Question

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

Answer 1

Given : A circle C (O,r) and a tangent AB at a point p.

To prove : OP is perpendicular to AB.

Construction : Take any point Q other than P , on the tangent AB . join OQ . Suppose OQ meets the circle at R .

Proof : we know that among all line segments joining the point O to a point on AB, the Shortest one is perpendicular to AB so, to prove that $OP\perp AB$

it is sufficient to prove that OP is shorter than any other segment joining O to any point of AB.

Clearly , OP+OR {Radii of the same circle}

$OQ=OR+RQ$

$\Rightarrow OQ>OR$

$\Rightarrow OQ>OP$

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

Hence proved.