Question

With the vertices A, B and C of a triangle ABС as centres, arcs are drawn with radii 5 cm each as shown in the given figure. If AB = 14 cm, BС = 48 cm and CA = 50 cm then find the area of the shaded region. [Use π = 3.14.]

Answer 1

Given:

Triangle ABC with AB = 14 cm, BC = 48 cm, CA = 50 cm.

Arcs drawn at A, B, C with radius r = 5 cm.

Use π = 3.14. 

Step-wise calculation:

1. Semi‑perimeter:

`s = (a + b + c)/2` 

= `(48 + 50 + 14)/2`

= 56 cm

2. Area of triangle (Heron’s formula): 

`Δ = sqrt(s(s - a)(s - b)(s - c))` 

= `sqrt(56 xx 8 xx 6 xx 42)` 

= `sqrt(112896)` 

= 336 cm²

3. Sum of areas of the three sectors (each of radius 5 cm).

Sum of sector angles = A + B + C = 180°

Sum of sectors = `(180/360) xx π xx r^2` 

= `1/2 xx π xx 25` 

= `(25π)/2`

With π = 3.14: `(25 xx 3.14)/2`

= 39.25 cm²

4. Shaded area = Area of triangle – Sum of sectors

= 336 – 39.25

= 296.75 cm²

Area of the shaded region = 296.75 cm².