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  • A line segment AB of length 64cm having C as its middle point, A B, A C and CB semicircles are described then radius of the circle inscribed in the space enclosed by three semic

A line segment AB of length 64cm having C as its middle point, A B, A C and CB semicircles are described then radius of the circle inscribed in the space enclosed by three semicircles is

Option: 1

8 cm


Option: 2

\frac{32}{3} \mathrm{~cm}


Option: 3

\frac{16}{3} \mathrm{~cm}


Option: 4

16cm


Answers (1)

best_answer

\mathrm{C M=\frac{1}{2} B C=\frac{1}{4} A B=16 \mathrm{~cm} }
Consider the   \mathrm{\triangle O M C }
\mathrm{O M^2=O C^2+C M^2 \\ }
\mathrm{ \Rightarrow(16+r)^2=(32-r)^2+16^2 }
\mathrm{ \Rightarrow(16+r)^2-(32-r)^2=16^2 \\ }
\mathrm{ \Rightarrow \quad(16+r+32-r)(16+r-32+r)=16^2 \\ }

\mathrm{ \Rightarrow 2 \cdot 48(r-8)=16^2 \\ }

\mathrm{ \Rightarrow r-8=\frac{8}{3} \Rightarrow r=\frac{32}{3} \mathrm{~cm} }.
 

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Rishi

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