question_answer

An equilateral triangle of side 9 cm is inscribed in a circle. The radius of the circle is

A) \[3\,\,cm\]                     

B) \[3\sqrt{2}\,cm\]

C) \[3\sqrt{3}\,cm\]

D) \[6\,\,cm\]

Correct Answer: C

Solution :

Let ABC be an equilateral triangle of side 9 cm.

Let AD be its median, then \[AD\bot BC\]and

\[BD=4.5\,\,cm\]

\[AD=\sqrt{A{{B}^{2}}-B{{D}^{2}}}\]

\[=\sqrt{{{9}^{2}}-{{\left( \frac{9}{2} \right)}^{2}}}=\frac{9\sqrt{3}}{2}cm\]

Let O be the centroid of\[\Delta ABC.\]

Then, \[\frac{AO}{OD}=\frac{2}{1}\]

Radius \[=AO=\frac{2}{3}AD=\frac{2}{3}\times \frac{9\sqrt{3}}{2}=3\sqrt{3}\,\,cm\]