The lengths (in meters) of the sides of a triangle are $2x + \dfrac{y}{2}, \dfrac{5x}{3} + y + \dfrac{1}{2} \text{ and } \dfrac{2}{3}x + 2y + \dfrac{5}{2}.$ If the triangle is equilateral, find its perimeter.

Since, triangle is equilateral hence all sides are equal.

$\therefore 2x + \dfrac{y}{2} = \dfrac{5x}{3} + y + \dfrac{1}{2} \\[1em] \Rightarrow 2x - \dfrac{5x}{3} + \dfrac{y}{2} - y = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{6x - 5x}{3} + \dfrac{y - 2y}{2} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{x}{3} - \dfrac{y}{2} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{2x - 3y}{6} = \dfrac{1}{2} \\[1em] \Rightarrow 2x - 3y = 3 …….(i)$

Similarly,

$\Rightarrow \dfrac{5x}{3} + y + \dfrac{1}{2} = \dfrac{2x}{3} + 2y + \dfrac{5}{2} \\[1em] \Rightarrow \dfrac{5x}{3} - \dfrac{2x}{3} + y - 2y = \dfrac{5}{2} - \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{3x}{3} - y = \dfrac{4}{2} \\[1em] \Rightarrow x - y = 2 ……..(ii)$

Multiplying (ii) by 2 we get,

2x - 2y = 4 …….(iii)

Subtracting (i) from (iii) we get,

⇒ 2x - 2y - (2x - 3y) = 4 - 3

⇒ -2y + 3y = 1

⇒ y = 1.

Substituting value of y in (i) we get,

⇒ 2x - 3(1) = 3

⇒ 2x = 6

⇒ x = 3.

Side 1 = $2x + \dfrac{y}{2} = 2(3) + \dfrac{1}{2} = 6 + 0.5$ = 6.5 m

∵ Triangle is equilateral, its other two sides will also be 6.5 m each.

∴ Perimeter = 3 x 6.5 = 19.5 m.

Hence, perimeter of triangle = 19.5 meters.