**Theorem 3.** If a line touches a circle and from the point of contact a chord is drawn, the angle which this chord makes with the given line are equal respectively to the angles formed in the corresponding alternate segments.

**Given:-** PQ is a tangent to circle with centre O at a point A, AB is a chord and C, D are points in the two segments of the circle formed by the chord AB.

To Prove:- (i)

(ii)

Construction:- A diameter AOE is drawn. BE is joined.

Proof: - In

From Theorem 3,

Again [Linear Pair]

and [Opposite angles of a cyclicquad]

**Theorem 4.** If a line is drawn through an end point of a chord of a circle so that the angle formed by it with the chord is equal to the angle subtend by chord in the alternate segment, then the line is a tangent to the circle.

**Given:- **A chord AB of a circle and a line PAQ. Such that where c is any point in the alternate segment ACB.

To Prove:- PAQ is a tangent to the circle.

Construction:- Let PAQ is not a tangent then let us draw P' AQ' another tangent at A.

Proof: - AS P ’AQ’ is tangent at A and AB is any chord

[theo.3]

But (given)

Hence AQ' and AQ are the same line i.e. P' AQ' and PAQ are the same line.

Hence PAQ is a tangent to the circle at A.

**Theorem 5. ** If two circles touch each other internally or externally, the point of contact lie on the line joining their centres.

**Given:-** Two circles with centres O_{1} and O_{2} touch internally in figure (i) and externally in figure (ii) at A.

To prove: - The points O_{1}, O_{2} and A lie on the same line.

Construction:- A common tangent PQ is drawn at A.

Proof: - In figure (i) (PA is tangent to the two circles)

O

_{1}, O_{2}and A are collinear.In figure (ii) (PA is tangent to the circles)

i.e. from a linear pair

O

_{1}, O_{2}and A lie on the same line.

**Example 5.** In the given figure TAS is a tangent to the circle, with centre O, at the point A. If , find the value of x and y.

**Solution:-**

Now TAS is tangent at A

[Angles in the alternateseg]

Subjects |
Maths (Part-1) by Mr. M. P. Keshari |

Chapter 9 |
Circle |

Chapter 10 |
Tangents to a circle |

Chapter 11 |
Geometrical Construction |

Chapter 12 |
Troigonometry |

Chapter 13 |
Height and Distance |

Chapter 14 |
Mensuration |

Chapter 15 |
Statistics |

Chapter 16 |
Probability |

Chapter 17 |
Co-ordinate Geometry |