Chapter 8, Problem 13CUR

To determine

To find: the value of x

Answer to Problem 13CUR

The value of x is $x=20$

Explanation of Solution

Given Information: A triangle with sides given in terms of x in the figure and angle bisector of an angle being extended to the opposite side to meet at a point.

Formula used:

In any triangle ABC, if $\angle A$ is bisected by a segment AD such that it meets the side opposite to the angle at D, it divides the opposite side into two parts which are in proportion with the remaining two sides of the triangle.

  

This can be expressed for the above triangle as,

  $\frac{c}{b}=\frac{x}{y}$

Calculation:

Consider the triangle ADC given below. It can be seen that the triangle is having the bisector of $\angle A$ meeting the opposite side AC at B. Now, applying the angle-bisector theorem in this case,

  

  $\begin{array}{l}\frac{x}{x+4}=\frac{x-5}{x-2}\\ x^2-2x=x^2-5x+4x-20\end{array}$

  $\begin{array}{l}-2x=-x-20\\ -x=-20\\ x=20\end{array}$