Q25 of 55 Page 1

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

Let a circle with center O is given.


Also, let XY be a tangent at point P to the circle.


We need to prove: OP ⊥ XY.



Proof:


So, firstly, take any point Q on XY other than P and connect OQ.


Then, point Q cannot lie inside the circle, because if so, then XY will not be a tangent to the circle.


So, OQ > OP


This happens with all other points on the line XY except the point P.


Therefore, OP is the shortest distance that connects to XY and shortest distance is always perpendicular.


∴ OP ⊥ XY.


Hence, proved.


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