Chapter 9.7, Problem 18WE

To determine

To Find: Find the value of PC from the given figure.

Answer to Problem 18WE

The value of x is 24

Explanation of Solution

Given information: We have given $PT=12,CD=18$ and a figure shown below

  

Let $PC=x$

As we know that when a secant segment and a tangent segments are drawn to a circle from an external point then the product of the secant segment and its external segment is equal to the square of the tangent segment.

Therefore, from given figure we must have

  $\begin{array}{c}P{T}^2=PD\times PC\\ {12}^2=\left(PC-CD\right)\times PC\\ 144=\left(x-18\right)\times x\\ 144=x^2-18x\\ x^2-18x-144=0\\ x^2-24x+6x-144=0\\ x\left(x-24\right)+6\left(x-24\right)=0\\ \left(x-24\right)\left(x+6\right)=0\end{array}$

This gives either $x=24orx=-6$

Since length is never negative. So we reject $x=-6$

Therefore, the value of x is 24