In this Circle -Theorems, the theorems are based on equal chord. Here, we shall also study theorems on subtended angles and the relationship between the lengths of chords and their distances from the center of the given circle.

1) Equal chords of a circle are equidistant from the center.

AB = CD and OL ⊥ CD.

i.e. OM = OL

1) OL ⊥ AB ⇒ AL = 1/2AB | 1) OM ⊥ CD ⇒ CM = 1/2CD |

2) AB = CD | 2) Given |

3) 1/2 AB = 1/2 CD | 3) Given |

4) AL = CM | 4) From AB = CD |

5) OA = OC | 5) Equal radius of the circle |

6) AL = CM | 6) From (4 ) |

7) ∠ALO = ∠CMO | 7) Each 90^{0} |

8) OL = OM | 8) CPCTC |

Statements |
Reasons |

1) AB = CD | 1) Given |

2) OA = OC | 2) Radius of same circle |

3) OB = OD | 3) Radius of same circle |

4) ΔAOB = ΔCOD | 4) SSS Postulate |

5) ∠AOB = ∠COD | 5) CPCTC |

3) Angles in the same segment of a circle are equal.

in the same segment of the circle.

Statements |
Reasons |

1) ∠POQ = 2∠PRQ | 1) Given |

2) ∠POQ = 2∠PSQ | 2) Given |

3) 2∠PRQ = 2∠PSQ | 3) From (1) and (2) |

4) ∴ ∠PRQ = ∠PSQ | 4) From (3) |

5) Reflex ∠POQ = 2∠PRQ and Reflex ∠POQ = 2∠PSQ | 5) From the diagram |

6) 2∠PRQ = 2∠PSQ | 6) From the diagram |

7) ∠PRQ = ∠PSQ | 7) From (6) |

• Theorems on Chord

• Theorems on Chord and Subtended Angle

• Theorems on Arc and Angle

• Theorems on Cyclic Quadrilateral

GMAT

GRE

1st Grade

2nd Grade

3rd Grade

4th Grade

5th Grade

6th Grade

7th grade math

8th grade math

9th grade math

10th grade math

11th grade math

12th grade math

Precalculus

Worksheets

Chapter wise Test

MCQ's

Math Dictionary

Graph Dictionary

Multiplicative tables

Math Teasers

NTSE

Chinese Numbers

CBSE Sample Papers