Question

In the given figure, ∆ABC is right-angled at A. Semi-circles are drawn on AB, AC and BC as diameters. It is given that AB = 3 cm and AC = 4 cm. Find the area of the shaded region.
Figure

Answer 1

Given: AB = 3 cm and AC = 4
Thus, BC = 5 cm (By Pythagoras' theorem)


Area of shaded region = {ar (ΔABC + ar(semi-circle APB) + ar(semi-circle AQC)} - ar(semi-circle BAC)
= $\left[\left(\frac{1}{2}\times 3\times 4\right)+\left(\frac{1}{2}\mathrm{\pi }\times \frac{3}{2}\times \frac{3}{2}\right)+\left(\frac{1}{2}\mathrm{\pi }\times 2\times 2\right)-\left(\frac{1}{2}\mathrm{\pi }\times \frac{5}{2}\times \frac{5}{2}\right)\right]$ cm²
= $\left\{6+\frac{1}{2}\mathrm{\pi }\left(\frac{9}{4}+4-\frac{25}{4}\right)\right\}$ cm²
= (6 + 0) cm² = 6 cm²