Two tangent segments BC and BD are drawn to a circle with center O such that ∠CBD = 120°. Prove that OB = 2BC.
Given : A circle with center O , BC and BD are two tangents such that ∠CBD = 120°
To Proof : OB = 2BC
Proof :
In △BOC and △BOD
BC = BD
[Tangents drawn from an external point are equal]
OB = OB
[Common]
OC = OD
[Radii of same circle]
△BOC ≅ △BOD [By Side - Side - Side criterion]
∠OBC = ∠OBD
[Corresponding parts of congruent triangles are congruent]
∠OBC + ∠OBD = ∠CBD
∠OBC + ∠OBC = 120°
2 ∠OBC = 120°
∠OBC = 60°
In △OBC
⇒OB = 2BC
Hence Proved !