2. Geometry

3. June 8, 2015Geometry 10.1 Tangents to Circles2 Goals  Know properties of circles.  Identify special lines in a circle.  Solve problems with special lines.

4. June 8, 2015Geometry 10.1 Tangents to Circles3 Circle: Set of points on a plane equidistant from a point (center). C This is circle C, or R CR is a radius. A B AB is a diameter. The diameter is twice the radius.

5. June 8, 2015Geometry 10.1 Tangents to Circles4 Terminology  One radius  Two radii  radii = ray-dee-eye

6. June 8, 2015Geometry 10.1 Tangents to Circles5 All Radii in a circle are congruent

7. June 8, 2015Geometry 10.1 Tangents to Circles6 Interior/Exterior A A is in the interior of the circle. B B is in the exterior of the circle. C C is on the circle.

8. June 8, 2015Geometry 10.1 Tangents to Circles7 Congruent Circles Radii are congruent.

9. June 8, 2015Geometry 10.1 Tangents to Circles8

10. June 8, 2015Geometry 10.1 Tangents to Circles9 Chord A chord is a segment between two points on a circle. A diameter is a chord that passes through the center.

11. June 8, 2015Geometry 10.1 Tangents to Circles10 Secant A secant is a line that intersects a circle at two points.

12. June 8, 2015Geometry 10.1 Tangents to Circles11 Tangent A tangent is a line that intersects a circle at only one point. It is called the point of tangency.

13. June 8, 2015Geometry 10.1 Tangents to Circles12 Tangent Circles Intersect at exactly one point. These circles are externally tangent.

14. June 8, 2015Geometry 10.1 Tangents to Circles13 Tangent Circles Intersect at exactly one point. These circles are internally tangent.

15. June 8, 2015Geometry 10.1 Tangents to Circles14 Can circles intersect at two points? YES!

16. June 8, 2015Geometry 10.1 Tangents to Circles15 Concentric Circles Have the same center, different radius.

17. June 8, 2015Geometry 10.1 Tangents to Circles16 Concentric Circles Have the same center, different radius.

18. June 8, 2015Geometry 10.1 Tangents to Circles17 Concentric Circles Have the same center, different radius.

19. June 8, 2015Geometry 10.1 Tangents to Circles18 Concentric Circles Have the same center, different radius.

20. June 8, 2015Geometry 10.1 Tangents to Circles19 Concentric Circles Have the same center, different radius.

21. June 8, 2015Geometry 10.1 Tangents to Circles20 Concentric Circles Have the same center, different radius.

22. June 8, 2015Geometry 10.1 Tangents to Circles21 Concentric Circles Have the same center, different radius.

23. June 8, 2015Geometry 10.1 Tangents to Circles22 Concentric Circles Have the same center, different radius.

24. June 8, 2015Geometry 10.1 Tangents to Circles23 Common External Tangents This is a common external tangent. And this is a common external tangent.

25. June 8, 2015Geometry 10.1 Tangents to Circles24 Common External Tangents in a real application…

26. June 8, 2015Geometry 10.1 Tangents to Circles25 Common Internal Tangents This is a common internal tangent. And this is a common internal tangent.

27. June 8, 2015Geometry 10.1 Tangents to Circles26

28. June 8, 2015Geometry 10.1 Tangents to Circles27 Theorem 10.1 (w/o proof) If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

29. June 8, 2015Geometry 10.1 Tangents to Circles28 Theorem 10.2 (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)

30. June 8, 2015Geometry 10.1 Tangents to Circles29 Example 1 RA T 5 12 13 YES TA = 13  RAT is a right triangle. 5 2 + 12 2 = 13 2 25 + 144 = 169 169 = 169 Is RA tangent to  T?

