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°

In ∆BOC,

∠BOC + ∠OBC + ∠OCB = 180°

∴ 120° + ∠OBC + ∠OBC = 180° [OB = OC ⇒∠OBC = ∠OCB]

⇒ 2 ∠OBC = 180° – 120° = 60°

⇒ ∠OBC = ∠OCB = 30°

In ∆OBD,

So, AB = BC = CA = cm

Thus, the length of each side of the equilateral triangle is cm.

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