Question

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 at the centre.​

Answer 1

Answer:

Draw a circle with center O and take a external point P. PA and PB are the tangents.

As radius of the circle is perpendicular to the tangent.

OA⊥PA

Similarly OB⊥PB

∠OBP=90o

∠OAP=90o

In Quadrilateral OAPB, sum of all interior angles =360o

⇒∠OAP+∠OBP+∠BOA+∠APB=360o

⇒90o+90o+∠BOA+∠APB=360o

∠BOA+∠APB=180o

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