The area of a triangle is 216 cm² and its sides are in the ratio 3 : 4 : 5. Find the perimeter of the triangle.

Given,

Area = 216 cm²

Sides = 3 : 4 : 5

Let the sides of a triangle be 3x, 4x and 5x.

Since 3 : 4 : 5 is a pythagorean triplet (3² + 4² = 5²)

Thus, the triangle is a right angled triangle, and sides containing right angle are 3x cm and 4x cm.

$\Rightarrow \text{Area of right angled triangle} = \dfrac{1}{2} \times \text{ (product of sides containg right angle)} \\[1em] \Rightarrow 216 = \dfrac{1}{2} \times 3x \times 4x \\[1em] \Rightarrow 432 = 12x^2 \\[1em] \Rightarrow x^2 = 36 \\[1em] \Rightarrow x = \sqrt{36} \\[1em] \Rightarrow x = 6.$

∴ Sides of a triangle are

⇒ 3x = 3 × 6 = 18 cm

⇒ 4x = 4 × 6 = 24 cm

⇒ 5x = 5 × 6 = 30 cm

Perimeter = Sum of all sides of a triangle

= 18 + 24 + 30

= 72 cm.

Hence, perimeter of the triangle = 72 cm.