Question

Prove that : A tangent to a circle is perpendicular to the radius of the circle​

Answer 1

Answer: A tangent to a circle is perpendicular to the radius through the point of contact.

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

Now, OP = OR (radii of the same circle)

Also, OQ = OR +RQ

⇒      OQ > OR

⇒      OQ >OP (∵ OP =OR)

⇒      OP < OQ

Thus, OP is shorter than any segment joining OP to any point on AB.

Hence, OP is perpendicular to AB.

Answer 2

Answer:

we are given a circle with centre O and a tangent XY to the circle at a point P.

take a point Q on XY other than P and join OQ

the point Q must lie outside the circle XY

therefore OQ is longer than the radius OF of the circle, that is OQ >OP

Hence proved

Hope it helps you