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1 Pract Quet 1. Prove th Equal chord of a circle ubted equal agle the cetre.. Prove th Chord of a circle which ubted equal agle the cetre are equal. 3. Prove th he perpedicular from the cetre of a circle to a chord biect the chord. 4. Prove th he lie draw through the cetre of a circle to biect a chord i perpedicular to the chord. 5. Prove th Chord equiditat from the cetre of a circle are equal i legth 6. Prove th Chord of a circle which are equiditat from the cetre are equal 7. Prove th Of ay two chord of a circle the the oe which i larger i earer to the cetre. 8. Prove th Of ay two chord of a circle the the oe which i earer to the cetre i larger. 9. Prove th lie joiig the midpoit of two equal chord of circle ubted equal agle with the chord. 10. Prove th if two chord of a circle biect each other they mut be diameter. 11. If two chord of a circle are equally iclied to the diameter through their poit of iterect, prove th the chord are equal. 1. Prove th he agle ubteded by a arc the cetre i double the agle ubteded by it ay poit o the remaiig part of the circle. 13. Prove th Agle i the ame egmet of a circle are equal. 14. Prove th Agle i a emicircle i a right agle. 15. Prove th Arc of a circle ubtedig a right agle ay poit of the circle i it altere egmet i a emicircle. 16. Prove th Ay agle ubteded by a mr arc i the altere egmet i acute ad ay agle ubteded by a major arc i the altere egmet i obtue. 17. Prove th If a lie egmet joiig two poit ubted equal agle two other poit lyig o the ame ide of the lie egmet, the four poit are cocyclic. 18. Prove th Circle draw o ay oe ide of the equal ide of a cele trailge a diameter biect the ide. 19. Prove th he um of either pair of oppoite agle of a cyclic quadrileral i 180º. 0. Prove th If the um of a pair of oppoite agle of a quadrileral i 180º, the quadrileral i cyclic. Page 1

2 1. Prove th If oe ide of the cyclic quadrileral i proded the the exterr agle i equal to the iterr oppoite agle.. Prove th If two ide of a cyclic quadrileral are parallel, the the remaiig two ide are equal ad the diagoal are alo equal. 3. Prove th If two oppoite ide of cyclic quadrileral are equal, the the other two ide are parallel. 4. Prove th If two o parallel ide of a trapezium are equal, it i cyclic. 5. Prove th he um of the agle i the four egmet exterr to a cyclic quadrileral i equal to 6 right agle. 6. wo circle with cetre A ad B iterect C ad D. Prove th ACB = ADB. 7. Biector AD of AC of ABC pae through the cetre of the circumcircle of ABC. Prove th AB = AC. 8. I the below figure A, B ad C are three poit o a circle h th agle ubteded by the chord AB ad AC the cetre O are 800 ad 100 repectively. Determie BAC. 9. I the above right-ided figure, P i the cetre of the circle. Prove th XPZ = ( XZY + YXZ). 30. Prove th the midpoit of the hypoteue of a right triagle i equiditat from it vert. 31. I the below figure ABCD i a cyclic quadrileral, O i the cetre of the circle. If BOD = 1600, fid BPD. 3. Prove th i a triagle if the biector of ay agle ad the perpedicular biector of it oppoite ide iterect, they will iterect o the circumcircle of the triagle. Page

3 33. he diagoal of a cyclic quadrileral are right agle. Prove th perpedicular from the poit of their iterect o ay ide whe proded backward biect the oppoite ide. 34. If two circle iterect two poit, prove th their cetre lie o the perpedicular biector of the commo chord. 35. If two iterectig chord of a circle make equal agle with the diameter paig through their poit of iterect, prove th the chord are equal. 36. wo circle of radii 5 cm ad 3 cm iterect two poit ad the ditace betwee their cetre i 4 cm. Fid the legth of the commo chord. 37. If two equal chord of a circle iterect withi the circle, prove th the egmet of oe chord are equal to correpodig egmet of the other chord. 38. If two equal chord of a circle iterect withi the circle, prove th the lie joiig the poit of iterect to the cetre make equal agle with the chord. 39. I the below figure, AB i a diameter of the circle, CD i a chord equal to the radiu of the circle. AC ad BD whe exteded iterect a poit E. Prove th AEB = I the above right-ided figure, ABCD i a cyclic quadrileral i which AC ad BD are it diagoal. If DBC = 55 ad BAC = 45, fid BCD. 41. Prove th the quadrileral formed (if poible) by the iteral agle biector of ay quadrileral i cyclic. 4. ABCD i a cyclic quadrileral whoe diagoal iterect a poit E. If DBC = 70, BAC i 30, fid BCD. Further, if AB = BC, fid ECD. 43. If diagoal of a cyclic quadrileral are diameter of the circle through the vert of the quadrileral, prove th it i a rectagle. 44. wo circle iterect two poit A ad B. AD ad AC are diameter to the two circle. Prove th B lie o the lie egmet DC. 45. Prove th the quadrileral formed (if poible) by the iteral agle biector of ay quadrileral i cyclic. 46. If the o-parallel ide of a trapezium are equal, prove th it i cyclic. Page 3

