Question

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

Answer 1

Referring to the figure:

$OA=OC$ (Radii of circle)

Now $OB=OC+BC$

$\therefore OB>OC$ ($OC$ being radius and $B$ any point on tangent)

$\Rightarrow OA<OB$

$B$ is an arbitrary point on the tangent.

Thus, $OA$ is shorter than any other line segment joining $O$ to any

point on tangent.

Shortest distance of a point from a given line is the perpendicular distance from that line.

Hence, the tangent at any point of circle is perpendicular to the radius.