# Arc and Chords

**Arc and Chords :**Arc is a part of a circle and chord is a segment whose end points are on the circle.

**Some Definitions related to Arc - Chords are :**

**Congruent circles :**Two circles are said to be congruent if and only if their radii are equal.

**Congruent arc :**Two arcs of a circle are said to be congruent if and only if they have the same degree of measure.

**Some results on congruent Arc and chords**

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

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

**Converse of the above is also true.**

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

**Converse of the above is also true.**

If MP = MQ then OM ⊥ PQ.

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

**Some solved problems on above theorems :**

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

**Given :**O is the center of two concentric circles.

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

**Prove that :**AB = CD

**Construction :**OM ⊥ BC and AD

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.

**Solution :**

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

^{2}= OP

^{2}+ AP

^{2}

5

^{2}= OP

^{2}+ 3

^{2}

⇒ OP

^{2}= 5

^{2}- 3

^{2}

⇒ OP

^{2}= 25 – 9 = 16

⇒ OP = 4 cm

In right triangle OCQ, by Pythagorean theorem

OC

^{2}= OQ

^{2}+ CQ

^{2}

⇒ 5

^{2}= OQ

^{2}- 4

^{2}= 9

⇒ OQ = 3

∴ PQ = PO – QO

PQ = 4 -3

PQ = 1 cm.

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**Circles**

• Circles

• Parts of Circle

• Arc and Chords

• Equal Chords of a Circle

• Arc and Angles

• Cyclic Quadrilaterals

• Tangent to Circle

• Circles

• Parts of Circle

• Arc and Chords

• Equal Chords of a Circle

• Arc and Angles

• Cyclic Quadrilaterals

• Tangent to Circle

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