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Arc and ChordsArc 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
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. __________________________________________________________________ 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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