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Geometry

Geometry. Other Angle Relationships in Circles. Goal. Use angles formed by tangents, secants, and chords to solve problems. Review. Note: in solving an equation with fractions, one of the first things to do is always “clear the fractions”. You do it. Solve:. Review.

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Geometry

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  1. Geometry Other Angle Relationships in Circles

  2. Goal • Use angles formed by tangents, secants, and chords to solve problems.

  3. Review Note: in solving an equation with fractions, one of the first things to do is always “clear the fractions”.

  4. You do it. Solve:

  5. Review The measure of an inscribed angle is equal to one-half the measure of the intercepted arc. 40 80 What if one side of the angle is tangent to the circle?

  6. If two lines intersect a circle, where can the lines intersect each other? Inside the circle. We already know how to do this. On the circle. Outside the circle.

  7. Theorem 12.13 (Inside the circle) If two chords intersect in a circle, then the measure of the angle is one-half the sum of the intercepted arcs. A C 1 B D

  8. b 1 a Simplified Formula

  9. Example 3 Find m1. A C 30 1 B 80 D

  10. Example 4 Solve for x. A C 20 60 B x 100 Check: 100 + 20 = 120 120 ÷ 2 = 60 D

  11. K 20 A C P 32 M 75 x 85 y O B D Your turn. Solve for x & y.

  12. Intersection Outside the Circle

  13. Secant-Secant A C 1 D B

  14. a 1 b Simplified Formula

  15. Secant-Tangent A C 1 B

  16. a b 1 Simplified Formula

  17. Tangent-Tangent A C 1 B

  18. 1 b a Simplified Formula

  19. Intersection Outside the Circle Secant-Secant Secant-Tangent Tangent-Tangent In all cases, the measure of the exterior angle is found the same way: One-half the difference of the larger and smaller arcs. (Click the titles above for Sketchpad Demonstrations.)

  20. Example 5 Find m1. 80 1 35 10

  21. Example 6 Find m1. 120 70 1 25

  22. Example 7 Find m1. k Rays k and m are tangent to the circle. 210 150 ? 1 30 m 360 – 210 = 150

  23. How to remember this: • If the angle vertex is on the circle, its measure is one-half the intercepted arc. • If an angle vertex is inside the circle, its measure if one-half the sum of the intercepted arcs. • If an angle vertex is outside the circle, its measure is one-half the difference of the intercepted arcs.

  24. Chords in a Circle Theorem 12.15 d a b c a  b = c  d

  25. Example 1 Find a. 10  4 = 8  a 40 = 8  a 5 = a 10 a 5 4 8

  26. Your Turn: Find x. 3x  x = 8  6 3x2 = 48 x2 = 16 x = 4 A D 3x 6 12 E x 4 B 8 C Check: 12  4 = 48 and 8  6 = 48

  27. Terminology This line is a secant. This segment is a secant segment.

  28. Terminology This segment is the external secant segment.

  29. Terminology This line is a tangent. This segment is a tangent segment.

  30. C B A D Terminology secant segment AC is a __________________. AB is the _________________________. AD is a _________________. external secant segment tangent segment

  31. C B A D Theorem 12.17 (tangent-secant)

  32. Theorem 12.17 (simplified) c2 = a(a + b) b a c

  33. C 6 B 4 A D Example 2 Find AD.

  34. Your Turn. Solve for x. 4 8 4 x

  35. Turn it up a notch… 4 x Now What? 5

  36. Quadratic Equation Set quadratic equations equal to zero.

  37. Quadratic Formula 1 a = 1 c = -25 b = 4

  38. Quadratic Formula 1 a = 1 c = -25 b = 4

  39. Solve it. x can’t be negative x  3.39

  40. All that for just one problem? Just do it!

  41. 3 x 2 Your Turn Solve for x. Equation: 32 = x(x + 2) x + 2

  42. 3 x 2 Solution 32 = x(x + 2) 9 = x2 + 2x 0 = x2 + 2x – 9 a = 1 b = 2 c = -9

  43. Theorem 12.16 (secant-secant) b a d c a(a+b) = c(c+d)

  44. 8 5 6 X Example 3 Solve for x. Solution: 5(5 + 8) = 6(6 + x) 5(13) = 36 + 6x 65 = 36 + 6x 29 = 6x x = 4 5/6 (or 4.83)

  45. 11 9 10 X Your Turn Solve for x.

  46. 11 9 10 X Solution

  47. 16 12 5 X Example 4 Solve for x. Equation: 5x = 4(16) Why? 5x = 64 x = 12.8 4

  48. b a a d c b c d ab = cd a(a+b) = c(c+d) b a c c2 = a(a + b) Formula Summary

  49. Practice Problems

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