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 :

Radius of circle = a units

\[\therefore \]Area of semi-circle \[=\frac{\pi {{a}^{2}}}{2}\text{sq}\,\,\text{units}\]

Both \[\Delta ABC\]and \[\Delta BCD\]are isosceles and equal.

\[\therefore \]Area of each triangle \[=\frac{1}{2}{{a}^{2}}\]

\[\therefore \]Area of both triangles \[=2\times \frac{1}{2}{{a}^{2}}={{a}^{2}}\,\,\text{sq}\,\,\text{units}\]

Area of shaded region

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