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 point of contact at the centre.

Answer 1

Given : $AP$ and $BP$ are tangent to circle with centre $O$

To prove $\angle P+\angle O={180}^{\circ }$

$\angle A=\angle B={90}^{\circ }$ (radius and tangent of circle are always perpendicular)

$\angle A+\angle B+\angle P+\angle O={360}^{\circ }$ (Angle sum property of quadrilateral)

${90}^{\circ }+{90}^{\circ }+\angle P+\angle O={180}^{\circ }$

$\Rightarrow \angle P+\angle O={180}^{\circ }$

Hence proved.