an equilateral triangle is inscribed in a cirle of radius 6 cm.Find its sides
Let ∆ABC be an equilateral triangle inscribed in the circle with center O.
Given, OA = OB = OC = 6 [Radius of the circle]
Draw AD ⊥ BC.
∠BAC = 60° [∵ ∆ABC is an equilateral triangle]
∠BOC = 2 ∠BAC (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 × 60° = 120°
∠BOC + ∠OBC + ∠OCB = 180°
∴ 120° + ∠OBC + ∠OBC = 180° [OB = OC ⇒∠OBC = ∠OCB]
⇒ 2 ∠OBC = 180° – 120° = 60°
⇒ ∠OBC = ∠OCB = 30°
So, AB = BC = CA = cm
Thus, the length of each side of the equilateral triangle is cm.