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Understanding Circle Geometry: Chord and Segment Lengths

This lesson focuses on the relationships between chord lengths and segments within circles. We explore theorems related to intersecting chords, secant segments, and tangent segments to help you solve for unknown values in circle geometry. You will learn how to apply Theorems 10.14, 10.15, and 10.16, which provide essential formulas for calculating segment lengths. Engage with examples and practice problems to solidify your understanding, preparing you for future mathematical challenges involving circles.

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Understanding Circle Geometry: Chord and Segment Lengths

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  1. Clickers Bellwork • If 10 is multiplied by 10 more than a number, the product is the square of 24. Find the number • Solve for x • 21(x-4)=(2x-7)(x+2) • 3x2-13x-7=0

  2. Bellwork Solutions • If 10 is multiplied by 10 more than a number, the product is the square of 24. Find the number

  3. Bellwork Solutions • Solve for x • 21(x-4)=(2x-7)(x+2) • 3x2-13x-7=0

  4. Section 10.6 Find Segment Lengths in Circles

  5. The Concept • After discussing most of the angle relationships that we can find in a circle, today we’re going to move on to chord lengths • The concepts that drive our theorems today are based on the relationships we’ve already seen within a circle, but just need to be defined

  6. Definition Segments of a chord The line segments formed when a chord is divided into two or more pieces Secant Segment Segment that contains a chord of a circle and has exactly one endpoint outside the circle. External Segment Exterior portion of secant segment C H G F E D A B

  7. Theorem Theorem 10.14 If two chords intersect in the interior of a circle, then the product of the lengths of the segment of one chord is equal to the product of the lengths of the segments of the other chord. C A E D B

  8. Example Solve for x A C x+1 x+5 3 E B x+3 D

  9. On your own • Solve for x A D x 8 x+8 6 C B

  10. On your own • Solve for x A D x x x+10 14 C B

  11. Theorem Theorem 10.15 If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment D A E C B

  12. Example Solve for x 3 4 A C 5 x E B D

  13. On your own • Solve for x A 3 4 E x 2 C B

  14. Theorems Theorem 10.16 If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment A D C B

  15. Example Find the measure of angle AEC 16 x 8

  16. On your own • What is the measure of angle ABC 15 x 14

  17. Homework • 10.6 • 1, 2-22 even

  18. Most Important Points • Segment Length formulas for inside a circle

  19. On your own • What is the measure of angle of E, if arc ABC is 200o A D B C E

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