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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