Heron's Formula — Question 5

Given,

Sides of a triangle are in the ratio of 12 : 17 : 25.

Let sides of the triangle are :

a = 12x, b = 17x and c = 25x.

Given,

Perimeter = 540 cm

∴ 12x + 17x + 25x = 540 cm

⇒ 54x = 540 cm

⇒ x = $\dfrac{540}{54}$

⇒ x = 10 cm

⇒ a = 12 × 10 = 120 cm,

⇒ b = 17 × 10 = 170 cm,

⇒ c = 25 × 10 = 250 cm.

Semi perimeter (s) = $\dfrac{\text{Perimeter of triangle}}{2} = \dfrac{540}{2}$ = 270 cm.

By Heron's formula,

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

Substituting values we get :

$A = \sqrt{270(270 - 120)(270 - 170)(270 - 250)} \\[1em] = \sqrt{270 \times 150 \times 100 \times 20} \\[1em] = \sqrt{81000000} \\[1em] = 9000 \text{ cm}^2.$

Hence, area of triangle = 9000 cm².