Question

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

Answer 1

Yes.

The tangent is always perpendicular to the radius.

The perpendicular distance is the shortest distance of all possible distances.

The radius is the shortest distance and therefore it must be perpendicular.

Answer 2

Answer:

$here \: is \: your \: answer$

Step-by-step explanation:

GIVEN:-

We are given a circle with centre O and a tangent XY to the circle at a point P.

TO PROVE:-

We need to prove that OP is perpendicular to XY.

PROOF:-

Take a point Q on XY other than P and join OQ as given in figure.

the point Q must lie outside the circle.

(Note : that if Q lies inside the circle, XY will become Secant and not a tangent to the circle ).

Therefore 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.