Int Math 2 Section 5-8 1011

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Properties of Circles

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Int Math 2 Section 5-8 1011

  1. 1. Section 5-8 Properties of CirclesWed, Feb 02
  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, navigationWed, Feb 02
  3. 3. Vocabulary 1. Circle: 2. Radius: 3. Chord: 4. Diameter: 5. Central Angle:Wed, Feb 02
  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:Wed, Feb 02
  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:Wed, Feb 02
  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:Wed, Feb 02
  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:Wed, Feb 02
  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 circleWed, Feb 02
  9. 9. Vocabulary 6. Arc: 7. Semicircle: 8. Minor Arc: 9. Major Arc: 10. Inscribed Angle:Wed, Feb 02
  10. 10. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: 8. Minor Arc: 9. Major Arc: 10. Inscribed Angle:Wed, Feb 02
  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:Wed, Feb 02
  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:Wed, Feb 02
  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:Wed, Feb 02
  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 containsWed, Feb 02
  15. 15. CircleWed, Feb 02
  16. 16. RadiusWed, Feb 02
  17. 17. ChordWed, Feb 02
  18. 18. DiameterWed, Feb 02
  19. 19. Central AngleWed, Feb 02
  20. 20. ArcWed, Feb 02
  21. 21. SemicircleWed, Feb 02
  22. 22. Minor ArcWed, Feb 02
  23. 23. Major ArcWed, Feb 02
  24. 24. Inscribed AngleWed, Feb 02
  25. 25. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB Wed, Feb 02
  26. 26. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132°Wed, Feb 02
  27. 27. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82°Wed, Feb 02
  28. 28. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° x°Wed, Feb 02
  29. 29. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° x° x°Wed, Feb 02
  30. 30. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° x° x°Wed, Feb 02
  31. 31. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 x° x°Wed, Feb 02
  32. 32. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 x° x°Wed, Feb 02
  33. 33. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 x° x°Wed, Feb 02
  34. 34. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 2 2 x° x°Wed, Feb 02
  35. 35. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 2 2 x° x° x = 73Wed, Feb 02
  36. 36. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 2 2 73° 73° x = 73Wed, Feb 02
  37. 37. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° 73° 73°Wed, Feb 02
  38. 38. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠ABC = (mAD + mCD 2 73° 73°Wed, Feb 02
  39. 39. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠ABC = (mAD + mCD 2 1 = (73 + 73) 2 73° 73°Wed, Feb 02
  40. 40. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠ABC = (mAD + mCD 2 1 1 = (73 + 73) = (146) 2 2 73° 73°Wed, Feb 02
  41. 41. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠ABC = (mAD + mCD 2 1 1 = (73 + 73) = (146) = 73° 2 2 73° 73°Wed, Feb 02
  42. 42. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° 73° 73°Wed, Feb 02
  43. 43. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠BCD = (mAD + mAB 2 73° 73°Wed, Feb 02
  44. 44. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠BCD = (mAD + mAB 2 1 = (73 +132) 2 73° 73°Wed, Feb 02
  45. 45. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠BCD = (mAD + mAB 2 1 1 = (73 +132) = (205) 2 2 73° 73°Wed, Feb 02
  46. 46. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠BCD = (mAD + mAB 2 1 1 = (73 +132) = (205) =102.5° 2 2 73° 73°Wed, Feb 02
  47. 47. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° 73° 73°Wed, Feb 02
  48. 48. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠CDA = (mBC + mAB 2 73° 73°Wed, Feb 02
  49. 49. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠CDA = (mBC + mAB 2 1 = (82 +132) 2 73° 73°Wed, Feb 02
  50. 50. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠CDA = (mBC + mAB 2 1 1 = (82 +132) = (214) 2 2 73° 73°Wed, Feb 02
  51. 51. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠CDA = (mBC + mAB 2 1 1 = (82 +132) = (214) =107° 2 2 73° 73°Wed, Feb 02
  52. 52. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° 73° 73°Wed, Feb 02
  53. 53. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠DAB = (mBC + mCD 2 73° 73°Wed, Feb 02
  54. 54. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠DAB = (mBC + mCD 2 1 = (82 + 73) 2 73° 73°Wed, Feb 02
  55. 55. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠DAB = (mBC + mCD 2 1 1 = (82 + 73) = (155) 2 2 73° 73°Wed, Feb 02
  56. 56. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠DAB = (mBC + mCD 2 1 1 = (82 + 73) = (155) = 77.5° 2 2 73° 73°Wed, Feb 02
  57. 57. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° m∠ABC = 73° m∠BCD =102.5° m∠CDA =107° 73° 73° m∠DAB = 77.5°Wed, Feb 02
  58. 58. Example 2 Identify the following for circle P. a. Diameter b. Radius c. Chord  d. mLM   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
  59. 59. Example 2 Identify the following for circle P. a. Diameter b. Radius JK c. Chord  d. mLM   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
  60. 60. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
  61. 61. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
  62. 62. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47°   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
  63. 63. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
  64. 64. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° g. m∠LKJ h. Central AngleWed, Feb 02
  65. 65. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° = 242° g. m∠LKJ h. Central AngleWed, Feb 02
  66. 66. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central AngleWed, Feb 02
  67. 67. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central Angle = 2 (62°) 1Wed, Feb 02
  68. 68. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central Angle = 2 (62°) = 31° 1Wed, Feb 02
  69. 69. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central Angle = 2 (62°) = 31° 1 ∠JPMWed, Feb 02
  70. 70. Problem SetWed, Feb 02
  71. 71. Problem Set 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 RochefoucauldWed, Feb 02

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