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Circle Segments and Volume

Circle Segments and Volume. Chords of Circles Theorem 1. In the same circle, or in congruent circles two minor arcs are congruent if and only if their corresponding chords are congruent. Chord Arcs Conjecture.

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Circle Segments and Volume

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  1. Circle Segments and Volume

  2. Chords of Circles Theorem 1

  3. In the same circle, or in congruent circles two minor arcs are congruent if and only if their corresponding chords are congruent.

  4. Chord Arcs Conjecture In the same circle, two minor arcs are congruent if and only if their corresponding chords are congruent. IFF G and IFF and

  5. Solve for x. 8x – 7 3x + 3 8x – 7 = 3x + 3 x = 2

  6. Example Find WX.

  7. Example Find 130º

  8. Chords of Circles Theorem 2

  9. If a diameter is perpendicular to a chord, then it also bisects the chord and its arc.This results in congruent arcs too.Sometimes, this creates a right triangle & you’ll use Pythagorean Theorem.

  10. If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Perpendicular Bisector of a Chord Conjecture H

  11. IN Q, KL  LZ. If CK = 2x + 3 and CZ = 4x, find x. x = 1.5 Q C Z K L

  12. In P, if PM  AT, PT = 10, and PM = 8, find AT. P A M MT = 6 T AT = 12

  13. Chords of Circles Theorem 3

  14. If one chord is a perpendicular bisector of another chord, then the bisecting chord is a diameter . Perpendicular Bisector to a Chord Conjecture • is a diameter of the circle.

  15. • , •  If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

  16. Chords of Circles Theorem 4

  17. In the same circle or in congruent circles two chords are congruent when they are equidistant from the center.

  18. Chord Distance to the Center Conjecture

  19. In K, K is the midpoint of RE. If TY = -3x + 56 and US = 4x, find the length of TY. U T K E x = 8 R S Y TY = 32

  20. Example 30 CE =

  21. Example LN = 96

  22. Segment Lengths in Circles

  23. Two chords intersect INSIDE the circle Type 1: part part part part Go down the chord and multiply

  24. Solve for x. 9 6 x 2 x = 3

  25. 12 2x 8 3x Find the length of DB. A D x = 4 C DB = 20 B

  26. Find the length of AC and DB. D A x – 4 x C 5 10 x = 8 AC = 13 DB = 14 B

  27. Two secants intersect OUTSIDE the circle Type 2:

  28. Ex: 3 Solve for x. B 13 A 7 E 4 C x D 7 (7 + 13) = 4 (4 + x) x = 31 140 = 16 + 4x 124 = 4x

  29. Ex: 4 Solve for x. B x A 5 D 8 6 C E 6 (6 + 8) = 5 (5 + x) 84 = 25 + 5x x = 11.8 59 = 5x

  30. Ex: 5 Solve for x. B 10 A x D 4 8 C E x (x + 10) = (8 + 4) 8 x2+10x = 96 x = 6 x2 +10x – 96 = 0

  31. Type 2 (with a twist): Secant and Tangent

  32. Ex: 5 Solve for x. x 12 24 (12 + x) 242 = 12 x = 36 576 = 144 + 12x

  33. Ex: 6 5 15 x (5 + 15) x2 = 5 x2 = 100 x = 10

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