The radius is always perpendicular to the radius at the point of conta

1.	The radius is always perpendicular to the radius at the point of contact
Answer» Given A radius OP of a circle C (O, r) and a line APB, perpendicular to OP.To Prove AB is a tangent to the circle at the point P.PROOF Take a point Q, different from P, on the line AB.since radius through the point of contact of tangent is perpendicular to it. Therefore,\xa0{tex} O P \\perp A B{/tex}.{tex}{/tex}We know that among all the line segments joining O to a point on AB, OP is the shortest one. Therefore,{tex}{/tex}\xa0OP <\xa0OQ{tex}{/tex}{tex}\\Rightarrow{/tex} OQ > OP{tex}\\Rightarrow{/tex}Q\xa0lies outside the circle.Therefore, every point on AB, other than P, lies outside the circle. This implies that AB meets the circle only at the point P.Hence, AB is a tangent to the circle at P.

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