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

Given: A circle C (O, r) and a tangent AB at a point P.

To Prove: OP is perpendicular to AB.

Construction: Take any point Q, other than P, on the tangent AB. Join OQ.

Since, Q is a point on the tangent AB, other than the point of contact P, so Q will be outside the circle.

Let OQ intersect the circle at R.

Then, OQ=OR+RQ

OQ>OR

OQ>OP (OR=OP=radius)

Thus, OP

But, among all the line segments, joining the point O to a point on AB, the shortest one is the perpendicular from O on AB.

Hence, OP is perpendicular to AB.