Q24 of 47 Page 1

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Given: XY is a tangent at point P to the circle join OQ.

To prove: OP ⊥ XY


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


Proof: If point Q lies inside the circle, then XY will become a secant and not a tangent to the circle


∴ OQ > OP


This happens with 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


∴ OP ⊥ XY …(Shortest side is the perpendicular)


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