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Integrated Math 2 Section 5-8

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Properties of Circles

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Integrated Math 2 Section 5-8

  1. 1. Section 5-8 Properties of Circles
  2. 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. 3. Vocabulary 1. Circle: 2. Radius: 3. Chord: 4. Diameter: 5. Central Angle:
  4. 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. 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. 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. 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. 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. 9. Vocabulary 6. Arc: 7. Semicircle: 8. Minor Arc: 9. Major Arc: 10. Inscribed Angle:
  10. 10. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: 8. Minor Arc: 9. Major Arc: 10. Inscribed Angle:
  11. 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. 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. 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. 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. 15. Circle
  16. 16. Radius
  17. 17. Chord
  18. 18. Diameter
  19. 19. Central Angle
  20. 20. Arc
  21. 21. Semicircle
  22. 22. Minor Arc
  23. 23. Major Arc
  24. 24. Inscribed Angle
  25. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 70. Homework
  71. 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

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