Equal chords of a circle are equidistant from the centre.

Chords of a circle that are equidistant from the centre are equal.

**Theorem 2. **The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

**Given: -** A circle with centre O and an arc AB subtending at the centre and on the remaining part of the circle at C.

To prove:-

Construction:- CO is joined and produced to P. OA and OB are joined

Proof:- In

OA = OC (Radii)

[Angle opposite to equal sides]

[Exterior angle of ]

[ ]

Similarity, by taking we have

Adding (i) and (ii) in fig (i) we have

Similarty, subtracting (i) from (ii) in fig. (ii) We have

Note: -

(i) Angle in a semi-circle is a right angle.

(ii) The circle drawn with hypotenuse of a right triangle as diameter pass through its opposite verdes or the arc of a circle subtending a right angle at any point on the remaining part of the circle is a semicircle.

**Theorem 3.** Angle in the same segment of a circle are equal.

**Given:-** A circle with centre O and are angles in the same segment.

To prove:-

Construction: - OA and OB are joined.

Proof:- In both the figure

from (i) and (ii) we get

Note:-It a line –segment joining two points subtends equal angles at two other points lying on the same side of the line containing the lime-segment, the four points lie on a circle.

**Example 3.** In the adjoining fig.

O is centre of the circle and the measure of are ABC is 100

^{0}. Determine .

**Solution:-** We know that the angle subtended by a chord at the centre of a circle is double the angle subtended at any other point on the circumference in the opposite segment.

Reflex

= 260

^{0}

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 |