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Understanding Tangents to Circles: Properties and Theorems Explained

This document delves into the theorem regarding tangents and chords in circles, outlining that two chords are congruent if and only if they are equidistant from the center. It provides examples that illustrate how to calculate lengths using the Pythagorean theorem when given different chord lengths and tangent segments. Additionally, it discusses key facts about tangents, including their perpendicular relationship with radius at the point of tangency and the congruence of tangent segments from a common exterior point.

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Understanding Tangents to Circles: Properties and Theorems Explained

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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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