Chapter 8.2, Problem 28WE

To determine

To find: the value of x

Answer to Problem 28WE

The value of x is $x=\frac{15}{7}$

Explanation of Solution

Given Information: A triangle with one right angle and an angle bisector of this angle being extended to the opposite side to meet at a point. The other sides of the triangle are 3 units and 4 units.

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,

  $\frac{c}{b}=\frac{x}{y}$

Now, when the triangle is right angled at A, the relation between the sides of the triangle is expressed using Pythagoras theorem as,

  ${\left(x+y\right)}^2=c^2+b^2$ ……….. (1)

Calculation: Consider the triangle given below. It can be seen that the triangle is having $\angle A=90°$ . Now, the hypotenuse can be calculated using the Pythagoras theorem in (1) as,

  

  $\begin{array}{c}B{C}^2=3^2+4^2\\ BC=\sqrt{3^2+4^2}\\ =5\end{array}$

Now, the angle bisector theorem can be used here as,

  $\frac{3}{4}=\frac{x}{5-x}$

Using cross-multiplication,

  $\begin{array}{l}4x=15-3x\\ 7x=15\\ x=\frac{15}{7}\end{array}$