Q36 of 52 Page 10

If the angle between two tangents drawn from an external point P to a circle of radius a and center O, is 60o, then find the length of OP. (CBSE 2017)


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°


⇒ 2∠OPA = 60°


⇒ ∠OPA = 30°


Also, OA ⏊ 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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