Question

A square and an equilateral triangle have equal perimeters. If the diagonal of the
is 12root2 cm, then area of the triangle is
​

Answer 1

Given :

A square and an equilateral triangle have equal perimeter

The diagonal of the square is 12√2 cm

To Find :

The area of triangle

Solution :

According to question

Let The side of square = S cm

Let The side of equilateral triangle = s cm

square and an equilateral triangle have equal perimeter

∵ The perimeter of square

• = 4 × side

• = 4 × S

And

The perimeter of equilateral triangle

• = 3 × side

• = 3 × s

∵ The diagonal of the square = 12√2cm ....(i)

∵ The diagonal of the square = side × √2 ....(ii)

From equation (i) and equation (ii)

12√2 cm = side × √2

So, Side of square = S= √2 cm

Since, A square and an equilateral triangle have equal perimeter

So, 4 × S = 3 × s

Or, 4 × √2 = 3 × s

$Or, s = \dfrac{4\sqrt{2}}{3}$

i.e The side of equilateral triangle:-

$s = \dfrac{4\sqrt{2}}{3} cm$

Since, The Area of equilateral triangle:-

$A = \dfrac{\sqrt{3}}{4} × (side)²$

$= \dfrac{\sqrt{3}}{4} × (\dfrac{4\sqrt{2}}{3} {)}^{2}$

$= \dfrac{8\sqrt{3}}{9} {cm}^{2}$

Hence, The Area of equilateral triangle

is:-

$\dfrac{8\sqrt{3}}{9} cm²$