question_answer

In the given figure\[AB=BC=CD=4\,\,cm.\]Whereas the unshaded portion in the upper semicircles is a semicircle with radius BC and center at C. Find the ratio of the area of the shaded region to that the unshaded region.

A) \[4:13\]                        

B) \[2:5\]

C) \[6:13\]                        

D) \[9:13\]

Correct Answer: B

Solution :

Area of shaded region \[=\left( \frac{1}{2}\pi \times {{6}^{2}}-\frac{1}{2}\pi \times {{4}^{2}} \right)\] \[=\frac{1}{2}\times \left( 36-18 \right)=\frac{20\pi }{2}=10\pi \] Radius of bigger circle\[=\frac{12}{2}=6\,\,cm\] Now, Area of unshaded \[=\frac{1}{2}\times \pi \times {{6}^{2}}+\frac{1}{2}\times \pi \times {{4}^{2}}\] \[=\frac{1}{2}\pi \left( 36+14 \right)=\frac{1}{2}\pi \times 50=25\pi \] Ratio\[=\frac{10\pi }{25\pi }=\frac{2}{5}=2:5\]