31. June 8, 2015Geometry 10.1 Tangents to Circles30 FOIL Find (x + 3) 2 (x + 3)(x + 3)

32. June 8, 2015Geometry 10.1 Tangents to Circles31 FOIL Find (x + 3) 2 (x + 3)(x + 3) x2x2

33. June 8, 2015Geometry 10.1 Tangents to Circles32 FOIL Find (x + 3) 2 (x + 3)(x + 3) x2x2 3x

34. June 8, 2015Geometry 10.1 Tangents to Circles33 FOIL Find (x + 3) 2 (x + 3)(x + 3) x 2 + 3x 3x

35. June 8, 2015Geometry 10.1 Tangents to Circles34 FOIL Find (x + 3) 2 (x + 3)(x + 3) 9 x 2 + 3x + 3x

36. June 8, 2015Geometry 10.1 Tangents to Circles35 FOIL Find (x + 3) 2 (x + 3)(x + 3) x 2 + 3x + 3x + 9

37. June 8, 2015Geometry 10.1 Tangents to Circles36 FOIL (x + 3) 2 = x 2 + 6x + 9

38. June 8, 2015Geometry 10.1 Tangents to Circles37 Expand (x + 9) 2  (x + 9)(x + 9)  F: x 2  O: 9x  I: 9x  L: 81  (x + 9) 2 = x 2 + 18x + 81

39. June 8, 2015Geometry 10.1 Tangents to Circles38 Example 2 BC is tangent to circle A at B. Find r. A B C r 16 24 D DC = 16 r AC = ? AC = r + 16 r 2 + 24 2 = (r + 16) 2

40. June 8, 2015Geometry 10.1 Tangents to Circles39 r 2 + 24 2 = (r + 16) 2 r 2 + 576 = (r + 16)(r + 16) r 2 + 576 = r 2 + 16r + 16r + 256 576 = 32r + 256 320 = 32r r = 10 Solve the equation. r 2 + 24 2 = (r + 16) 2

41. June 8, 2015Geometry 10.1 Tangents to Circles40 A B C 10 16 24 D AC = 26 10 Here’s where the situation is now. Check: 10 2 + 24 2 = 26 2 100 + 576 = 676 676 = 676 26 r = 10

42. June 8, 2015Geometry 10.1 Tangents to Circles41 Theorem 10.3  If two segments from the same exterior point are tangent to a circle, then the segments are congruent. Theorem Demo

43. June 8, 2015Geometry 10.1 Tangents to Circles42 Example 3 HE and HA are tangent to the circle. Solve for x. H A E 12x + 15 9x + 45

44. June 8, 2015Geometry 10.1 Tangents to Circles43 Solution 12x + 15 = 9x + 45 3x + 15 = 45 3x = 30 x = 10 H A E 12x + 15 9x + 45 9(10) + 45 90 + 45 = 135 12(10) + 15 120 + 15 = 135

45. June 8, 2015Geometry 10.1 Tangents to Circles44 Try This: The circle is tangent to each side of  ABC. Find the perimeter of  ABC. A B C 7 2 5 2 5 7 9 7 12 7 + 12 + 9 = 28

46. June 8, 2015Geometry 10.1 Tangents to Circles45 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?

47. June 8, 2015Geometry 10.1 Tangents to Circles46 Practice Problems Skip

48. June 8, 2015Geometry 10.1 Tangents to Circles47 Practice Problem 1 MD and ME are tangent to the circle. Solve for x. D E M 4x  12 2x + 12 4x – 12 = 2x + 12 2x – 12 = 12 2x = 24 x = 12

49. June 8, 2015Geometry 10.1 Tangents to Circles48 Practice Problem 2 x 2 + 4 2 = (4 + 12) 2 x 2 + 16 = 256 x 2 = 240 x = 415  15.5 R T 4 12 x Solve for x.

50. June 8, 2015Geometry 10.1 Tangents to Circles49 Practice Problem 3 x 2 + 8 2 = (x + 6) 2 x 2 + 64 = x 2 + 12x + 36 64 = 12x + 36 28 = 12x x = 2.333… R T 6 8 x Solve for x. x