4 47. wo circle iterect two poit B ad C. hrough B, two lie egmet ABD ad PBQ are draw to iterect the circle A, D ad P, Q repectively. Prove th ACP = QCD. 48. If circle are draw takig two ide of a triagle a diameter, prove th the poit of iterect of thee circle lie o the third ide. 49. Prove th the circle draw with ay ide of a rhombu a diameter, pae through the poit of iterect of it diagoal. 50. I the adjoiig figure, A, B, C ad D are four poit o a circle. AC ad BD iterect a poit E h th BEC = 130 ad ECD = 0. Fid BAC. 51. I the above right-ided figure, PQR = 100, where P, Q ad R are poit o a circle with cetre O. Fid OPR. 5. ABCD i a parallelogram. he circle through A, B ad C iterect CD (proded if eceary) E. Prove th AE = AD. 53. AC ad BD are chord of a circle which biect each other. Prove th (i) AC ad BD are diameter, (ii) ABCD i a rectagle. 54. A chord of a circle i equal to the radiu of the circle. Fid the agle ubteded by the chord a poit o the mr arc ad alo a poit o the major arc. 55. Prove th the circle draw with ay ide of a rhombu a a diameter, pae through the poit of it diagoal. 56. Biector of agle A, B ad C of a triagle ABC iterect it circumcircle D, E ad F A B C repectively. Prove th the agle of DDEF are 900, 900 ad Prove th the lie of cetre of two iterectig circle ubted equal agle the two poit of iterect. Page 4

5 58. I the adjoiig Fig., ABC = 69, ACB = 31, fid BDC. 59. I the above right-ided figure, A,B ad C are three poit o a circle with cetre O h th BOC = 30 ad AOB = 60. If D i a poit o the circle other tha the arc ABC, fid ADC. 60. I the below figure, AB ad CD are two equal chord of a circle with cetre O. OP ad OQ are perpedicular o chord AB ad CD, repectively. If POQ = 150, the fid APQ. 61. I the above right ided figure, if OA = 5 cm, AB = 8 cm ad OD i perpedicular to AB, the fid CD. 6. wo chord AB ad CD of legth 5 cm ad 11 cm repectively of a circle are parallel to each other ad are o oppoite ide of it cetre. If the ditace betwee AB ad CD i 6 cm, fid the radiu of the circle. 63. wo cogruet circle iterect each other poit A ad B. hrough A ay lie egmet PAQ i draw o th P, Q lie o the two circle. Prove th BP = BQ. 64. I ay triagle ABC, if the agle biector of A ad perpedicular biector of BC iterect, prove th they iterect o the circumcircle of the triagle ABC. 65. If arc AXB ad CYD of a circle are cogruet, fid the r of AB ad CD. 66. If the perpedicular biector of a chord AB of a circle PXAQBY iterect the circle P ad Q, prove th arc PXA Arc PYB. 67. A, B ad C are three poit o a circle. Prove th the perpedicular biector of AB, BC ad CA are cocurret. Page 5

6 68. AB ad AC are two equal chord of a circle. Prove th the biector of the agle BAC pae through the cetre of the circle. 69. I the below figure, if OAB = 400, the fid ACB 70. I the above right ided figure, if DAB = 600, ABD = 500 the fid ACB. 71. I the below figure, BC i a diameter of the circle ad BAO = 600 the fid ADC I above right ided figure, AOB = 900 ad ABC = 300, the fid CAO 73. he legth of two parallel chord of a circle are 6 cm ad 8 cm. If the maller chord i ditace 4 cm from the cetre, wh i the ditace of the other chord from the cetre? 74. A, B, C D are four coecutive poit o a circle h th AB = CD. Prove th AC = BD. 75. If a lie egmet joiig mid-poit of two chord of a circle pae through the cetre of the circle, prove th the two chord are parallel. 76. ABCD i h a quadrileral th A i the cetre of the circle paig through B, C ad D. Prove 1 th CBD + CDB = BAD 77. O i the circumcetre of the triagle ABC ad D i the mid-poit of the bae BC. Prove th BOD = A. 78. O a commo hypoteue AB, two right triagle ACB ad ADB are itued o oppoite ide. Prove th BAC = BDC. Page 6

7 1 arc(byc). Fid BOC 79. I the below figure, AOC i a diameter of the circle ad arc(axb) = 80. I the above right ided figure, ABC = 450, prove th OA OC. 81. wo chord AB ad AC of a circle ubted agle equal to 90 ad 150, repectively the cetre. Fid BAC, if AB ad AC lie o the oppoite ide of the cetre. 8. If BM ad CN are the perpedicular draw o the ide AC ad AB of the triagle ABC, prove th the poit B, C, M ad N are cocyclic. 83. If a lie i draw parallel to the bae of a cele triagle to iterect it equal ide, prove th the quadrileral o formed i cyclic. 84. If a pair of oppoite ide of a cyclic quadrileral are equal, prove th it diagoal are alo equal. 85. he circumcetre of the triagle ABC i O. Prove th OBC + BAC = A chord of a circle i equal to it radiu. Fid the agle ubteded by thi chord a poit i major egmet. 87. I the below figure, ADC = 130 ad chord BC = chord BE. Fid CBE. 88. I the above right ided figure, ACB = 400. Fid OAB. 89. A quadrileral ABCD i icribed i a circle h th AB i a diameter ad ADC = Fid BAC. 90. wo circle with cetre O ad O iterect two poit A ad B. A lie PQ i draw parallel to OO through A(or B) iterectig the circle P ad Q. Prove th PQ = OO Page 7

8 91. I the below figure, AOB i a diameter of the circle ad C, D, E are ay three poit o the emicircle. Fid the value of ACD + BED. 9. I the above right ided figure, AB = 300 ad OCB = 570. Fid BOC ad AOC. 93. I the below figure, O i the cetre of the circle, BCO = 300, fid x ad y. 94. I the above right ided figure, O i the cetre of the circle BD = OD ad CD AB. Fid CAB. 95. Let the vertex of a agle ABC be loced outide a circle ad let the ide of the agle iterect equal chord AD ad CE with the circle. Prove th ABC i equal to half the differece of the agle ubteded by the chord AC ad DE the cetre. Page 8

21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle.

21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle. 22. Prove that If two sides of a cyclic quadrilateral are parallel, then