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

It is given in the question that, O is the centre of the circle and l is the tangent at point A

We have to prove that OA is perpendicular to tangent l

Construction: First of all take a point B on the tangent and join AB. Suppose that OB meets the circle at C

Proof: We know that radius of the circle is equal to each other

∴ OA = OC

Also, OB = OC + BC

Clearly, OB > OC

This will happen with every pint on the line l except the point C also OC is the shortest of all the distances of the point O to the points of tangent (l)

Hence, OC is perpendicular to tangent (l) as we know that the shortest side is the perpendicular