Question

Prove that a tangent to a circle is perpendicular to the radius drawn from the point of contact.

Answer 1

Given: Line $l$ is tangent to the $\odot (0,r)$ at point $A$.
To prove: $OA\perp l$
Proof: Let $P\in l,P\ne A$
If $P$ is in the interior of $\odot (0,r)$, then the line $l$ will be a secant of the circle and not a tangent. But $l$ is a tangent of the circle, so $P$ is not in the interior of the circle. Also $P\ne A$.
$\therefore$ $P$ is the point in the exterior of the circle.
$\therefore$ $OP>OA$ ($OA$ is the radius of the circle)
Therefore each point $P\in l$ except $A$ satisfies the inequality $OP>OA$
Therefore $OA$ is the shortest distance of the line $1$ from $O$.
$\therefore$ $OA\perp l$