Question

If an equilateral triangle is drawn inside a circle such that the circle is the circum-circle of the triangle, find the relation between the length of the triangle and the radius of the circle.

• A. r/3
• B. 3r
• C. r/√3
• D. √3 x r

Answer 1

The correct option is D

√3 x r

Here we need to find the relation between the circumradius and the side length of the equilateral triangle.

Let us consider the radius of the circle is ‘r’ and the side length of the triangle is ‘a’.

Construction: Join OA. Draw OD perpendicular to AB. Since OD is perpendicular to AB it bisects the chord. Therefore, AD =$\frac{1}{2}\times AB$

In the triangle AOD,

$\angle AOD=\frac{1}{2}\angle AOB$ (triangle AOB is isosceles triangle)

Because, angle subtended by a chord A the centre O is double the angle subtended by the chord at any point on the circumference.

$\angle AOB=2\times \angle ACB$ $\angle ACB$ is internal angle of the equilateral triangle)

= $2\times {60}^{\circ }$

=${120}^{\circ }$ ​

Therefore, $\angle AOD=\frac{1}{2}\times {120}^{\circ }$ ​

$\angle AOD={60}^{\circ }$

$\angle ADO={90}^{\circ }$ (OD perpendicular to AB)

Applying trigonometric properties in rt triangle AOD,

$\frac{AD}{AO}$ = sin 60

AD = AO Sin 60

AD = $\frac{\sqrt{3}}{2}\times AO$ (AO = r)

AD = $\frac{\sqrt{3}r}{2}$

AB = 2 x AD

= 2 x$\frac{\sqrt{3}r}{2}$

a = $\sqrt{3}\times r$

The relation between the side of the equilateral triangle and the radius of circumcircle is, a = $\sqrt{3}\times r$.