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

STEP 1 : Construction

Let $PA$ and $PB$ be two tangents drawn from an external point $P$.

Join $OA$ and $OB$ such that $OA$ and $OB$ are the radius of the circle with centre $O$.

STEP 2 : Applying angle sum property in quadrilateral $AOBP$

We know that,

$\angle A=\angle B=90°$ (Angle between the tangent and the radius)

The sum of the interior angles of a quadrilateral is $360°$

$\Rightarrow \angle A+\angle O+\angle B+\angle P=360°$

$\Rightarrow 90°+\angle O+90°+\angle P=360°$

$\Rightarrow \angle O+\angle P+180°=360°$

$\Rightarrow \angle O+\angle P=360°-180°=180°$

$\Rightarrow \angle O+\angle P=180°$

Hence, 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