Question

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

Answer 1

Answer:

Given :

A circle with centre O and a tangent AB to the circle at a point P.

To prove :

OP is perpendicular to AB.

Construction :

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

Proof :

Since all line segments joining to the point O to a point AB, the shortest one is perpendicular to AB. So, in order to prove that OP is perpendicular to AB, it is sufficient to prove that OP is shorter than any other segment joining O to any point AB.

Clearly, OP = OR [ Radii of the same circle]

Now, OQ = OR + RQ

=> OQ > OR

=> OQ > OP [ OP = OR ]

=> OP < OQ

Thus,OP is shorter than any other segment joining O to any point of AB.

Hence, OP is perpendicular to AB.