1) If two arcs of a circle are congruent, then corresponding chords are equal.

If arc PQ = arc RS then chord PQ = chord RS

If chord PQ = chord RS then arc PQ = arc RS.

2) The perpendicular from the center of a circle to a chord bisects the chord.

If OM ⊥ PQ then MP = MQ

If MP = MQ then OM ⊥ PQ.

3) If two chords of a circle AB and AC of a circle with center O are such that center O lies on the bisector of ∠BAC, then AB = AC ( chords are equal ).

4) If two circles intersect in two points, then the line through the centers is perpendicular bisector of the common chord.

OP ⊥ bisector of AB

1) Line l intersect two concentric circles whose common center is ‘O’ at the points A,B, C and D. Show that AB = CD.

Line l intersect two circles in A,B, C and D.

Statements |
Reasons |

1) OM ⊥ BC | 1) By construction |

2) BM = CM | 2) Perpendicular drawn from the center bisects the chord. |

3) OM ⊥ AD | 3) By construction |

4) AM = DM | 4) Perpendicular drawn from the center bisects the chord. |

5) AM - BM = DM - CM | 5) Subtraction property (4) and (2) |

6) AB = CD | 6) Subtraction property |

2) O is the center of the circle of radius 5 cm. OP ⊥ AB,OQ ⊥ CD, AB || CD, AB = 6 cm and CD = 8 cm. Find PQ.

Join OA and OC

As perpendicular drawn from the center, bisects the chord AB and CD at P and Q respectively.

AP = PB = ½ AB = 3 cm

And CQ = QD = ½ CD = 4 cm

In right triangle OAP, by Pythagorean theorem

OA

5

⇒ OP

⇒ OP

⇒ OP = 4 cm

In right triangle OCQ, by Pythagorean theorem

OC

⇒ 5

⇒ OQ = 3

∴ PQ = PO – QO

PQ = 4 -3

PQ = 1 cm.

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• Circles

• Parts of Circle

• Arc and Chords

• Equal Chords of a Circle

• Arc and Angles

• Cyclic Quadrilaterals

• Tangent to Circle