$ABC$ is an equilateral $\triangle $ of side $24$cm. If the centroid $O$ is the centre of the circle then find the radius of the circle.

Question

$ABC$ is an equilateral $\triangle $ of side $24$cm. If the centroid $O$ is the centre of the circle then find the radius of the circle.

I couldn't get any idea to solve. How can I approach such type of problems?

Answer 1

In the equilateral triangle $ABC$, construct the medians $AD$, $BE$, $CF$. (See figure on the page Properties of Equilateral Triangles.) We know the following important properties of an equilateral triangle:

• The medians are also altitudes, angle bisectors, and perpendicular bisectors of sides.

• The medians intersect at the point $O$ which is simultaneously the centroid, orthocenter, incenter, and circumcenter.

Note that in the right triangle $BOD$ we have $$BD={BC\over2}=12, \quad \angle BOD=60^\circ, \quad \angle DBO=30^\circ.$$ Hence, using the trigonometric functions tan and cos, we find $$ \mbox{ the radius of inscribed circle (inradius) } DO = BD \tan 30^\circ = 4\sqrt3; $$ $$ \mbox{ the radius of circumscribed circle (circumradius) } BO = {BD \over \cos30^\circ} = 8\sqrt3. $$