Chapter 9.7, Problem 13WE

To determine

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

Answer to Problem 13WE

The value of x is 4 or 12

Explanation of Solution

Given information: We have given $AP=6,BP=8,CD=16$ and a figure shown below

  

Let $DP=x$ then $CP=16-x$

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

  $\begin{array}{l}AP\times PB=CP\times DP\\ 6\times 8=\left(16-x\right)\times x\\ 48=16x-x^2\\ x^2-16x+48=0\\ x^2-12x-4x+48=0\\ x\left(x-12\right)-4\left(x-12\right)=0\\ \left(x-12\right)\left(x-4\right)=0\end{array}$

This gives either $x=12or4$

Therefore, the value of x is 4 or 12