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With the vertices A, B and C of a triangle ABC as centres, arcs are drawn with radii 5 cm each as shown in Fig. 11.15. If AB = 14 cm, BC = 48 cm and CA = 50 cm, then find the area of the shaded region. (Use π = 3.14)

Solution:

Given, arcs are drawn with radius 5 cm with vertices A, B and C of a triangle ABC as centres.

The side lengths of the triangle are

AB = 14 cm

BC = 48 cm

CA = 50 cm

We have to find the area of the shaded region.

Area of sector = πr²θ/360°

Area of sector with A as centre = π(5)²(∠A/360°)

= (∠A/360°)25π

Area of sector with B as centre = π(5)²(∠B/360°)

= (∠B/360°)25π

Area of sector with C as centre = π(5)²(∠C/360°)

= (∠C/360°)25π

Area of 3 sectors = (∠A/360°)25π + (∠B/360°)25π + (∠C/360°)25π

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

In triangle ABC,

∠A + ∠B + ∠C = 180°

So, area of 3 sectors = 25π(∠A + ∠B + ∠C)/360°

= 25π(180°/360°)

= 25(3.14)(1/2)

= 39.25 cm²

Area of the shaded region = area of triangle - area of 3 sectors

Considering triangle ABC,

By pythagoras theorem,

AB² + BC² = AC²

LHS: AB² + BC²

= (14)² + (48)²

= 196 + 2304

= 2500

RHS: AC²

= (50)²

= 2500

LHS = RHS

So, ABC is a right triangle with B at right angle.

Area of triangle = (1/2) × base × height

Area of triangle ABC = (1/2) × BC × AB

= (1/2)(48)(14)

= (48)7

= 336 cm²

Area of the shaded region = 336 - 39.25

= 296.75 cm²

Therefore, the area of the shaded region is 296.75 cm²

✦ Try This: With the vertices D, E and F of a triangle DEF as centres, arcs are drawn with radii 5 cm each. If DE = 13 cm, EF = 28 cm and FD = 40 cm, then find the area of the shaded region. (Use π = 3.14).

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

NCERT Exemplar Class 10 Maths Exercise 11.4 Sample Problem 2

With the vertices A, B and C of a triangle ABC as centres, arcs are drawn with radii 5 cm each as shown in Fig. 11.15. If AB = 14 cm, BC = 48 cm and CA = 50 cm, then find the area of the shaded region. (Use π = 3.14)

Summary:

With the vertices A, B and C of a triangle ABC as centres, arcs are drawn with radii 5 cm each as shown in Fig. 11.15. If AB = 14 cm, BC = 48 cm and CA = 50 cm, then the area of the shaded region is 296.75 cm²

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