Find AB. 2. Find OC. 5. Find AE. 4. Find AD.
(7.31)
14, AE
CHAPTER 7 Similarity
Fig. 7-64
Fig. 7-65
7.47. CD is the altitude to hypotenuse AB in Fig. 7-66. (a) If p (b) If p 2 and q 4 and a 6, find a and h. 6, find c and h. (c) If p (d) If b 16 and h 12 and q 8, find q and b. 6, find p and h.
(7.32)
Fig. 7-66
7.48. In a right triangle whose arms have lengths a and b, find the length of the hypotenuse c when (a) a (b) a 15, b 15, b 20 36 (c) a (d) a 5, b 5, b 4 5 23 (e) a 7, b 7
(7.33)
7.49. In the right triangle in Fig. 7-67, find the length of each missing arm when (a) a (b) b 12, c 6, c 20 8 (c) b (d) a 15, c 2, c 17 4 (e) a (f) a 5 22, c 25, c 10 2 22
(7.33)
Fig. 7-67
7.50. Find the lengths of the arms of a right triangle whose hypotenuse has length c if these arms have a ratio of (a) 3:4 and c 15; (b) 5:12 and c 26; (c) 8:15 and c 170; (d) 1: 2 and c 10. (7.34) 7.51. In a rectangle, find the length of the diagonal if its sides have lengths (a) 9 and 40; (b) 5 and 10. (7.33)
7.52. In a rectangle, find the length of one side if the diagonal has length 15 and the other side has length (a) 9; (b) 5; (c) 10. (7.33) 7.53. Of triangles having sides with lengths as follows, which are right triangles (a) 33, 55, 44 (b) 120, 130, 50 (c) 4, 72, 82 (d) 25, 7, 24
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