question_answer

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

Answer:

Given, a tangent AB at point P of the circle with centre O.

To prove:           .

Construction: Join OQ where Q is a point (other than P) on AB.

Proof: Since Q is a point on the tangent AB (other than P).

Q lies outside the circle.

Let OQ intersect the circle at R.

.

But                   .         (Radii of the circle)

.

Thus, OP is the shortest distance than any other line segment joining O to any point of AB.

But, we know that the shortest distance between a point and a line is the perpendicular distance

Hence Proved.