Sides of a right triangular field are 25m, 24m and 7m. At the three corners of the field, a cow, a buffalo and a horse are tied separately with ropes of 3.5 m each to graze in the field. Find the area of the field that cannot be grazed by these animals.

A
64.75 m²
B
54.75 m²
C
64.75 cm²
D
84.75 m²

Solution:

Given that sides of the triangular field are 24m, 25m, 7m and radii of three circles = 3.5 m.

We know that, Area of triangle = ½ × base × height.
Here, base, BC = 7 m and height, AB = 24 m.

Area of Δ ABC = 12 × BC × AB Area of Δ ABC = 12 × 24 × 7 Area of Δ ABC = 84 m 2 … … (1)

We can find the area of three sectors which are grazed by all animals using the below formula,
Area of sector =

θ ˙ 360 ° × π r 2

.
Here θ = sum of three angles of a triangle = 180⁰.
Area of three sectors = πr²/360° x (sum of three angles of the triangle)
Area of three sectors =

180 ° 360 ° × π r 2

Area of three sectors

= 12 × π r 2

Area of three sectors

= 12 × 227 × (3.5) 2 [r = 3.5 m]

Area of three sectors

= 12 × 227 × 3.5 × 3.5

Area of three sectors

= 11 × 0.5 × 3.5

Area of three sectors

= 5.5 × 3.5

Area of three sectors grazed by all animals = 19.25 m² …….(2)
Let us subtract eq 2 from eq 1 to find the area of the field that cannot be grazed by animals.
Area of the field that cannot be grazed by animals = Area of Triangle

Δ ABC

- Area of three sectors grazed by all animals.
Area of the field that cannot be grazed by animals = 84 – 19.25 = 64.75 m².
Hence, the area of the field that cannot be grazed by given three animals is 64.75 m².
The correct option is (1).