A square and an equilateral triangle have equal perimeters. If the diagonal of the square is 12√ 2 cm, then area of the triangle is

Question

A square and an equilateral triangle have equal perimeters. If the diagonal of the square is 12\(\sqrt2\) cm, then area of the triangle is 

A. 24\(\sqrt2\) cm² 

B. 24\(\sqrt3\) cm² 

C. 48\(\sqrt3\) cm² 

D. 64\(\sqrt3\) cm²

Answer 1

Let each side of square = y cm and each side of an equilateral triangle = x cm 

Perimeter of square = perimeter of an equilateral triangle

4y = 3x …………(1) 

Diagonal of square = 12\(\sqrt2\) cm 

Therefore using Pythagorous theorem: 

y² + y² = \((12\sqrt2)^2\)

2y² = 288

y = 12 cm

Therefore substituting value of y in equation (1) we get: x = 16 cm

Area of an equilateral triangle = \(\frac{\sqrt3a^2}{4}\) = \(\frac{\sqrt{3}\times {16} \times{16}}{4}\) = \(64\sqrt3\)cm²