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

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 3a, 4a and 5a.

The perimeter of the triangle is 144 cm.

Perimeter = sum of all sides of triangle

⇒ 144 = 3a + 4a + 5a

⇒ 144 = 12a

⇒ a = $\dfrac{144}{12}$

⇒ a = 12

So, the sides of triangle = 3a, 4a and 5a

= 3 x 12, 4 x 12 and 5 x 12

= 36, 48 and 60

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

The semi-perimeter s:

$∵ s = \dfrac{a + b + c}{2}\\[1em] = \dfrac{36 + 48 + 60}{2}\\[1em] = \dfrac{144}{2}\\[1em] = 72$

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

= $\sqrt{72(72 - 36)(72 - 48)(72 - 60)}$ cm²

= $\sqrt{72 \times 36 \times 24 \times 12}$ cm²

= $\sqrt{746,496}$ cm²

= 864 cm²

Hence, the area of the triangle is 864 cm².