Question

An equilateral triangle of side 6cm is inscribed in a circle. Find the radius of the circle.
This question is from math, topic-circles.

Answer 1

Hello,
Let ∆ABC be an equilateral triangle inscribed in the circle with center O.

Given,radius  of the circle:
 OA = OB = OC=6 cm

Draw AD ⊥ BC.
so ∆ABC is an equilateral triangle.
we have that:
∠BAC = 60°     
the angle subtended by the arc at the centre is twice the angle subtended by it at any point on the remaining part of the circle
 ∠BOC = 2 ∠BAC    
then
∠BOC = 2 × 60° = 120°

In ∆BOC:
∠BOC + ∠OBC + ∠OCB = 180°
being OB = OC ⇒∠OBC = ∠OCB,
120° + ∠OBC + ∠OBC = 180°        
2 ∠OBC = 180° – 120° = 60°
∠OBC=60°/2=30°
so
 ∠OBC = ∠OCB = 30°

In ∆OBD,
cos∠OBD= BD/OB;
cos30°=BD/6;
√3/2=BD/6;
BD=3√3 cm

then
BC=2×BD=2×3√3=6√3 cm

So,
AB = BC = CA = 6√3 cm

Thus, the length of each side of the equilateral triangle is  6√3 cm

bye :-)