Question

An equilateral triangles of side $9cm$ is inscribed in a circle. Radius of circle is

• A. $\sqrt{3}cm$
• B. $2\sqrt{3}cm$
• C. $3\sqrt{3}cm$
• D. $4\sqrt{3}cm$

Answer 1

The correct option is C $3\sqrt{3}cm$

Let $ABC$ be an equilateral triangle of side $9$cm and let $AD$ be one of its medians.

Let $G$ be the centroid of $△ABC$.

Then $AG:GD=2:1$

WKT in an equilateral triangle centroid coincides with the circum center.

$\therefore ,G$ is the center of the circumference with circum radius $GA$

Also $G$ is the center and $GD$ perpendicular to $BC$ .

$\therefore$, In right $△ADB$, we have

${AB}^2={AD}^2+{DB}^2$

$\Rightarrow 9^2={AD}^2+{DB}^2$

$\Rightarrow AD=\sqrt{81-\frac{81}{4}}=\frac{9\sqrt{3}}{2}$cm

$\therefore$ Radius$=AG=\frac{2}{3}AD=\frac{2}{3}\times \frac{9\sqrt{3}}{2}=3\sqrt{3}$cm