Question

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

Answer 1

Given: A circle with centre O with tangent t at the point of contact P.

To prove: OP is perpendicular to tangent t.

Proof: Let A be a point on tangent t.
OA > OB
OA > OP (as OB = OP = radii)

Similarly, assume point C and E on tangent t.
As above, we get that
OC > OP
and
OE > OP

So, same could be proved for all the points lying on the tangent t.
Therefore, we can say that OP is the smallest line that connects origin to the tangent.
Hence, OP is perpendicular to tangent t.
(a perpendicular distance is the smallest distance between a point and a line)