Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. (CBSE 2011,2014)

Given: XY is a tangent at point P to the circle.

To prove: OP ⊥ XY

Construction: Take a point Q on XY other than P and join OQ.

Proof: If point Q lies inside the circle, then XY will become a secant and not a tangent to the circle. So the point Q must lie outside the circle.

∴ OQ > OP

This happens with every point on the line XY except the point P.

So, OP is the shortest of all the distances of the point O to the points of XY.

∴ OP ⊥ XY

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