The sides of a triangular field are in the ratio 5 : 3 : 4 and its perimeter is 180 m. Find :

(i) its area.

(ii) altitude of the triangle corresponding to its largest side.

(iii) the cost of levelling the field at the rate of ₹ 10 per square metre.

(i) Given:

The sides of a triangular field are in the ratio 5 : 3 : 4.

Perimeter = 180 m

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

Perimeter = Sum of all sides of triangular field

⇒ 180 = 5a + 3a + 4a

⇒ 180 = 12a

⇒ a = $\dfrac{180}{12}$

⇒ a = 15

Thus, sides of field = 5a , 3a and 4a

= 5 x 15, 3 x 15 and 4 x 15

= 75 m, 45 m and 60 m

Let a = 75 m, b = 45 m and c = 60 m.

$∵ s = \dfrac{a + b + c}{2}\\[1em] = \dfrac{75 + 45 + 60}{2}\\[1em] = \dfrac{180}{2}\\[1em] = 90$

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

= $\sqrt{90(90 - 75)(90 - 45)(90 - 60)}$ m²

= $\sqrt{90 \times 15 \times 45 \times 30}$ m²

= $\sqrt{18,22,500}$ m²

= 1,350 m²

Hence, the area is 1,350 m².

(ii) Area = $\dfrac{1}{2}$ x base x altitude

Base = 75 m

Let h be the altitude of triangle corresponding to the largest side 75 m,

⇒ 1350 = $\dfrac{1}{2}$ x 75 x h

⇒ h = $\dfrac{1350 \times 2}{75}$

⇒ h = $\dfrac{2,700}{75}$

⇒ h = 36 m

Hence, the altitude of the triangle corresponding to its largest side is 36 m.

(iii) Cost of levelling = ₹ 10 per square metre

Total cost = Area x Cost of levelling

= ₹ 1,350 x 10

= ₹ 13,500

Hence, the total cost of levelling is ₹ 13,500.