Question

Sides of a triangular field are 15 m, 16 m and 17 m. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length 7 m each to graze in the field. Find the area of the field which cannot be grazed by the three animals.

Answer 1

According to the question,

Sides of the triangle are 15 m, 16 m, and 17 m.

Now, perimeter of the triangle = (15+16+17) m = 48 m

∴ Semi-perimeter of the triangle = s = 48/2 = 24 m

By Heron’s formula,

Area of the triangle = √(s(s – a)(s – b)(s – c)) ,here a, b and c are the sides of triangle

= √(24(24 – 15)(24 – 16)(24 – 17))

= 109.982 m²

Let B, C and H be the corners of the triangle on which buffalo, cow and horse are tied respectively with ropes of 7 m each.

So, the area grazed by each animal will be in the form of a sector.

∴ Radius of each sector = r = 7 m

Let x, y, z be the angles at corners B, C, H respectively.

∴ Area of sector with central angle x,

= ½ (x/180) × πr² = (x/360) × π(7)²

Area of sector with central angle y,

= ½ (y/180) × πr² = (y/360) × π(7)²

Area of sector with central angle z,

= ½ (z/180) × πr² = (z/360) × π(7)²

Area of field not grazed by the animals = Area of triangle – (area of the three sectors)

= 109.892 – 77

= 32.982 cm²