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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

Solution:

Given, sides of a triangular field are 15 m, 16 m and 17 m.

At the three corners, a cow, a buffalo and a horse are tied separately with ropes of length 7 m each to gaze in the field.

We have to find the area of the field which cannot be gazed by the three animals.

From the figure, CBH is the triangular field.

The length of the sides are

CB = 15 m

BH = 16 m

CH = 17 m

The area gazed by the animals represents the sector of a circle.

Area of the field which cannot be gazed by the three animals = area of triangle - area of 3 sectors.

By Heron’s formula,

Area of triangle = √s(s - a)(s - b)(s - c)

Where, s = a + b + c/2

s = (15 + 16 + 17)/2

s = 48/2

s = 24 cm

Area of triangle = √24(24 - 15)(24 - 16)(24 - 17)

= √24(9)(8)(7)

= √12096

= 109.98 m²

Area of sector = πr²θ/360°

Area of field gazed by cow = π(7)²(∠C/360°)

= (∠C/360°)49π

Area of field gazed by buffalo = π(7)²(∠B/360°)

= (∠B/360°)49π

Area of field gazed by horse = π(7)²(∠H/360°)

= (∠H/360°)49π

Area of 3 sectors = (∠C/360°)49π + (∠B/360°)49π + (∠H/360°)49π

= 49π(∠C + ∠B + ∠H)/360°

We know that the sum of all three interior angles of a triangle is always equal to 180°

∠C + ∠B + ∠H = 180°

So, area of 3 sectors = 49π(180°/360°)

= 49π(1/2)

= 49(22/7)(1/2)

= 49(11/7)

= 539/7

= 77 m²

Area of the field which cannot be gazed by the three animals = 109.98 - 77

= 32.98 m²

Therefore, the area of the field which cannot be gazed by the three animals is 32.98 m²

✦ Try This: Sides of a triangular field are 8 m, 9 m and 10 m. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length 5 m each to graze in the field. Find the area of the field which cannot be grazed by the three animals.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 12

NCERT Exemplar Class 10 Maths Exercise 11.4 Problem 3

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

Summary:

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. The area of the field which cannot be grazed by the three animals is 32.98 m²

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