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.   r/3 3r r/√3 √3 x r

Solution

## 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 =12×AB In the triangle AOD,                      ∠AOD=12∠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.                      ∠AOB=2×∠ACB         ∠ACB is internal angle of the equilateral triangle)                                 = 2×60∘                                    =120∘   ​ Therefore,      ∠AOD=12×120∘   ​                       ∠AOD=60∘                          ∠ADO=90∘                   (OD perpendicular to AB) Applying trigonometric properties in rt triangle AOD,              ADAO = sin 60                    AD = AO Sin 60                    AD = √32×AO                            (AO = r)                      AD = √3r2          AB = 2 x AD                 =  2 x√3r2            a = √3×r The relation between the side of the equilateral triangle and the radius of circumcircle is,  a = √3×r.

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