Question

The measure of the sides (in cms) of a triangle are $\frac{5}{3}x+y+\frac{1}{2},2x+\frac{1}{2}y$ and $\frac{2}{3}x+2y+\frac{5}{2}$. For what values of $x$ and $y$ the triangle is equilateral? Also find the measures of the sides of the triangle.

Answer 1

Given measures of sides of triangle are

$\frac{5}{3}x+y+\frac{1}{2}$ ....(1)

$2x+\frac{1}{2}y$ ....(2)

and $\frac{2}{3}x+2y+\frac{5}{2}$ ....(3)

The triangle becomes equilateral if all three sides are equal

i.e. if $\frac{5}{3}x+y+\frac{1}{2}=2x+\frac{1}{2}y$ and $2x+\frac{1}{2}y=\frac{2}{3}x+2y+\frac{5}{2}$

By solving above equations we get the following equations

$2x-3y=3$ ....... $\left(4\right)$

$8x-9y=15$ ........ $\left(5\right)$

Solving $\left(4\right)$ and $\left(5\right)$ simultaneously we get

$x=3$

Substituting $x$ in $\left(4\right)$ we get

$y=1$

We need to check that, for $x=3$ and $y=1$, whether the three given sides are equal or not

Substituting $x$ and $y$ in $\left(1\right),\left(2\right)$ and $\left(3\right)$

$⟹\frac{5}{3}x+y+\frac{1}{2}=\frac{5}{3}\times 3+1+\frac{1}{2}=\frac{13}{2}$ ........ First side

$2x+\frac{1}{2}y=2\times 3+\frac{1}{2}\times 1=\frac{13}{2}$ ...... Second side

$\frac{2}{3}x+2y+\frac{5}{2}=\frac{2}{3}\times 3+2\times 1+\frac{5}{2}=\frac{13}{2}$ .......... Third side

So, we get all three sides equal to $\frac{13}{2}$

Thus, all three sides are equal for $x=3$ and $y=1$

Hence, the triangle is an equilateral triangle for $x=3$ and $y=1$