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1) In a circle equal-chords are equidistant from the center.

If AB = CD then OM = OL

If OM = OL then AB = CD

2) Equal-chords of congruent circles are equidistant from the corresponding centers.

If two circles are congruent and AB = CD then OL = PM.

If two circles are congruent and OL = PM then AB = CD.

3)In a circle equal chords subtend equal angles at the center.

In a circle, if AB = CD then ∠AOB = ∠COD

In a circle, if ∠AOB = ∠COD then AB = CD.

1) If two chords of a circle are equally inclined to the diameter through their point of intersection, prove that the chords are equal.

Statements |
Reasons |

1)OL ⊥ AB and OM ⊥ AC | 1) By construction |

2)∠OLA = ∠OMA | 2)Each 90^{0} |

3) OA = OA | 3) Reflexive (common ) |

4) ∠OAL = ∠OAM | 4) Given |

5) ΔOLA = ΔOMA | 5) AAS postulate |

6) OL = OM | 6) CPCTC |

7) AB = CD | 7) Chords are equidistant from center O |

2) If two equal-chords of a circle intersect within the circle, prove that the line joining the point of intersection to the center makes equal angles with the chords.

Statements |
Reasons |

1) OV = OU | 1) Equal-chords of a circle are equidistant from the center |

2)∠OVT = ∠OUT | 2)Each 90° |

3) OT = OT | 3) Reflexive (common) |

4) ΔOVT ≅ ΔOUT | 4) HL postulate or (RHS theorem) |

5) ∠OTV = ∠OTU | 5) CPCTC |

• Circles

• Parts of Circle

• Arc and Chords

• Equal Chords of a Circle

• Arc and Angles

• Cyclic Quadrilaterals

• Tangent to Circle

Russia-Ukraine crisis update - 3rd Mar 2022

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