Question

Prove that : In a plane of circle, 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 $⨀(0,r)$ at point $A$.
To prove: $\overline{OA}\perp l$
Proof: Let $P\in l,P\ne A$.
If $P$ is in the interior of $⨀(0,r)$, 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.
$\therefore OP>OA$ .......... ($\overline{OA}$ is the radius of circle)
Therefore each point $P\in l$ except A satisfies the inequality $OP>OA$.
Therefore, $OA$ is the shortest distance of line $l$ from $O$.
$\therefore \overline{OA}\perp l$.