Chapter 12, Problem 2CR

To determine

To verify : The given triangle $\Delta ABC$ is scalene, isosceles or equilateral.

Explanation of Solution

Given information :

Perimeter of $\Delta ABC$ is 28 units.

Sides are given in equations form as $AB=2x+3,BC=4x-5\text{ and }CA=8x+19$

Formula used:

Perimeter of the triangle $\Delta ABC$ is $AB+BC+CA$

Verification:

Given perimeter of $\Delta ABC$ is 28 units

  $\begin{array}{l}AB+BC+CA=28\\ \Rightarrow \left(2x+3\right)+\left(4x-5\right)+\left(8x+19\right)=28\\ \Rightarrow \left(2x+4x+8x\right)+\left(3-5+19\right)=28\\ \Rightarrow \left(14x\right)+\left(17\right)=28\\ \Rightarrow 14x=11\\ \Rightarrow x=\frac{11}{14}\end{array}$

Length of the sides are

  $\begin{array}{l}AB=\left|2\left(\frac{11}{14}\right)+3\right|=\left|\frac{11}{7}+\frac{21}{7}\right|=\left|\frac{32}{7}\right|=\frac{32}{7}\\ BC=\left|4\left(\frac{11}{14}\right)-5\right|=\left|\frac{22}{7}-\frac{35}{7}\right|=\left|-\frac{17}{7}\right|=\frac{17}{7}\\ CA=\left|8\left(\frac{11}{14}\right)+19\right|=\left|\frac{44}{7}+\frac{133}{7}\right|=\left|\frac{177}{7}\right|=\frac{177}{7}\end{array}$

The given triangle $\Delta ABC$ is Scalene since all the side in lengths are different.