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Understanding Congruent Chords and Segment Lengths in Circles

This worksheet focuses on concepts of congruent chords, corresponding arcs, and segment lengths in circles. It guides students through problems involving two chords intersecting inside a circle, using flow charts and proofs to establish relationships effectively. Problems cover both chord and secant lengths, ensuring a solid understanding of these fundamental geometric principles.Students will find various exercises, including establishing midpoint calculations, applying the perpendicular bisector theorem, and solving algebraic equations to find segment lengths.

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Understanding Congruent Chords and Segment Lengths in Circles

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  1. Daily Check

  2. Homework Review

  3. How do you know when two chords are congruent? • corresponding arcs are congruent B A M P b. equidistant from the center L C LP  PM ALP = BMP = 90 D

  4. Flow Chart Proof

  5. 2x x + 40 2x = x + 40 x = 40

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

  7. IF AC is the perpendicular bisector of segment DB, then… D It’s the DIAMETER!!! Arcs DC and BC are congruent!!! A C B

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

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

  10. Your turn!  ÐUTV ÐXTW. Find WX.___________ 130º 11 Find ___________

  11. Your turn!  Find the length of each chord. 96 30 LN = _______ CE = _______

  12. Segment Lengths in Circles • Find the lengths of segments of chords • Find the lengths of segments of tangents and secants

  13. Secant and Tangent 2 Secants intersecet outside Chords intersect inside

  14. Two chords intersect INSIDE the circle Type 1: a ab = cd d c b

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

  16. 12 2x 8 3x Find the length of DB. x = 4 DB = 20 A D C B

  17. Find the length of each chord. D x = 8 AC = 13 DB = 14 A x - 4 x C 5 10 B

  18. Two secants intersect OUTSIDE the circle Type 2: E A B C D EA•EB = EC•ED

  19. 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

  20. 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

  21. Type 2 (with a twist): Secant and Tangent C B E A EA2= EB • EC

  22. Ex: 5 Solve for x. C B x 12 E 24 A (12 + x) 242 = 12 x = 36 576 = 144 + 12x

  23. Ex: 6 5 B E 15 C x A (5 + 15) x2 = 5 x2 = 100 x = 10

  24. Homework Practice Worksheet

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