The lengths of the sides of a triangle are in the ratio 4 : 5 : 3 and its perimeter is 96 cm. Find its area.

Given:

Perimeter of the triangle = 96 cm

The lengths of the sides of a triangle are in the ratio 4 : 5 : 3.

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

∵ Perimeter of triangle = sum of sides of the triangle.

⇒ 4a + 5a + 3a = 96 cm

⇒ 12a = 96 cm

⇒ a = $\dfrac{96}{12}$ cm

⇒ a = 8 cm

So, the lengths of the sides are 4a, 5a and 3a.

= 4 x 8, 5 x 8 and 3 x 8

= 32, 40 and 24

Let a = 32 m, b = 40 m and c = 24 m.

$∵ s = \dfrac{a + b + c}{2}\\[1em] = \dfrac{32 + 40 + 24}{2}\\[1em] = \dfrac{96}{2}\\[1em] = 48$

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

= $\sqrt{48(48 - 32)(48 - 40)(48 - 24)}$ cm²

= $\sqrt{48 \times 16 \times 8 \times 24}$ cm²

= $\sqrt{1,47,456}$ cm²

= 384 cm²

Hence, the area of the triangle is 384 cm².