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Integrated Math 2 Section 5-8
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Integrated Math 2 Section 5-8

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Properties of Circles

Properties of Circles

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  • 1. Section 5-8 Properties of Circles
  • 2. Essential Questions • What are the relationships among parts of a circle? • What are the properties of circles and how do you apply them? • Where you’ll see this: • Market research, food service, art, recreation, navigation
  • 3. Vocabulary 1. Circle: 2. Radius: 3. Chord: 4. Diameter: 5. Central Angle:
  • 4. Vocabulary 1. Circle: All points that are the same distance from a fixed center point; 360° total 2. Radius: 3. Chord: 4. Diameter: 5. Central Angle:
  • 5. Vocabulary 1. Circle: All points that are the same distance from a fixed center point; 360° total 2. Radius: A segment whose endpoints are the center of a circle and on the circle 3. Chord: 4. Diameter: 5. Central Angle:
  • 6. Vocabulary 1. Circle: All points that are the same distance from a fixed center point; 360° total 2. Radius: A segment whose endpoints are the center of a circle and on the circle 3. Chord: A segment where both endpoints are on the circle 4. Diameter: 5. Central Angle:
  • 7. Vocabulary 1. Circle: All points that are the same distance from a fixed center point; 360° total 2. Radius: A segment whose endpoints are the center of a circle and on the circle 3. Chord: A segment where both endpoints are on the circle 4. Diameter: A chord that goes through the center of a circle 5. Central Angle:
  • 8. Vocabulary 1. Circle: All points that are the same distance from a fixed center point; 360° total 2. Radius: A segment whose endpoints are the center of a circle and on the circle 3. Chord: A segment where both endpoints are on the circle 4. Diameter: A chord that goes through the center of a circle 5. Central Angle: An angle where the vertex is the center of the circle
  • 9. Vocabulary 6. Arc: 7. Semicircle: 8. Minor Arc: 9. Major Arc: 10. Inscribed Angle:
  • 10. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: 8. Minor Arc: 9. Major Arc: 10. Inscribed Angle:
  • 11. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: An arc that is half of the circumference; half a circle 8. Minor Arc: 9. Major Arc: 10. Inscribed Angle:
  • 12. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: An arc that is half of the circumference; half a circle 8. Minor Arc: An arc that is less than half the circumference; same measure as the central angle 9. Major Arc: 10. Inscribed Angle:
  • 13. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: An arc that is half of the circumference; half a circle 8. Minor Arc: An arc that is less than half the circumference; same measure as the central angle 9. Major Arc: An arc that is more than half the circumference 10. Inscribed Angle:
  • 14. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: An arc that is half of the circumference; half a circle 8. Minor Arc: An arc that is less than half the circumference; same measure as the central angle 9. Major Arc: An arc that is more than half the circumference 10. Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle; half the measure of the arc it contains
  • 15. Circle
  • 16. Radius
  • 17. Chord
  • 18. Diameter
  • 19. Central Angle
  • 20. Arc
  • 21. Semicircle
  • 22. Minor Arc
  • 23. Major Arc
  • 24. Inscribed Angle
  • 25. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫
  • 26. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 132°
  • 27. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 132° 82°
  • 28. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 132° 82° x°
  • 29. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 132° 82° x° x°
  • 30. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ x + x +132 + 82 = 360 132° 82° x° x°
  • 31. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 x° x°
  • 32. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 x° x°
  • 33. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 x° x°
  • 34. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 2 2 x° x°
  • 35. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 2 2 x° x° x = 73
  • 36. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 2 2 73° 73° x = 73
  • 37. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 132° 82° 73° 73°
