# Two equal sides of an isosceles triangle are ( 3 x - 1 ) and ( 2 x + 2 ) units, and third...

## Question:

Two equal sides of an isosceles triangle are {eq}(3x-1) {/eq} and {eq}(2x+2) {/eq} units, and third sides is {eq}2x {/eq} units. Find {eq}x {/eq} and the perimeter of the triangle.

## Sides of an isosceles triangle as a function of x.

The equal sides of the isosceles triangle are described as functions of an unknown variable. An equation relating the two sides solves for the unknown variable and eventually the sides of the triangle which add up to give the perimeter.

The two sides defined as (3x-1) and (2x+2) are equal.

\begin{align} 3x-1&=2x+2\\[0.3cm] 3x&=2x+3 &&\left[\text{Add 1 on both sides}\right]\\[0.3cm] x&=3 &&\left[\text{Subtract 2x from both sides}\right]\\[0.3cm] \end{align}

The equal sides are

\begin{align} 3(3)-1&=2(3)+2&&\left[\text{Substitute value of x}\right]\\[0.3cm] 9-1&=6+2\\[0.3cm] 8&=8\,units\\[0.3cm] \end{align}

Thus, the third side is,

\begin{align} 2x&=2(3)&&\left[\text{Substitute value x}\right]\\[0.3cm] &=6\,units\\[0.3cm] \end{align}

The perimeter is the sum of the three sides

\begin{align} 8+8+6&=22\,units\\[0.3cm] \end{align}

{eq}x{/eq} is {eq}3{/eq} while the perimeter of the triangle is {eq}22\,units{/eq}.