Q. 194.6( 5 Votes )

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 segments joining the points of contact at the center.

Answer :



Let us consider a circle with center O and PA and PB are two tangents to the circle from an external point P


To Prove : Angle between two tangents is supplementary to the angle subtended by the line segments joining the points of contact at center, i.e. APB + AOB = 180°


Proof :


As AP and BP are tangents to given circle,


We have,


OA AP and OB BP


[Tangents drawn at a point on circle is perpendicular to the radius through point of contact]


So, OAP = OBP = 90°


In quadrilateral AOBP, By angle sum property of quadrilateral, we have


OAP + OBP + AOB + APB = 360°


90° + 90° + AOB + APB = 360°


AOB + APB = 180°


Hence Proved


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