The tangent at any point of a circle is perpendicular to the radius through the point of contact

The tangent at any point of a circle is perpendicular to the radius th

1.	The tangent at any point of a circle is perpendicular to the radius through the point of contact
Answer» GIVEN\xa0: A circle C (0, r) and a tangent AB at a point P.TO PROVE :\xa0{tex}O P \\perp A B{/tex}CONSTRUCTION Take any point Q, other than P, on the tangent AB. Join OQ. Suppose OQ meets the circle at R.PROOF\xa0: We know that among all line segments joining the point O to a point on AB, the shortest one is perpendicular to AB.To prove that {tex}O P \\perp A B{/tex}, it is sufficient to prove that OP is shorter than any other segment joining O to any point of AB.Clearly,\xa0{tex}OP = OR{/tex}Now,\xa0{tex}OQ = OR + RQ{/tex}{tex}\\Rightarrow \\quad O Q > O R{/tex}{tex}\\Rightarrow \\quad O Q > O P \\quad [ \\because O P = O R ]{/tex}{tex}\\Rightarrow \\quad O P < O Q{/tex}{tex}\\therefore{/tex}\xa0{tex}O P \\perp A B{/tex}