question_answer

The length of each side of an equilateral triangle is \[14\sqrt{3}\,.cm\] The area of the incircle, in \[c{{m}^{2}},\] is

A) 450                              

B) 308

C) 154                  

D) 77

Correct Answer: C

Solution :

\[BD=DC=7\sqrt{3}\,cm\] \[AD=\sqrt{A{{B}^{2}}-B{{D}^{2}}}\] \[=\sqrt{{{(14\sqrt{3})}^{2}}-{{(7\sqrt{3})}^{2}}}\] \[=\sqrt{(14\sqrt{3}+7\sqrt{3})(14\sqrt{3}-7\sqrt{3})}\] \[=\sqrt{21\sqrt{3}\times 7\sqrt{3}}=21\,cm\] \[\therefore \]      OD = Radius of circle             \[=\frac{1}{3}\times 21=7\,cm\] \[\therefore \] Area of circle \[=\pi {{r}^{2}}\]             \[=\frac{22}{7}\times 7\times 7=154\,sq\,cm\]