# Prove that the ta

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.

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