# Tangent to a Circle

The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.

**Statement 1**

The shortest segment that can be drawn from a given point to a given line is the perpendicular from the given point to the given line

**Theorem**

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

**Given **

P is a point of contact of a tangent AB to the circle C (O, r).

**To Prove **

OP âŠ¥ AB

**Construction **

Let Q be a point on AB, other than P. Join O to Q.

**Proof **

Q is a point on the tangent AB other than the point of contact P cutting the circle at P'.

Now, OQ = OP' + P'Q

= OP + P'Q [OP = OP' = r]

OQ > OP, i.e. OP < OQ

From statement 1, OP is the shortest of all the distances thus we conclude

OP âŠ¥ AB.