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 the shaded region.

Answer 1

BC is the diameter of the circle passing through the centre O.

Now,

∆ABC is a right angles triangle, right angled at A.      (Angle subtended by the diameter on the circumference of the circle is 90^°)

In right ∆ABC,

BC² = AB² + AC²

= (7)² + (24)²

= 49 + 576

= 625

∴ BC = 25 cm

Also,

∠COD + ∠BOD = 180^° (Linear pair angles)

⇒∠COD = 180^° − 90^° = 90^°

Now,
Area of the shaded region = Area of sector OCABDO − Area of ∆ABC

`= 270^@/360^@ xx pi((BC)/2)^2 - 1/2 xx AB xx AC`  (∵ 360^° - 90^° = 270^° )

`= 3/4 xx 3.14 xx (25/2)^2 - 1/2 xx 7 xx 24`

= 367.97 - 84

= 283.97 cm² (Approx)

Hence the area of shaded region is approximately 283.97 cm²