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

Referring to the figure:
OA=OC(Radii of circle)
Now OB=OC+BC
∴OB>OC(OC being radius and B any point on tangent)
⇒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.