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An equilateral triangle of side $9\ cm$ is inscribed in a circle. Find the radius of the circle.
Given:
An equilateral triangle of side $9\ cm$ is inscribed in a circle.
To do:
We have to find the radius of the circle.
Solution:
Steps of construction:
1. Draw a line segment $BC = 9\ cm$.
2. With centres $B$ and $C$, draw arcs of $9\ cm$ radius which intersect each other at $A$.
3. Join $AB$ and $AC$.
$\triangle ABC$ is the required triangle.
4. Draw perpendicular bisectors of sides $AB$ and $BC$ which intersect each other at $O$.
5. With centre $O$ and radius $OB$, draw a circle which passes through $A, B$ and $C$.
This is the require circle in which $\triangle ABC$ is inscribed.
On measuring the radius is $5.2\ cm$.
Radius $=\frac{2}{3} \mathrm{AD}$
$=\frac{2}{3} \times \frac{\sqrt{3}}{2} \times \operatorname{side}$
$=\frac{2}{3} \times \frac{\sqrt{3}}{2} \times 9$
$=3 \sqrt{3} \mathrm{~cm}$
$=3 \times 1.732$
$=5.196$
$=5.2 \mathrm{~cm}$
The radius of the circle is $5.2\ cm$.
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