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Circles

Circles. Points & Circle Relationships. Inside the circle THE circle Outside the circle. B. G. A. F. E. D. C. Parts of a Circle. R. Center Radius Diameter Chord Is a diameter a chord?. A. C. P. B. D. Parts of a Circle. R. Center: P Radius: PR Diameter: AB

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Circles

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  1. Circles

  2. Points & Circle Relationships • Inside the circle • THE circle • Outside the circle B G A F E D C

  3. Parts of a Circle R • Center • Radius • Diameter • Chord Is a diameter a chord? A C P B D

  4. Parts of a Circle R • Center: P • Radius: PR • Diameter: AB • Chord: CD & AB Is a diameter a chord? YES A C P B D

  5. Construct a Regular Hexagon • With a compass – make a circle • DO NOT CHANGE compass measure

  6. Construct a Regular Hexagon • Place point of compass on the circle

  7. Construct a Regular Hexagon • Make an arc to the left and right side of the compass on the circle

  8. Construct a Regular Hexagon • Move compass to arcs and repeat 4 & 5 until you have 6 marks

  9. Construct a Regular Hexagon • Connect the consecutive marks

  10. Major & Minor Arcs • An Arc is part of a circle. • Minor Arc is less than half • Major Arc is more than half Identify the Minor Arcs and The Major Arcs… B A E C

  11. Major & Minor Arcs B A Identify the Minor Arcs and The Major Arcs… • Minor Arcs: AB, BC, AC • Major Arcs: ABC, BCA, BAC E ) ) ) ) ) ) C

  12. Semicircles • An arc that is exactly half the circle. D E F

  13. Measure of Arc Arcs are measured in two ways • Degrees • Length

  14. Arc Measure: Degrees • The arc measure corresponds the the central angle. What is the mAB? B ) A 120° 95° P C

  15. Arc Measure: Degrees ) • What is the mAB? B A 120° 95° P C

  16. Arc Measure: Degrees ) • What is the mAB? 120° B A 120° 95° P C

  17. Arc Measure: Degrees ) • What is the mAB? 120° • What is the mBC? ) B A 120° 95° P C

  18. Arc Measure: Degrees ) • What is the mAB? 120° • What is the mBC? 95° ) B A 120° 95° P C

  19. Arc Measure: Degrees ) • What is the mAB? 120° • What is the mBC? 95° • What is the mAC? ) B ) A 120° 95° P C

  20. Arc Measure: Degrees ) • What is the mAB? 120° • What is the mBC? 95° • What is the mAC? 145° ) B ) A 120° 95° P C

  21. Arc Measure: Degrees ) • What is the mAB? 120° • What is the mBC? 95° • What is the mAC? 145° • What is the mACB? ) ) B A 120° ) 95° P C

  22. Arc Measure: Degrees ) • What is the mAB? 120° • What is the mBC? 95° • What is the mAC? 145° • What is the mACB? 240° ) ) B A 120° ) 95° P C

  23. Arc Measure: Length • The length is part of the circumference… so you would have to know the radius. B A 120° 95° P C

  24. Arc Measure: Length • The length is part of the circumference… so you would have to know the radius. And the formula Length = 2pr B A 120° degree° 360 95° 5cm P · C

  25. Arc Measure: Length degree° 360 Length = 2pr AB = · ) B A 120° 95° 5cm P C

  26. Arc Measure: Length degree° 360 Length = 2pr AB = 2p5(120/360) · ) B A 120° 95° 5cm P C

  27. Arc Measure: Length degree° 360 Length = 2pr AB = 2p5(120/360) = 10.47 cm · ) B A 120° 95° 5cm P C

  28. Arc Measure: Length degree° 360 Length = 2pr AC = · ) B A 120° 95° 5cm P C

  29. Arc Measure: Length degree° 360 Length = 2pr AC = 2p5(145/360) · ) B A 120° 95° 5cm P C

  30. Arc Measure: Length degree° 360 Length = 2pr AC = 2p5(145/360) = 12.65 cm · ) B A 120° 95° 5cm P C

  31. Chords and Arcs Theorem • What would you think if 2 chords of a circle had equal length? B A P C D

  32. Chords and Arcs Theorem • What would you think if 2 chords of a circle had equal length? B A P ) ) AC @ BD ? C D

  33. Chords and Arcs Theorem • What would you think if 2 chords of a circle had equal length? B A P ) ) AC @ BD ? Prove it! C D

  34. Chords and Arcs Theorem • Draw lines to each point • What do you know about the dotted lines? B A P C D

  35. Chords and Arcs Theorem • AP @ BP (radii of the same O are @) • CP @ DP B A P C D

  36. Chords and Arcs Theorem • AP @ BP (radii of the same O are @) • CP @ DP B A What do you know about the triangles? P C D

  37. Chords and Arcs Theorem • The D‘s are @ by SSS B A P C D

  38. Chords and Arcs Theorem • The D‘s are @ by SSS B A P So where does this lead… C D

  39. Chords and Arcs Theorem • What do you know about angles 1 & 2? B A P 1 2 C D

  40. Chords and Arcs Theorem • What do you know about angles 1 & 2? B A P <1 @ <2 by CPCTC 1 2 C D

  41. Chords and Arcs Theorem • So the Central angles are @ • And the arcs formed are @ B A P C D

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