Question

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

Answer 1

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.