**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.

**Given**:

A circle with centre O .

Line PQ is tangent, touches the circle at P.

**Proof**

$Therefore,OQradius$

$Thisistrueforallthepointsonlineof\mathrm{tan}gentPQ.$

$SoOPisshortestdis\mathrm{tan}ceoflinePQfromP.\phantom{\rule{0ex}{0ex}}$ $MeansOPisperpendiculartoPQ$

**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.

**Given**:

A circle with centre O .

$LetLineXY\perp OPatpointP.$**To Prove**

XY is tangent to the circle.

**Proof**

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

$\u2206OPQisrightangletrianglewith\angle P={90}^{\circ}$So OQ is hypotenuse

We can say

$OQOP\phantom{\rule{0ex}{0ex}}$ $OQradius$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.

**Given**:

O is centre of the circle.

PQ, PR are two tangents drawn from point .

**To Prove**

PQ = PR

**Proof**

…………………..(R.H.S)

PQ = PR ………….(CPCT)

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