The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm. Find the area of the triangle.

It is given that the lengths of the sides of a triangle are in the ratio 3 : 4 : 5.

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

The perimeter of the triangle is 144 cm.

Perimeter = sum of all sides of triangle

⇒ 144 = 3x + 4x + 5x

⇒ 144 = 12x

⇒ x = $\dfrac{144}{12}$

⇒ x = 12.

So the sides of a triangle are

⇒ 3x = 3 × 12 = 36 cm

⇒ 4x = 4 × 12 = 48 cm

⇒ 5x = 5 × 12 = 60 cm

Let a = 36 cm, b = 48 cm, c = 60 cm.

s = $\dfrac{1}{2}(a + b + c) = \dfrac{1}{2}(36 + 48 + 60) = \dfrac{144}{2}$ = 72 cm.

(s - a) = (72 - 36) cm = 36 cm.

(s - b) = (72 - 48) cm = 24 cm.

(s - c) = (72 - 60) cm = 12 cm.

We know that,

$\Rightarrow \text{Area of triangle} = \sqrt{s(s - a)(s - b)(s - c)} \\[1em] \Rightarrow \sqrt{72 × 36 × 24 × 12} \\[1em] \Rightarrow \sqrt{746496} \\[1em] \Rightarrow 864 \text{ cm}^2. \\[1em]$

Hence, area of triangle = 864 cm².