Chapter 16, Problem 7RP

To determine

To find: The length of the third side of the given triangle.

Answer to Problem 7RP

The length of the third side of the given triangle is $\frac{10\sqrt{3}}{3}\text{units}$ .

Explanation of Solution

Given information:

The given figure is,

  

Consider the following figure by using the bis theorem as,

  

The value of $x$ is calculated by using Stewart’s Theorem as,

  $\begin{array}{l}{\left(6\right)}^2\left(2x\right)+{\left(4\right)}^2\left(3x\right)={\left(4\right)}^2\left(5x\right)+5x\left(2x\right)\left(3x\right)\\ 72x+48x=80x+30x^3\\ x=\frac{2\sqrt{3}}{3}\end{array}$

The length of the third side of the given triangle is calculated as,

  $\begin{array}{c}l=2x+3x\\ =5x\\ =5\left(\frac{2\sqrt{3}}{3}\right)\\ =\frac{10\sqrt{3}}{3}\text{units}\end{array}$

Thus, the length of the third side of the given triangle is $\frac{10\sqrt{3}}{3}\text{units}$ .