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Question

Two semi-circles are drawn on the diameter of a semi-circle with radius 18cm. A circle with centre C is drawn such that it touches the two semi-circles. Find area of shaded region.
872173_3792f15951d6467ca1d601b3e59c0e0a.png

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Solution

Draw RMAB

AM=MB=AB2=18 cm

AP=PM=MQ=QB=182=9 cm

MR=AM=18

CM=RMCR=18r

PC=PE+EC=9+r

In triangle CMP

PC2=CM2+PM2(9+r)2=(18r)2+92

81+18r+r2=32436r+r2+8154r=324r=6 cm

Area of shaded region=area of semi circle AB2 area of semi circle-area of circle with C as centre

=12π×1822×12×π×92π×62

=162π81π36π=45π cm2

949530_872173_ans_a006625f3b474c65af372cf163923e1e.png

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