Chapter 7.6, Problem 23WE

To determine

The value of AC in the given figure.

Answer to Problem 23WE

Value of $AC=78$ .

Explanation of Solution

Given:

The given figure,

  

Here, $AB=2x-12$ , $BC=x$ , $CD=x+5$ , and $AD=2x-4$ .

In the give figure BD is the angle bisector of triangle ADC.

On applying angle bisector theorem,

  $\begin{array}{l}\frac{2x-12}{2x-4}=\frac{x}{x+5},\\ \Rightarrow \left(2x-12\right)\left(x+5\right)=x\left(2x-4\right),\\ \Rightarrow 2x^2+10x-12x-60=2x^2-4x,\\ \Rightarrow 10x-12x+4x=60,\\ \Rightarrow 2x=60,\\ \Rightarrow x=30.\end{array}$

Now,

  $AC=AB+BC=2x-12+x=3x-12$ ,

  $\Rightarrow AC\text{ = 3}x-12=3\left(30\right)-12=78$ .

Hence, $AC=78$ .