Prove that angle between the two tangents to a circle drawn from an external point is supplementary

👉Q. Prove that the angle between the two tangents to a circle drawn from an external point is supplementary to the angle subtended by the line segment joining the points of contact at the centre. (CBSE 2008 C)

Or

Q. Two tangents PA and PB are drawn from an external point P to a circle with centre O. Prove that AOBP is a cyclic quadrilateral.
(CBSE Sample Paper 2011)

#class10 #circleclass10

sol-
As radius of the circle is perpendicular to the tangent.

OA⊥PA

Similarly OB⊥PB

∠OBP=90

∠OAP=90

In Quadrilateral OAPB, sum of all interior angles =360

⇒∠OAP+∠OBP+∠BOA+∠APB=360

⇒90+90+∠BOA+∠APB=360

∠BOA+∠APB=180

It proves the angle between the two tangents drawn from an external point to a circle supplementary to the angle subtented by the line segment