Question

If the angle between two tangents drawn from an external point P to a circle of radius "A" and center O is 60 degrees. find the length of OP.

Answer 1

We know that tangent is always perpendicular to the radius at the point of contact.

So, ∠OAP = 90

We know that if 2 tangents are drawn from an external point, then they are equally inclined to the line segment joining the centre to that point.

So, ∠OPA = 1/2∠APB = 1/2×60° = 30°

According to the angle sum property of triangle-

In ∆AOP,∠AOP + ∠OAP + ∠OPA = 180°⇒∠AOP + 90° + 30° = 180°⇒∠AOP = 60°

So, in triangle AOP

sin angle APO = OA/OP

sin 30= r/OP
1/2=r/OP
OP =2r
this is the right answer

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