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.

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
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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

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