Q. 8

Prove that the perpendicular at the point of contact of the tangent to a circle passes through the center.

Answer :


Let us consider a circle with center O and XY be a tangent


To prove : Perpendicular at the point of contact of the tangent to a circle passes through the center i.e. the radius OP XY


Proof :


Take a point Q on XY other than P and join OQ .


The point Q must lie outside the circle. (because if Q lies inside the circle, XY


will become a secant and not a tangent to the circle).


OQ is longer than the radius OP of the circle. That is,


OQ > OP.


Since this happens for every point on the line XY except the point P, OP is the


shortest of all the distances of the point O to the points of XY.


So OP is perpendicular to XY.


[As Out of all the line segments, drawn from a point to points of a line not passing through the point, the smallest is the perpendicular to the line.]


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