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(: Angle subtended by an arc at the centre is twice the, angle subtended by it in the remaining portion of the circle, Mathematic, 600, . AOBD is equilateral, ZBOD= 60°, (LPA's), ZAOD+ ZBOD= 180°, ZAOD+ 60° 180°, 120°, ZAOD =, 1, ZAOD, ZACD =, 1, x120° = 60°, In AAMC,, (Angles on the same line), ZAMC + ZACM+ ZCAM= 180°, 90° + 60° + CAM= 180°, ZCAM= 180° –, -150° 30°, ZCAB = 30° Ans., NCERT Exercises with Solutions, NCERT EXERCISE 10.1, 1. Suppose you are given a circle. Give a construction to find its centre., If two circles intersect at two points, prove that their centres lie on the perpendicular, bisector of the common chord., 2., Solutions:, Steps of construction:, (i) Mark any three points A, B and C on the circle., 1., (ii) Join AB and BC., (iii) Draw the perpendicular bisector of AB. Let it, be PQ., (iv) Draw the perpendicular bisector of BC. Let it, be RS., (v) PQ and RS intersect at point O., (vi) O is the centre of given circle., 2. Given : Circles C(0, r) and C(D, r,) intersect at A, P., and B., Fig. 10.124., To prove : OD is perpendicular bisector of, A, АВ., Construction : Join OA, AD, OB and BD., Proof : In AAOD and ABOD,, OA = OB (Radii of the circle), DA = DB (Radii of the circle), OD = OD, (common), Fig. 10.125., 1

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Crcles, 601, AMOD ABOD, (By SSS), (By cpct), and, In AOAC and AOBC,, OA OB, (Radii of the circle), OC OC, (Common), (Proved), 21= 22, AAOC AOBC, (By SAS), (By cpct), (Ву срсt), 23 4, de, and, АС ВС, 23+ 4 180°, (LPA's), 243 180°, 180°, 23 =, = 90°, 2, . OD is perpendicular bisector of AB., NCERT EXERCISE 10.2, 1. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their, centres is 4 cm. Find the length of the common chord., 2. If two equal chords of a circle intersect within the circle, prove that the segments of one, chord are equal to corresponding segment of the other, chord., If two equal chords of a circle intersect within the circle,, prove that the line joining the point of intersection to the, centre makes equal angles with the chords., 4. If a line intersects two.concentric circles (circles within, the same centre) with centre O at A, B, Cand D, prove that, AB CD (see Fig. 10.126)., 5. Three girls Reshma, Salma and Mandip are playing a game, by standing on a circle of radius 5 m drawn in a park., Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance, between Reshma and Salma and between Salma and Mandip is 6 m each, what is the, distance between Reshma and Mandip?, B, Fig. 10.126., 5. A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are, sitting at equal distance on its boundary each having a toy telephone in his hands to talk, cach other. Find the length of the string of each phone., Solutions:, 1. Given: Let O and O, be the centre of bigger, and smaller circle respectively, OA OB-5 cm, 0,4=0,B-3 cm, 00,- 4 cm, To find: AB, Construction: Join OA, OB, O,A and O,B join,, B, AB also., Fig. 10.127., 2

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5. Given: OR =OM=5 m and SR= SM=6 m., Constrution : Join OR, OM and OS. Draw ON1 SR., In circle II, BC is the chord and OP 1 BC., 603, Ordes, (D respectively. Join OP, OP = OP, (Common), OE = OF, (: AB = CD), ZF = ZE, (Each 90°), AOPF = AOPE, (By R.H.S.), PE = PF, (By cpct) .(1), (Given), and so, AB = CD, D., Fig. 10.129., 1, CD, AB =, 2, 2, BE = CF, .(2), AE= DF, and, .(3), Adding (1) and (3), we get,, PE+ AE= PF+ DF, AP= DP, Subtracting (1) and (2) we get,, BE- PE= CF-PF, BP= CP, 3. Given: AB and CD are two equal chords of a circle which, intersect at E., To prove:, 21 = 22, Construction : Draw OL LAB and OM1 CD. Join OE., E, Proof: In AOLE and AOME,, M, OE = OE, (Common), (: AB = CD), W7 =77, (Each 90°), AOLE = AOME (By R.H.S.), (Bу срс), D, Fig. 10.130., and so, 21 = 2, To prove : AB = CD, Construction : Draw OP 1 AD., AP= DP, .(1), II, .(2), BP = CP, Substracting (1) and (2), we get, B., KA, АP- ВР3D DР - СР, Fig. 10.131., AB = CD, To find : MR, 4