Question

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Answer 1

Here,

$OA=OC=radius$

and, $OB=OC+CB$

$\Rightarrow OB>OC$

$\Rightarrow OB>OA$

Here, B is any point on the tangent. Thus $OA$ is shortest among all the line segment joining O to any point on the tangent.

We also know that shortest distance between a line & a point is $\perp$ distance.

Hence, the tangent at any point of circle is $\perp$ to the radius.