Given: Circle k(O), O∈ PL KE - tangent at E KE=18, PL=15 Find: KP

Question

Given: Circle k(O), O∈

PL

   

KE

 - tangent at E

KE=18, PL=15

    Find:  KP

Answer 1

Theorem: If a secant segment (KL) and a tangent segment (KE) share an endpoint outside a circle, then the product of the length of the secant segment (KL) and the length of its external segment (KP) equals the square of the length of the tangent segment (KE).

In other words: KE**2 = KP * KL

We are given KE = 18 and PL = 15. We want to find KP so let KP = x and KL = PL + x = 15 + x.

18**2 = x * (15 + x)

324 = 15x + x**2

x**2 + 15x - 324 = 0

Using the quadratic formula with a = 1, b = 15 and c = -324

You will find that x = 12 or -27. x=12 is the only value that makes sense (length is positive), so KP = 12.