Tangent to the circle: 

A line that intersects the circle at only one point is called a tangent.

* There is only one tangent at a point on the circle.

* There are two tangents to a circle from a point outside the circle.

Length of tangent : 

The length of tangent from the external point and point of contact with the circle is called the length of the tangent.

Theorem 1

The tangent at any point of a circle is perpendicular to the radius through the point of contacts.



A circle with centre O .

Line PQ  is tangent, touches the circle at  P.


     As point Q   is outside the circle. 


Therefore,  OQ>radius    


 This is true for all the points on line of tangent PQ. 


So OP  is shortest distance of line PQ  from  P.   Means OP  is perpendicular to PQ

Theorem 2

Converse of the theorem:

Given a line which is perpendicular to radius at the end point of the radius then that line is tangent to the circle.



A circle with centre O .

Let Line XY  OP at point P.

To Prove

XY is tangent to the circle.



Take any point Q on XY . Q is appoint other than P.

OPQ  is right angle triangle with P = 90

So OQ  is hypotenuse

We can say

OQ > OP OQ > radius

Q is point outside the circle.

So all the points on line  XY are outside the circle.

Only point P  is on the circle.

Point of intersection of line and circle is only point  P.

Line  XY is tangent to the circle.

Theorem 3

The lengths of tangents drawn from an external point to a circle are equal.



O is centre of the circle.

PQ, PR are two tangents drawn from point .

To Prove



OQ  PQ , OR  PR OPR, OPQ are right angled triangles OP = OP .......(common) OQ = OR ....(Radii of same circle) OPR  OPQ


PQ = PR  ………….(CPCT)

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