question_answer

The area of the shaded region in the figure given below is

A) \[\frac{{{a}^{2}}}{2}\left( \frac{\pi }{2}-1 \right)\,\,\text{sq}\,\,\text{units}\]

B) \[{{a}^{2}}(\pi -1)\,\,\text{sq}\,\,\text{units}\]

C) \[{{a}^{2}}\left( \frac{\pi }{2}-1 \right)\,\,\text{sq}\,\,\text{units}\]

D) \[\frac{{{a}^{2}}}{2}(\pi -1)\,\,\text{sq}\,\,\text{units}\]

Correct Answer: C

Solution :

Area of semi-circle \[=\frac{\pi {{a}^{2}}}{2}\]

Now, area of \[\Delta ABD=\frac{1}{2}\times 2a\times a\]

\[={{a}^{2}}\]

\[\therefore \]Area of shaded region

\[=\frac{\pi {{a}^{2}}}{2}-{{a}^{2}}\]

\[={{a}^{2}}\left( \frac{\pi }{2}-1 \right)\,\,\text{sq}\,\,\text{units}\]