Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact to the centre.


fig.20


In the fig.20, PA and PB are the two tangents drawn from an external point P at the point of contacts A and B on the circle with centre O respectively.


OA PA and OB PB


[ radius of a circle is always to the tangent at the point of contact.]


OAP = OBP = 90°


we know that –


Sum of all the angles of a quadrilateral = 360°


In quadrilateral OAPB,


OAP + OBP + APB + AOB = 360°


180° + APB + AOB = 360°


APB + AOB = 180°


Hence, the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact to the centre.


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