Question

An equilateral triangle of side $9\text{cm}$ is inscribed in a circle. Find the radius of the circle.

• A. $3\sqrt{3}\text{cm}$
• B. $6\text{cm}$
• C. $3\sqrt{2}\text{cm}$
• D. $3\text{cm}$

Answer 1

The correct option is A $3\sqrt{3}\text{cm}$

Let triangle ABC be an equilateral triangle of side $9\text{cm}$, and AD be the median, where G is the centroid of the triangle that divides the median into 2:1.
GD is perpendicular to BC.
In the right-angled triangle ADB,
$A{D}^2+D{B}^2=A{B}^2$, where $AB=9\text{cm}$ and $BD=4.5\text{cm}$.
$A{D}^2+{\left(\frac{9}{2}\right)}^2=9^2$
$A{D}^2=9^2-{\left(\frac{9}{2}\right)}^2$
$A{D}^2=81-\left(\frac{81}{4}\right)$
$AD=\frac{9\sqrt{3}}{2}\text{cm}$
Radius $=\frac{2}{3}\left(AD\right)=\frac{2}{3}\left(\frac{9\sqrt{3}}{2}\right)=3\sqrt{3}\text{cm}$