Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

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Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

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

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