Question

An equilateral triangle of side 9 cm is inscribed in a circle. The radius of the circle is
(a) 3 cm
(b) $3\sqrt{2}\mathrm{cm}$
(c) $3\sqrt{3}\mathrm{cm}$
(d) 6 cm
Figure

Answer 1

(c) $3\sqrt{3}\mathrm{cm}$

Let ΔABC be an equilateral triangle of side 9 cm.
Let AD be one of its medians.

Then AD ⊥ BC and BD = 4.5 cm
∴ $\mathrm{AD}=\sqrt{{\mathrm{AB}}^2-{\mathrm{BD}}^2}=\sqrt{{\left(9\right)}^2-{\left(\frac{9}{2}\right)}^2}=\sqrt{81-\frac{81}{4}}=\sqrt{\frac{324-81}{4}}=\sqrt{\frac{243}{4}}=\frac{9\sqrt{3}}{2}\mathrm{cm}$
Let G be the centroid of ΔABC.
Then AG : GD = 2 : 1
∴ Radius = AG = $\frac{2}{3}\mathrm{AD}=\left(\frac{2}{3}\times \frac{9\sqrt{3}}{2}\right)\mathrm{cm}=3\sqrt{3}\mathrm{cm}$