Theorem of Equal Chords Subtending Angles at the Center
Two semi-circ...
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.
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Solution
Draw RM⊥AB
AM=MB=AB2=18cm
AP=PM=MQ=QB=182=9cm
MR=AM=18㎝
CM=RM−CR=18−r
PC=PE+EC=9+r
In triangle CMP
PC2=CM2+PM2⇒(9+r)2=(18−r)2+92
⇒81+18r+r2=324−36r+r2+81⇒54r=324⇒r=6cm
∴Area of shaded region=area of semi circle AB−2 area of semi circle-area of circle with C as centre