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

Question

Philoid · Accepted Answer

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

We have to prove that OP ⊥ AB.

Construction: Take any point Q, other than P on the tangent AB. Join OQ. Suppose OQ meets the circle at R.

Proof:

We know that among all line segments joining the point O to a point on AB, the shortest one is perpendicular to AB.

So, to prove that OP ⊥ AB, it is sufficient to prove that OP is shorter than any other segment joining O to any point of AB.

⇒ OP = OR [Radii of the same circle]

Now, OQ = OR + RQ

⇒ OQ > OR

⇒ OQ > OP [∵ OP = OR]

∴ OP is shorter than any other segment joining O to any point on AB.