The perimeter of a triangular field is 540 m and its sides are in the ratio 25 : 17 : 12. Find the area of the triangle. Also, find the cost of cultivating the field at ₹ 24.60 per 100 m².

It is given that the sides of a triangular field are in the ratio 25 : 17 : 12.

Let the lengths of the sides be 25x, 17x and 12x.

Given,

The perimeter of a triangular field is 540 m.

Perimeter = sum of all sides of triangle

⇒ 540 = 25x + 17x + 12x

⇒ 540 = 54x

⇒ x = $\dfrac{540}{54}$

⇒ x = 10.

So the sides of the triangle are

⇒ 25x = 25 × 10 = 250 m

⇒ 17x = 17 × 10 = 170 m

⇒ 12x = 12 × 10 = 120 m

Let a = 250 m, b = 170 m, c = 120 m.

s = $\dfrac{1}{2}(a + b + c) = \dfrac{1}{2}(250 + 170 + 120) = \dfrac{540}{2}$ = 270 m.

(s - a) = (270 - 250) m = 20 m.

(s - b) = (270 - 170) m = 100 m.

(s - c) = (270 - 120) m = 150 m.

We know that,

$\Rightarrow \text{Area of triangle} = \sqrt{s(s - a)(s - b)(s - c)} \\[1em] \Rightarrow \sqrt{270 × 20 × 100 × 150} \\[1em] \Rightarrow \sqrt{81000000} \\[1em] \Rightarrow 9000 \text{ m}^2. \\[1em]$

Rate = ₹ 24.60 per 100 m²

= ₹ $\dfrac{24.60}{100}$ per m².

Cost = Area of triangle × Rate

= 9000 × $\dfrac{24.60}{100}$

= 24.60 × 90 = ₹ 2,214.

Hence, area of field = 9000 m² & cost of cultivating the field = ₹ 2,214.