Question

In the adjoining figure, the crescent is formed by two circles which touch at the point A. O is the centre of bigger circle. If CB = 9 cm and DE = 5 cm, find the area of the shaded portion.

Answer 1

Let R and r be the radii of two circles. Then, 2(R - r) = 9. 
Join AD and CD,

ΔAOB ∼ ΔDOC
∴ `"OD"/"OC" = "OA"/"OD"`

⇒ OD² = OA.OC
⇒ (R - 5)² = R(R - 9)
⇒ R² + 25 - 10R = R² - 9R
∴ R = 25
So, 2(25 - r) = 9
∴ r = 20.5 cm

Area of the shaded portion = πR² - πr² 
= π[ (25)² - (20.5)² ] cm²
= `22/7` [ 625 - 420.25 ] cm²
= `22/7` x 204.75 cm²
= 643.5 cm².