Arc and Chords
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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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