Question

Prove that the angle between two tangent 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

Let $PA$ and $PB$ be two tangents drawn from an external point $P$ to a circle with centre $O$.

We have to prove that angles $\angle AOB$ and $\angle APB$ are supplementary

i.e. $\angle AOB+\angle APB={180}^o$

In right $\Delta OAP$ and $\Delta OBP$, we have
$PA=PB$ [tangents drawn from an external point are equal]
$OA=OB$ [each equal to radius]
$OP=OP$

So, by $SSS-$criterion of congruence, we have
$△OAP\cong △OBP$

$\Rightarrow \angle OPA=\angle OPB$

$\Rightarrow$ $\angle AOP=\angle BOP$

$\Rightarrow \angle ABP=2\angle OPA$

$\Rightarrow$ $\angle AOB=2\angle AOP$

But, $\angle AOP={90}^o-\angle OPA$ [$\because △OAP$ is right triangle]
$\therefore 2\angle AOP={180}^o-2\angle OPA$
$\Rightarrow \angle AOB={180}^o-\angle APB$

$\Rightarrow \angle AOB+\angle APB={180}^o$