A) \[{{(\pi +2)}^{2}}:2{{\pi }^{2}}\]
B) \[2{{\pi }^{2}}:{{(\pi +2)}^{2}}\]
C) \[{{(\pi +2)}^{2}}:4{{\pi }^{2}}\]
D) \[4{{\pi }^{2}}:{{(\pi +2)}^{2}}\]
Correct Answer: C
Solution :
Let radius of circle A be \[{{r}_{1}}\]units. Radius of semi-circle = \[{{r}_{2}}\]units. According to the question, \[2\pi {{r}_{1}}=\pi {{r}_{2}}+2{{r}_{2}}\] \[\Rightarrow \,\,2\pi {{r}_{1}}={{r}_{2}}(\pi +2)\Rightarrow \frac{{{r}_{1}}}{{{r}_{2}}}=\frac{\pi +2}{2\pi }\] \[\therefore \] Ratio of their areas\[=\frac{\pi r_{1}^{2}}{\pi r_{2}^{2}}\] \[=\frac{{{(\pi +2)}^{2}}}{{{(2\pi )}^{2}}}={{(\pi +2)}^{2}}:4\pi {{r}^{2}}\]
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