Question

An equilateral triangle is inscribed in a circle of radius $6$ cm. Find its side.

Answer 1

Let $ABC$ be an equilateral triangle inscribed in a circle of radius 6 cm . Let $O$ be the centre of the circle . Then ,

$OA=OB=OC=6cm$
Let $OD$ be perpendicular from $O$ on side $BC$ . Then , $D$ is the mid - point of $BC$. $OB$ and $OC$ are bisectors of $\angle B$ and $\angle C$ respectively.

Therefore, $\angle OBD={30}^o$

In triangle $OBD$, right angled at $D$, we have $\angle OBD={30}^o$ and $OB=6cm.$

Therefore, $\cos \left(OBD\right)=\frac{BD}{OB}$

$⟹\cos \left({30}^o\right)=\frac{BD}{6}$

$⟹BD=6\cos {30}^0$

$⟹BD=6\times \frac{\sqrt{3}}{2}=3\sqrt{3}cm$

$⟹BC=2BD=2\left(3\sqrt{3}\right)=6\sqrt{3}cm$

Hence, the side of the equilateral triangle is $6\sqrt{3}cm$