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Geometry Section 10-5 1112

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Geometry Section 10-5 1112

1. 1. Section 10-5 TangentsMonday, May 21, 2012
2. 2. Essential Questions • How do you use properties of tangents? • How do you solve problems involving circumscribed polygons?Monday, May 21, 2012
3. 3. Vocabulary 1. Tangent: 2. Point of Tangency: 3. Common Tangent:Monday, May 21, 2012
4. 4. Vocabulary 1. Tangent: A line in the same plane as a circle and intersects the circle in exactly one point 2. Point of Tangency: 3. Common Tangent:Monday, May 21, 2012
5. 5. Vocabulary 1. Tangent: A line in the same plane as a circle and intersects the circle in exactly one point 2. Point of Tangency: The point that a tangent intersects a circle 3. Common Tangent:Monday, May 21, 2012
6. 6. Vocabulary 1. Tangent: A line in the same plane as a circle and intersects the circle in exactly one point 2. Point of Tangency: The point that a tangent intersects a circle 3. Common Tangent: A line, ray, or segment that is tangent to two diﬀerent circles in the same planeMonday, May 21, 2012
7. 7. Theorems Theorem 10.10 - Tangents: Theorem 11.11 - Two Segments:Monday, May 21, 2012
8. 8. Theorems Theorem 10.10 - Tangents: A line is tangent to a circle IFF it is perpendicular to a radius drawn to the point of tangency Theorem 11.11 - Two Segments:Monday, May 21, 2012
9. 9. Theorems Theorem 10.10 - Tangents: A line is tangent to a circle IFF it is perpendicular to a radius drawn to the point of tangency Theorem 11.11 - Two Segments: If two segments from the same exterior point are tangent to a circle, then the segments are congruentMonday, May 21, 2012
10. 10. Example 1 Draw in the common tangents for each ﬁgure. If no tangent exists, state no common tangent. a. b.Monday, May 21, 2012
11. 11. Example 1 Draw in the common tangents for each ﬁgure. If no tangent exists, state no common tangent. a. b. No common tangentMonday, May 21, 2012
12. 12. Example 1 Draw in the common tangents for each ﬁgure. If no tangent exists, state no common tangent. a. b. No common tangentMonday, May 21, 2012
13. 13. Example 1 Draw in the common tangents for each ﬁgure. If no tangent exists, state no common tangent. a. b. No common tangentMonday, May 21, 2012
14. 14. Example 2 KL is a radius of ⊙K. Determine whether LM is tangent to ⊙K. Justify your answer.Monday, May 21, 2012
15. 15. Example 2 KL is a radius of ⊙K. Determine whether LM is tangent to ⊙K. Justify your answer. a +b =c 2 2 2Monday, May 21, 2012
16. 16. Example 2 KL is a radius of ⊙K. Determine whether LM is tangent to ⊙K. Justify your answer. a +b =c 2 2 2 20 + 21 = 29 2 2 2Monday, May 21, 2012
17. 17. Example 2 KL is a radius of ⊙K. Determine whether LM is tangent to ⊙K. Justify your answer. a +b =c 2 2 2 20 + 21 = 29 2 2 2 400 + 441 = 841Monday, May 21, 2012
18. 18. Example 2 KL is a radius of ⊙K. Determine whether LM is tangent to ⊙K. Justify your answer. a +b =c 2 2 2 20 + 21 = 29 2 2 2 400 + 441 = 841 Since the values hold for a right triangle, we know that LM is perpendicular to KL, making it a tangent.Monday, May 21, 2012
19. 19. Example 3 In the ﬁgure, WE is tangent to ⊙D at W. Find the value of x.Monday, May 21, 2012
20. 20. Example 3 In the ﬁgure, WE is tangent to ⊙D at W. Find the value of x. a +b =c 2 2 2Monday, May 21, 2012
21. 21. Example 3 In the ﬁgure, WE is tangent to ⊙D at W. Find the value of x. a +b =c 2 2 2 x + 24 = ( x + 16) 2 2 2Monday, May 21, 2012
