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.
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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