Given: With OP Joining TS, the circle has a centre O at point P and a tangent to TS.
to prove: OP is perpendicular to TS passing through the centre of the circle
Construction: Draw a line OR intersecting the circle at Q and meeting the tangent TS at R
Proof:
$OP=OQ$ (radius of the circle)
And $OQ<OR$
$\Rightarrow OP<OR$
We can similarly prove that $OP<$ every line drawn from $O$ to $TS$ .
$OP$ is the shortest
$OP$ is perpendicular to $TS$
Therefore, the perpendicular through P will be passing through the centre of the circle
– Hence proved