Question

The length of the sides of a triangle are $2x+\frac{y}{2},5\frac{x}{3}+y+\frac{1}{2},2\frac{x}{3}+2y+\frac{5}{2}$, if the triangle is equilateral. Then its perimeter is $\frac{m}{2}$. Find $m$

Answer 1

$2x+\frac{y}{2}=5\frac{x}{3}+y+\frac{1}{2}$

$\Rightarrow \frac{4x+y}{2}=\frac{10x+6y+3}{6}$

$\Rightarrow 12x+3y=10x+6y+3$

$\Rightarrow 2x=3+3y$

$\Rightarrow x=\frac{3+3y}{2}\frac{5x}{3}+y+\frac{1}{2}=\frac{2x}{3}+2y+\frac{5}{2}$

$\Rightarrow \frac{5(3+3y)}{6}+y+\frac{1}{2}=\frac{2(3+3y)}{6}+2y+\frac{5}{2}$

$\Rightarrow 15+15y+6y+3=6+6y+12y+15$

$\Rightarrow 21y+18=18y+21$

$\Rightarrow 3y=3$

$\Rightarrow y=1x=\frac{3y+3}{2}=\frac{3\times 1+3}{2}=3$

Perimeter $=2x+\frac{y}{2}+\frac{5x}{3}+y+\frac{1}{2}+\frac{2x}{3}+2y+\frac{5}{2}$

$=6+\frac{1}{2}+5+1+\frac{1}{2}+2+2+\frac{5}{2}=16+\frac{7}{2}=\frac{39}{2}$

Perimeter $=\frac{m}{2}=\frac{39}{2}$

$\Rightarrow m=39$