Question

An equilateral triangle of side 9 cm is inscribed in a circle. The radius of the circle is

(a) 3 cm

(b) $3\sqrt{2}$ cm

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

(d) 6 cm

Answer 1

ANSWER:
(c) $3\sqrt{3}cm$
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
∴ $A{D}^2=A{B}^2-B{D}^2$

$\Rightarrow A{D}^2=9^2-(\frac{9}{2}{)}^2$

$\Rightarrow A{D}^2=\frac{4\ast 9^2-9^2}{4}$

$\Rightarrow A{D}^2=\frac{3\ast 9^2}{4}$

$\Rightarrow AD=\frac{9\sqrt{3}}{2}cm$
Let G be the centroid of ΔABC.

Now, AD=AG+GD
Also, AG : GD = 2 : 1

This means that $AG=\frac{2}{3}\ast AD$ or $AG=\frac{2}{3}\ast \frac{9\sqrt{3}}{2}cm$ or $\Rightarrow AG=3\sqrt{3}cm$

$\therefore$The radius of the circle is $3\sqrt{3}cm$