Chapter 9.7, Problem 14WE

To determine

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

Answer to Problem 14WE

The value of x is 3 or 8

Explanation of Solution

Given information: We have given $CD=10,CP=6,AB=11$ and a figure shown below

  

Let $AP=x$ then $PB=11-x$

As we know that when two chords intersect insides in a circle, then the product of the segments of one chord equals the product of the segments of the other chord.

Therefore, from given figure we must have

  $\begin{array}{l}AP\times PB=CP\times DP\\ x\times \left(11-x\right)=6\times \left(10-6\right)\\ 11x-x^2=24\\ x^2-11x+24=0\\ x^2-8x-3x+24=0\\ x\left(x-8\right)-3\left(x-8\right)=0\\ \left(x-8\right)\left(x-3\right)=0\end{array}$

This gives either $x=8or3$

Therefore, the value of x is 3 or 8