  • 38. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ª ª ) 132° 82° m∠ABC = (mAD + mCD 2 73° 73°
  • 39. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ª ª ) 132° 82° m∠ABC = (mAD + mCD 2 1 = (73 + 73) 2 73° 73°
  • 40. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ª ª ) 132° 82° m∠ABC = (mAD + mCD 2 1 1 = (73 + 73) = (146) 2 2 73° 73°
  • 41. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ª ª ) 132° 82° m∠ABC = (mAD + mCD 2 1 1 = (73 + 73) = (146) = 73° 2 2 73° 73°
  • 42. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 132° 82° 73° 73°
  • 43. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ª ª ) 132° 82° m∠BCD = (mAD + mAB 2 73° 73°
  • 44. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ª ª ) 132° 82° m∠BCD = (mAD + mAB 2 1 = (73 +132) 2 73° 73°
  • 45. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ª ª ) 132° 82° m∠BCD = (mAD + mAB 2 1 1 = (73 +132) = (205) 2 2 73° 73°
  • 46. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ª ª ) 132° 82° m∠BCD = (mAD + mAB 2 1 1 = (73 +132) = (205) =102.5° 2 2 73° 73°
  • 47. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 132° 82° 73° 73°
  • 48. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ∫ ª ) 132° 82° m∠CDA = (mBC + mAB 2 73° 73°
  • 49. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ∫ ª ) 132° 82° m∠CDA = (mBC + mAB 2 1 = (82 +132) 2 73° 73°
  • 50. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ∫ ª ) 132° 82° m∠CDA = (mBC + mAB 2 1 1 = (82 +132) = (214) 2 2 73° 73°
  • 51. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ∫ ª ) 132° 82° m∠CDA = (mBC + mAB 2 1 1 = (82 +132) = (214) =107° 2 2 73° 73°
  • 52. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 132° 82° 73° 73°
  • 53. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ∫ ª ) 132° 82° m∠DAB = (mBC + mCD 2 73° 73°
  • 54. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ∫ ª ) 132° 82° m∠DAB = (mBC + mCD 2 1 = (82 + 73) 2 73° 73°
  • 55. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ∫ ª ) 132° 82° m∠DAB = (mBC + mCD 2 1 1 = (82 + 73) = (155) 2 2 73° 73°
  • 56. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 1 ∫ ª ) 132° 82° m∠DAB = (mBC + mCD 2 1 1 = (82 + 73) = (155) = 77.5° 2 2 73° 73°
  • 57. Example 1 ª ≅ CD . Find the measures of the ª In circle O, AD angles of quadrilateral ABCD, when ª =132° and mBC = 82°. mAB ∫ 132° 82° m∠ABC = 73° m∠BCD =102.5° m∠CDA =107° 73° 73° m∠DAB = 77.5°
  • 58. Example 2 Identify the following for circle P. a. Diameter b. Radius c. Chord ª d. mLM ) º e. mLMK f. mLJ g. m∠LKJ h. Central Angle
  • 59. Example 2 Identify the following for circle P. a. Diameter b. Radius JK c. Chord ª d. mLM ) º e. mLMK f. mLJ g. m∠LKJ h. Central Angle
  • 60. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord ª d. mLM ) º e. mLMK f. mLJ g. m∠LKJ h. Central Angle
  • 61. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord ª d. mLM KL ) º e. mLMK f. mLJ g. m∠LKJ h. Central Angle
  • 62. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord ª d. mLM KL = 62° + 47° ) º e. mLMK f. mLJ g. m∠LKJ h. Central Angle
  • 63. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord ª d. mLM KL = 62° + 47° =109° ) º e. mLMK f. mLJ g. m∠LKJ h. Central Angle
  • 64. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord ª d. mLM KL = 62° + 47° =109° ) º e. mLMK f. mLJ = 62° +180° g. m∠LKJ h. Central Angle
  • 65. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord ª d. mLM KL = 62° + 47° =109° ) º e. mLMK f. mLJ = 62° +180° = 242° g. m∠LKJ h. Central Angle
  • 66. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord ª d. mLM KL = 62° + 47° =109° ) º e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central Angle
  • 67. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord ª d. mLM KL = 62° + 47° =109° ) º e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central Angle 1 = 2 (62°)
  • 68. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord ª d. mLM KL = 62° + 47° =109° ) º e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central Angle = 2 (62°) = 31° 1
  • 69. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord ª d. mLM KL = 62° + 47° =109° ) º e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central Angle = 2 (62°) = 31° 1 ∠JPM
  • 70. Homework
  • 71. Homework p. 228 #1-25 odd “We are so accustomed to disguise ourselves to others that in the end we become disguised to ourselves.” - Francois de La Rochefoucauld