Question

10. 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.


please answer this question in image form
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Answer 1

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=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