Answer :

Let l be a tangent to the circle having centre O. l touches the circle at P. Let m be the perpendicular line to l from P.

We have to prove that m passes through O i.e. O ∈ m.

Proof:

If O m, then we can find such M m that O and M are in the same half plane of l.

T ∈ l is a distinct point from P.

∴ ∠MPT = 90° and ∠OPT = 90°

M and O are points of the same half plane so this is impossible.

Thus, our assumption is wrong.

∴ O ∈ m

∴The perpendicular drawn to a tangent to the circle at the point of contact of the tangent passes through the centre of the circle.

Hence proved.

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