The perimeter of a triangle is 450 m and its sides are in the ratio 12 : 5 : 13. Find the area of the triangle.

Given:

Perimeter of the triangle = 450 m

Ratio of the sides = 12 : 5 : 13.

Let the sides of triangle be 12a, 5a and 13a.

Perimeter of the triangle = Sum of sides of the triangle

⇒ 12a + 5a + 13a = 450

⇒ 30a = 450

⇒ a = $\dfrac{450}{30}$

⇒ a = 15

Thus, sides are 12 x 15, 5 x 15 and 13 x 15 m

= 180 m, 75 m and 195 m

The sides of the triangle are:

a = 180 m, b = 75 m and c = 195 m.

The semi-perimeter s:

$∵ s = \dfrac{a + b + c}{2}\\[1em] = \dfrac{180 + 75 + 195}{2}\\[1em] = \dfrac{450}{2}\\[1em] = 225$

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

= $\sqrt{225(225 - 180)(225 - 75)(225 - 195)}$ m²

= $\sqrt{225 \times 45 \times 150 \times 30}$ m²

= $\sqrt{45,562,500}$ m²

= 6,750 m²

Hence, the area of the triangle is 6,750 m².