Chapter 3.6, Problem 12PSB

To determine

To Show: In the given figure $\Delta ABC$is scalene

Explanation of Solution

Given information :

We are given a figure in which $AB=x+3,AC=3x+2,BC=2x+3\text{ and Perimeter of }\Delta ABC\text{ =20}$

Formula used : Perimeter of a triangle is equal to sum of all its sides.

Proof:

As per the question we have:

  $\begin{array}{l}AB=x+\mathrm{3.........................................}(\text{i)}\\ AC=3x+\mathrm{2......................................}(\text{ii)}\\ BC=2x+\mathrm{3......................................}\text{(iii) }\\ \text{Perimeter of }\Delta ABC\text{ =20}\end{array}$

Using above formula for the given triangle we will get:

  $\begin{array}{l}\text{Perimeter of }\Delta ABC\text{ = }AB\text{ +}AC\text{ + }BC\\ \Rightarrow \text{ 20 = }x+3\text{ + 3}x\text{ +2+ }2x+3\\ \Rightarrow \text{ 20 = 6}x\text{ + 8 }\\ \Rightarrow \text{ 20 - 8 = 6}x\\ \Rightarrow \text{ 12 = 6}x\\ \Rightarrow x=\text{ 2}\end{array}$

Substituting this value of $x\text{ = 2 in (i),(ii),and (iii) }$ we will get:

  $\begin{array}{l}AB=x+3=2+3=\text{ 5}\\ AC=3x+2\text{ = (3}\times \text{2) + 2 = 6 + 2 = 8}\\ BC=2x+3\text{ = ( 2}\times \text{ 2) + 3 = 4 + 3 = 7}\\ \\ \text{ Here as }AB\ne BC\ne CA\end{array}$

So, the given triangle is a scalene triangle which proves the required result.