If the lengths of the sides of a triangle are in the ratio 3: 4 : 5 and its perimeter is 48 cm, find its area.

Let a, b and c be the sides of the triangle.

Given,

Ratio of the sides are 3 : 4 : 5.

Let a = 3x cm, b = 4x cm and c = 5x cm.

Given,

⇒ Perimeter = 48 cm

⇒ a + b + c = 48

⇒ 3x + 4x + 5x = 48

⇒ 12x = 48

⇒ x = $\dfrac{48}{12}$ = 4.

Substituting value of x,

a = 3x = 3 × 4 = 12 cm,

b = 4x = 4 × 4 = 16 cm,

c = 5x = 5 × 4 = 20 cm.

We know that,

Semi perimeter (s) = $\dfrac{(a + b + c)}{2}$.

$s = \dfrac{12 + 16 + 20}{2} \\[1em] = \dfrac{48}{2} \\[1em] = 24 \text{ cm}.$

Area of triangle = $\sqrt{s(s - a)(s - b)(s - c)}$

Substituting values we get,

$A = \sqrt{24(24 - 12)(24 - 16)(24 - 20)} \\[1em] = \sqrt{24 \times 12 \times 8 \times 4} \\[1em] = \sqrt{9216} \\[1em] = 96 \text{ cm}^2.$

Hence, area of triangle = 96 cm².