# If the angle betw

Let AP and BP are two tangents from an external point P on a circle with center O, at points A and B respectively.

Angle with AP and BP,

APB = 60° [Given]

In ΔAOP and ΔBOP

AP = BP [Tangents drawn from an external point to a circle are equal]

OP = OP [Common]

OA = OB [Radii of same circle]

ΔAOP ΔBOP [By Side-Side-Side Criterion]

OPA = OPB [Corresponding parts of congruent triangles are equal]

Also,

OPA + OPB = APB

OPA + OPA = 60°

2OPA = 60°

OPA = 30°

Also, OA ā AP [Tangent drawn at a point on the circle is perpendicular to the radius through point of contact]

In ΔOAP,

[As OA = radius of circle = a]

OP = 2a

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