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.

Ex. 2: IN K, K is the midpoint of RE 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

3.) Find the length of CV

2 Facts about Tangents

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

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

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

Example – find x ANSWER: x = 8 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

Example Find KY a2 + b2 = c2 102 + b2 = 242 100 + b2 = 576 476 = b2

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

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.

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

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

Example Find length of tangent

Find NP N T S P R Q