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