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Properties of Chords, Secants, and Tangents in Circle Geometry

This segment explores critical theorems related to chords, secants, and tangents in circle geometry, specifically focusing on the relationships between their segments when they intersect. It covers Theorems 10.15, 10.16, and 10.17, detailing formulas for solving for unknowns in geometric problems involving these lines. Examples illustrate the application of theorems to find unknown values, employing algebraic techniques, including the quadratic formula for solving equations resulting from geometric principles.

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Properties of Chords, Secants, and Tangents in Circle Geometry

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  1. 10.5 Segment Lengths in Circles p. 629

  2. Thm 10.15 If 2 chords intersect in the interior of a circle, then AC * CD = BC * CE B A C D F E

  3. Ex: Solve for x. 12 * 9 = 18 * x 108 = 18x 6 = x x 18 12 9

  4. R Definitions S Tangent segment – a piece of a tangent with one endpt. at the pt. of tangency. Secant segment – a piece of a secant containing a chord, with one endpt. in the exterior of the circle & the other on the circle. External secant segment – the piece of a secant seg. that is outside the circle. SP Q RP P PQ

  5. Thm 10.16 If 2 secant segs. share the same endpt. outside the circle, then AB * AC = AE * AD Exterior parts Whole secant seg. C D B E A

  6. Ex: Solve for x. 11 * 21 = 12 (12 + x) 231 = 144 + 12x 87 = 12x 7.25 = x 10 11 x 12

  7. Thm 10.17 If a secant seg. & a tangent seg. share an endpt. outside of a circle, then (AB)2 = AC * AD Tangent Ext. secant seg. Whole secant seg B A C D

  8. Ex: solve for x. 24 302 = x (x + 24) 900 = x2 + 24x x2 +24x – 900 = 0 How do you solve for x? Use the quadratic formula!! x = 20.31 x = -44.31 30 x a = 1 b = 24 c = -900

  9. Assignment

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