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

Let AB be the tangent drawn at the point P on the circle with O.

If possible, let PQ be perpendicular to AB, not passing through O. Join OP.

$$\therefore$$ Tangent at a point to a circle is perpendicular to the radius through the point,

$$\therefore$$ $$AB \ ⊥ \ OP$$
$$\Rightarrow ∠ \ OPB \ = \ 90°$$

$$∠ \ QPB \ = \ 90°$$ (Construction)

$$\therefore ∠ \ QPB \ = \ ∠ \ OPB$$ , which is not possible.

$$\therefore$$ It contradicts our supposition.

Hence, the perpendicular at the point of contact to the tangent to a circle passes through the centre.