Question

In the given figure, O is the centre of the circle with AC = 24 cm, AB = 7 cm and ∠BOD = 90°. Find the area of shaded region.
[Use π = 3.14] [CBSE 2012]

Answer 1

In right triangle ABC
BC² = AB² + AC²
= (7)² + (24)²
= 49 + 576
= 625
∴ BC² = 625
⇒ BC = 25
Now, ∠COD + ∠BOD = 180° (Linear pair angles)
⇒∠COD = 180° − 90° = 90°
Now, Area of the shaded region = Area of sector having central angle (360° − 90°) − Area of triangle ABC
$$\begin{gathered} =\frac{270°}{360°}\mathrm{\pi }{\left(\frac{\mathrm{BC}}{2}\right)}^2-\frac{1}{2}\mathrm{AB}\times \mathrm{AC} \\ =\frac{3}{4}\times 3.14{\left(\frac{25}{2}\right)}^2-\frac{1}{2}\times 7\times 24 \\ =367.97-84 \\ =283.97{\mathrm{cm}}^2 \end{gathered}$$
Hence, the area of shaded region is 283.97 cm²