Question

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

Answer 1

Let AB be the tangent to the circle at point P with centre O.
We have to prove that PQ passes through the point O (center).
Suppose that PQ doesn't pass through point O. Join OP.
Through O, draw a straight line CD parallel to the tangent AB.
PQ intersect CD at R and also intersect AB at P.
As, CD || AB, PQ is the transversal line.
$\angle$ ORP = $\angle$ RPA (Alternate interior angles)
but also,
$\angle RPA={90}^{\circ }(PQ\perp AB)$
$\Rightarrow \angle ORP={90}^{\circ }$
$\angle ROP+\angle OPA={180}^{\circ }$ (Co-interior angles)
$\Rightarrow \angle ROP+{90}^{\circ }={180}^{\circ }$
$\Rightarrow \angle ROP={90}^{\circ }$
Thus, the $\Delta$ ORP has 2 right angles i.e., $\angle$ ORP and $\angle$ ROP which is not possible.
Hence, our supposition is wrong.
$\therefore$ PQ passes through the point center O.