22. 22. Example 3 In the ﬁgure, WE is tangent to ⊙D at W. Find the value of x. a +b =c 2 2 2 x + 24 = ( x + 16) 2 2 2 x + 576 = x + 32x + 256 2 2Monday, May 21, 2012
23. 23. Example 3 In the ﬁgure, WE is tangent to ⊙D at W. Find the value of x. a +b =c 2 2 2 x + 24 = ( x + 16) 2 2 2 x2 + 576 = x 2 + 32x + 256 2 2 −x −xMonday, May 21, 2012
24. 24. Example 3 In the ﬁgure, WE is tangent to ⊙D at W. Find the value of x. a +b =c 2 2 2 x + 24 = ( x + 16) 2 2 2 x2 + 576 = x 2 + 32x + 256 2 2 −x −256 −x −256Monday, May 21, 2012
25. 25. Example 3 In the ﬁgure, WE is tangent to ⊙D at W. Find the value of x. a +b =c 2 2 2 x + 24 = ( x + 16) 2 2 2 x2 + 576 = x 2 + 32x + 256 2 2 −x −256 −x −256 320 = 32xMonday, May 21, 2012
26. 26. Example 3 In the ﬁgure, WE is tangent to ⊙D at W. Find the value of x. a +b =c 2 2 2 x + 24 = ( x + 16) 2 2 2 x2 + 576 = x 2 + 32x + 256 2 2 −x −256 −x −256 320 = 32x x = 10Monday, May 21, 2012
27. 27. Example 4 AC and BC are tangent to ⊙Z. Find the value of x.Monday, May 21, 2012
28. 28. Example 4 AC and BC are tangent to ⊙Z. Find the value of x. 3x + 2 = 4x − 3Monday, May 21, 2012
29. 29. Example 4 AC and BC are tangent to ⊙Z. Find the value of x. 3x + 2 = 4x − 3 x=5Monday, May 21, 2012
30. 30. Example 5 Some round cookies are marketed in a triangular package to pique the consumer’s interest. If ∆QRS is circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and SM = 10 cm, ﬁnd the perimeter of ∆QRS.Monday, May 21, 2012
31. 31. Example 5 Some round cookies are marketed in a triangular package to pique the consumer’s interest. If ∆QRS is circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and SM = 10 cm, ﬁnd the perimeter of ∆QRS. 2 8 10Monday, May 21, 2012
32. 32. Example 5 Some round cookies are marketed in a triangular package to pique the consumer’s interest. If ∆QRS is circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and SM = 10 cm, ﬁnd the perimeter of ∆QRS. 2 10 8 10Monday, May 21, 2012
33. 33. Example 5 Some round cookies are marketed in a triangular package to pique the consumer’s interest. If ∆QRS is circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and SM = 10 cm, ﬁnd the perimeter of ∆QRS. 2 10 2 8 10Monday, May 21, 2012
34. 34. Example 5 Some round cookies are marketed in a triangular package to pique the consumer’s interest. If ∆QRS is circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and SM = 10 cm, ﬁnd the perimeter of ∆QRS. 2 10 2 8 10 6Monday, May 21, 2012
35. 35. Example 5 Some round cookies are marketed in a triangular package to pique the consumer’s interest. If ∆QRS is circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and SM = 10 cm, ﬁnd the perimeter of ∆QRS. 2 10 2 8 10 6 6Monday, May 21, 2012
36. 36. Example 5 Some round cookies are marketed in a triangular package to pique the consumer’s interest. If ∆QRS is circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and SM = 10 cm, ﬁnd the perimeter of ∆QRS. 2 P = 10 + 10 + 2 + 2 + 6 + 6 10 2 8 10 6 6Monday, May 21, 2012
37. 37. Example 5 Some round cookies are marketed in a triangular package to pique the consumer’s interest. If ∆QRS is circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and SM = 10 cm, ﬁnd the perimeter of ∆QRS. 2 P = 10 + 10 + 2 + 2 + 6 + 6 10 2 P = 36 cm 8 10 6 6Monday, May 21, 2012
38. 38. Check Your Understanding p. 721 #1-8Monday, May 21, 2012
39. 39. Problem SetMonday, May 21, 2012
40. 40. Problem Set p. 722 #9-31 odd, 45, 49, 55 “Stop thinking whats good for you, and start thinking whats good for everyone else. It changes the game.” - Stephen Basilone & Annie MebaneMonday, May 21, 2012