question_answer

In a right-angled triangle, the lengths of the sides containing the right angle are a and b. With the midpoint of each side as centre, three semicircular areas are drawn outside the triangle. The total area enclosed is

A) \[\frac{\pi }{2}({{a}^{2}}+{{b}^{2}})+\frac{1}{8}{{(a+b)}^{2}}\]

B)  \[\frac{\pi }{2}({{a}^{2}}+{{b}^{2}})+\frac{1}{2}ab\]

C)  \[\frac{\pi }{2}({{a}^{2}}+{{b}^{2}})+\frac{1}{8}{{(a+b)}^{2}}\]

D)  \[\frac{\pi }{4}({{a}^{2}}+{{b}^{2}})+\frac{1}{2}ab\]  

Correct Answer: D

Solution :

 Total area \[=I+II+III+IV\] \[=\frac{1}{2}\pi {{\left( \frac{a}{2} \right)}^{2}}+\frac{1}{2}\pi {{\left( \frac{b}{2} \right)}^{2}}+\frac{1}{2}\pi {{\left( \frac{\sqrt{{{a}^{2}}+{{b}^{2}}}}{2} \right)}^{2}}+\frac{1}{2}ab\] \[=\frac{\pi }{8}({{a}^{2}}+{{b}^{2}})+\frac{\pi }{8}({{a}^{2}}+{{b}^{2}})+\frac{1}{2}ab\] \[=\frac{\pi }{4}({{a}^{2}}+{{b}^{2}})+\frac{1}{2}ab\]