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Tangents to Circles

Tangents to Circles . Theorem: Two chords are congruent IFF they are equidistant from the center. B. AD  BC IFF LP  PM. A. M. P. L. C. D. Ex. 1: IN A, PR = 2x + 5 and QR = 3x –27. Find x. R. x. x. A. P. Q. x = 32.

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Tangents to Circles

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  1. Tangents to Circles

  2. Theorem:Two chords are congruent IFF they are equidistant from the center. B AD  BC IFF LP  PM A M P L C D

  3. Ex. 1: IN A, PR = 2x + 5 and QR = 3x –27. Find x. R x x A P Q x = 32

  4. Ex. 2: IN K, K is the midpoint of RE. If TY = -3x + 56 and US = 4x, find x. U T K E R S Y x = 8

  5. 3.) Find the length of CV

  6. 2 Facts about Tangents

  7. Fact #1 • A tangent line is ALWAYS perpendicular to the radius of the circle drawn to the point of tangency. radius 90 degrees = perpendicular tangent

  8. What this fact means…. • What this means is that you can make a right triangle and use the pythagorean theorem to find distances. • The right angle will always be the one on the outside of the circle radius tangent

  9. Example – Find the length of AC a2 + b2 = c2 52+ 82= c2 25 + 64 = c2 89 = c2 = c

  10. Example – find x Since a radius of the circle is 5, any radius is 5… Since it is a radius drawn to a point of tangency, it is perpendicular to the tangent. 5 a2 + b2 = c2 122 + 52 = c2 144 + 25 = c2 169 = c2 13 = c ? This whole length is 13. x + 5 = 13 x = 8 5 12 ANSWER: x = 8

  11. Example • Find KY a2 + b2 = c2 102+ b2= 242 100 + b2 = 576 476 = b2 = b

  12. Example • Does this picture show a tangent? • It must satisfy Pythagorean Theorem a2 + b2 = c2 72+ 242= (18+7)2 625 = 625 Yes!

  13. Fact #2 • If two segments from the same exterior point are tangent to a circle, then they are congruent. tangent #1 exterior point tangent #2 They are congruent.

  14. What this fact means…. • What this means is that you can set the 2 tangents equal to each other • Tangent 1 = tangent 2 tangent #1 tangent #2

  15. Example exterior point Because of Fact #2, x=14.

  16. Example • Find length of tangent

  17. Find NP N T S P R Q

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