IF ANGLE BETWEEN TWO TANGENTS DRAWN FROM A POINT P TO A CIRCLE OF RADIUS A AND CENTER O IS 60 DEGREE THEN PROVE THAT AP = A UNDER ROOT 3.

We have the following situation-

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 = 12APB = 12×60° = 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

Hence proved.

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