question_answer

In the adjoining figure\[AB=CD=\]\[2BC=2BP=2CQ\]. In the middle, a circle with radius 1 cm is drawn. In the rest figure all are the semicircular arcs. What is the perimeter of the whole figures?

A) \[4\pi \]                        

B)  \[8\pi \]

C)  \[10\pi \]                     

D)  None of these.

Correct Answer: C

Solution :

\[AB=CD=2BC=2BP=2CQ\] \[\because \]\[BC=1\times 2=2\,\,cm\] \[\Rightarrow \]\[AB=2BC\] \[\therefore \]\[AB=2\times 2=4\,\,cm\] Now, \[AB=2BP\] \[\therefore \]\[BP=\frac{AB}{2}=2\,\,cm\] Now, \[CD=2BC\] \[\therefore \]\[CD=4\,\,cm\] Now, \[\therefore \]\[CQ=\frac{4}{2}=2\,\,cm\] Total perimeter \[=\left( \frac{1}{2}\times 2\pi \times \frac{2}{2} \right)\times 4+\left( \frac{1}{2}\times 2\pi \times 2 \right)\times 2+2\pi {{(1)}^{2}}\] \[=4\pi +4\pi +2\pi =10\pi \]