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Geometry

Geometry. Circles 10.1. Goals. Know properties of circles. Identify special lines in a circle. Solve problems with special lines. CR is a radius. AB is a diameter. Circle: Set of points on a plane equidistant from a point (center). B. This is circle C, or. C. R. A.

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Geometry

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  1. Geometry Circles 10.1

  2. Goals • Know properties of circles. • Identify special lines in a circle. • Solve problems with special lines.

  3. CR is a radius. AB is a diameter. Circle: Set of points on a plane equidistant from a point (center). B This is circle C, or C R A The diameter is twice the radius.

  4. Terminology • One radius • Two radii • radii = ray-dee-eye

  5. All Radii in a circle are congruent

  6. A B Interior/Exterior A is in the interior of the circle. C is on the circle. C B is in the exterior of the circle.

  7. Congruent Circles Radii are congruent.

  8. Lines in a circle.

  9. Chord A chord is a segment between two points on a circle. A diameter is a chord that passes through the center.

  10. Secant A secant is a line that intersects a circle at two points.

  11. Tangent • A tangent is a line that intersects a circle at only one point. • It is called the point of tangency.

  12. Tangent Circles Intersect at exactly one point. These circles are externally tangent.

  13. Tangent Circles Intersect at exactly one point. These circles are internally tangent.

  14. Can circles intersect at two points? YES!

  15. Concentric Circles Have the same center, different radius.

  16. Concentric Circles Have the same center, different radius.

  17. Concentric Circles Have the same center, different radius.

  18. Concentric Circles Have the same center, different radius.

  19. Concentric Circles Have the same center, different radius.

  20. Concentric Circles Have the same center, different radius.

  21. Concentric Circles Have the same center, different radius.

  22. Concentric Circles Have the same center, different radius.

  23. Common External Tangents And this is a common external tangent. This is a common external tangent.

  24. Common External Tangents in a real application…

  25. Common Internal Tangents And this is a common internal tangent. This is a common internal tangent.

  26. Tangent Theorems

  27. Theorem 12.1 (w/o proof) If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

  28. Theorem 12.2 (w/o proof) If a line drawn to a circle is perpendicular to a radius, then the line is a tangent to the circle. (The converse of 10.1)

  29. Is RA tangent to T? Example 1 YES R 12 A 5 52 + 122 = 132 25 + 144 = 169 169 = 169 13 T TA = 13 RAT is a right triangle.

  30. FOIL Find (x + 3)2 (x + 3)(x + 3)

  31. FOIL Find (x + 3)2 (x + 3)(x + 3) x2

  32. FOIL Find (x + 3)2 (x + 3)(x + 3) 3x x2

  33. FOIL Find (x + 3)2 (x + 3)(x + 3) 3x x2 + 3x

  34. FOIL Find (x + 3)2 (x + 3)(x + 3) 9 x2 + 3x + 3x

  35. FOIL Find (x + 3)2 (x + 3)(x + 3) x2 + 3x + 3x + 9

  36. FOIL (x + 3)2 = x2 + 6x + 9

  37. Expand (x + 9)2 • (x + 9)(x + 9) • F: x2 • O: 9x • I: 9x • L: 81 • (x + 9)2 = x2 + 18x + 81

  38. A D 16 r C B 24 DC = 16 Example 2 BC is tangent to circle A at B. Find r. AC = r + 16 AC = ? r r2 + 242 = (r + 16)2

  39. Solve the equation. r2 + 242 = (r + 16)2 r2 + 576 = (r + 16)(r + 16) r2 + 576 = r2 + 16r + 16r + 256 576 = 32r + 256 320 = 32r r = 10 r2 + 242 = (r + 16)2

  40. Here’s where the situation is now. 26 A 10 D 16 10 C B 24 Check: 102 + 242 = 262 100 + 576 = 676 676 = 676 AC = 26 r = 10

  41. Theorem 12.3 • If two segments from the same exterior point are tangent to a circle, then the segments are congruent. Theorem Demo

  42. Example 3 HE and HA are tangent to the circle. Solve for x. A 12x + 15 H 9x + 45 E

  43. Solution 12(10) + 15 120 + 15 = 135 12x + 15 = 9x + 45 3x + 15 = 45 3x = 30 x = 10 A 12x + 15 H 9x + 45 9(10) + 45 90 + 45 = 135 E

  44. Try This: The circle is tangent to each side of ABC. Find the perimeter of ABC. 7 + 12 + 9 = 28 A 2 2 9 7 7 5 C B 7 5 12

  45. Can you… • Identify a radius, diameter? • Recognize a tangent or secant? • Define Concentric circles? Internally tangent circles? Externally tangent? • Tell the difference between internal and external tangents? • Solve problems using tangent properties?

  46. Practice Problem 1 MD and ME are tangent to the circle. Solve for x. D 4x – 12 = 2x + 12 2x – 12 = 12 2x = 24 x = 12 4x  12 M 2x + 12 E

  47. R x 4 T 12 Solve for x. Practice Problem 2 x2 + 42 = (4 + 12)2 x2 + 16 = 256 x2 = 240 x = 415 15.5

  48. R 8 x 6 x T Solve for x. Practice Problem 3 x2 + 82 = (x + 6)2 x2 + 64 = x2 + 12x + 36 64 = 12x + 36 28 = 12x x = 2.333…

  49. Practice Problems

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