Let there is a circle having radius r and center at O.

Let we have to draw a tangent t from the point A on the circle.

Now, take a point B, on the tangent t.

Now, join OB and let it meets the circle at point C.

Since among all line segment joining the point O to a point on tangent t, the perpendicular is shortest to tangent t.

Again, OA = OB  {radii of the circle}

Again, OB = OC + BC

=> OB > OC

=> OB > OA

=> OA < OB

Thus, OA is the shorter than any other line segment joining O to any point on tangent t.

Hence, OA is perpendicular to tangent t.

So, the tangents at any point of circle is perpendicular to the radius through the point of contact.