Question

In the given figure, ∆ ABC is a right-angled triangle in which ∠ A is 90°. Semicircles are drawn on AB, AC and BC as diameters. Find the area of the shaded region

Answer 1

In right angled ΔABC

`AB^2 + AC^2 = BC^2`

`=> BC  =  sqrt(AB^2 + AC^2)= sqrt(3^2 + 4^2)`

`=> BC = sqrt(9 + 16) = sqrt25 = 5 cm`

Area of shaded region = {ar (ΔABC) + ar(semicircle  on side AB) + ar(semicircle on side AC) } - ar(semicircle on side BC)

Area of shaded region = `[(1/2 xx 3 xx 4)+(1/2pi xx (3/2)^2) +(1/2pi xx (2)^2)] - (1/2pi xx (5/2)^2) cm^2`

`= 6 + 1/2pi(9/4 + 4 - 25/4) cm^2`

= 6 cm²

Hence, area of shaded region is 6 cm²