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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.
Solution
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.
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