Chapter 9.7, Problem 15WE

To determine

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

Answer to Problem 15WE

The value of x is 6

Explanation of Solution

Given information: We have given $AB=12,CP=9,DP=4$ and a figure shown below

  

Let $BP=x$ then $AP=12-x$

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

  $\begin{array}{c}AP\times PB=CP\times DP\\ \left(12-x\right)\times x=9\times 4\\ 12x-x^2=36\\ x^2-12x+36=0\\ x^2-6x-6x+36=0\\ x\left(x-6\right)-6\left(x-6\right)=0\\ \left(x-6\right)\left(x-6\right)=0\\ {\left(x-6\right)}^2=0\\ \left(x-6\right)=0\\ x=6\end{array}$

Therefore, the value of x is 6