CBSE Class 9 Circles Sure Shot Questions

CBSE Class 9 Circles Sure Shot Questions. There are many more useful educational material which the students can download in pdf format and use them for studies. Study material like concept maps, important and sure shot question banks, quick to learn flash cards, flow charts, mind maps, teacher notes, important formulas, past examinations question bank, important concepts taught by teachers. Students can download these useful educational material free and use them to get better marks in examinations.  Also refer to other worksheets for the same chapter and other subjects too. Use them for better understanding of the subjects.

1. Prove that “Equal chords of a circle subtend equal angles at the centre”.

2. Prove that “Chords of a circle which subtends equal angles at the centre are equal”.

3. Prove that “The perpendicular from the centre of a circle to a chord bisects the chord.”

4. Prove that “The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord”.

5. Prove that “Chords equidistant from the centre of a circle are equal in length”

6. Prove that “Chords of a circle which are equidistant from the centre are equal”

7. Prove that “Of any two chords of a circle then the one which is larger is nearer to the centre.”

8. Prove that “Of any two chords of a circle then the one which is nearer to the centre is larger.”

9. Prove that “line joining the midpoints of two equal chords of circle subtends equal angles with the chord.”

10. Prove that “if two chords of a circle bisect each other they must be diameters.

11. If two chords of a circle are equally inclined to the diameter through their point of intersection, prove that the chords are equal.

12. Prove that “The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.”

13. Prove that “Angles in the same segment of a circle are equal.”

14. Prove that “Angle in a semicircle is a right angle.”

15. Prove that “Arc of a circle subtending a right angle at any point of the circle in its alternate segment is a semicircle.”

16. Prove that “Any angle subtended by a minor arc in the alternate segment is acute and any angle subtended by a major arc in the alternate segment is obtuse.”

17. Prove that “If a line segment joining two points subtends equal angles at two other points lying on the same side of the line segment, the four points are concyclic.”

18. Prove that “Circle drawn on any one side of the equal sides of an isosceles trainlge as diameter bisects the side.”

19. Prove that “The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.”

20. Prove that “If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.”

 

 

 

 

48. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

 

49. Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals.

 

50. In the adjoining figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC = 130° and ∠ECD = 20°. Find ∠BAC.

 

51. In the above right-sided figure, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.

 

52. ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD.

 

53. AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD arediam eters, (ii) ABCD is a rectangle.

 

54. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

 

55. Prove that the circle drawn with any side of a rhombus as a diameter, passes through the point of its diagonals.

 

56. Bisectors of angles A, B and C of a triangles ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of DDEF are 900 –A/2, 900-B/2 and 900-C/2

 

57. Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

58. In the adjoining Fig., ∠ABC = 69°, ∠ACB = 31°, find ∠BDC.

 

59. In the above right-sided figure, A,B and C are three points on a circle with centre O such that ∠BOC = 30° and ∠AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.

 

60. In the below figure, AB and CD are two equal chords of a circle with centre O. OP and OQ are perpendiculars on chords AB and CD, respectively. If ∠POQ = 150؛, then find ∠APQ.

 

61. In the above right sided figure, if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, thenfind  CD.

 

62. Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.

 

63. Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ.

 

64. In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC.

 

65. If arcs AXB and CYD of a circle are congruent, find the ratio of AB and CD.

 

66. If the perpendicular bisector of a chord AB of a circle PXAQBY intersects the circle at P and Q, prove that arc PXA ≈Arc PYB.

 

67. A, B and C are three points on a circle. Prove that the perpendicular bisectors of AB, BC and CA are concurrent.

68. AB and AC are two equal chords of a circle. Prove that the bisector of the angle BAC passes through the centre of the circle.

 

69. In the below figure, if ∠OAB = 400, then find ∠ACB

 

70. In the above right sided figure, if ∠DAB = 600, ∠ABD = 500 then find  ∠ACB.

 

71. In the below figure, BC is a diameter of the circle and ∠BAO = 60⁰ then find ∠ADC In above right sided figure, ∠AOB = 900 and ∠ABC = 300, then find ∠CAO

 

73. The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre?

 

74. A, B, C D are four consecutive points on a circle such that AB = CD. Prove that AC = BD.

 

75. If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel.

 

76. ABCD is such a quadrilateral that A is the centre of the circle passing through B, C and D. Prove that ∠CBD + ∠CDB = 1/2 ∠BAD

 

77. O is the circumcentre of the triangle ABC and D is the mid-point of the base BC. Prove that ∠BOD = ∠A.

 

78. On a common hypotenuse AB, two right triangles ACB and ADB are situated on opposite sides. Prove that ∠BAC = ∠BDC.

79. In the below figure, AOC is a diameter of the circle and arc(AXB) = 1/2 arc(BYC). Find ∠BOC

 

80. In the above right sided figure, ∠ABC = 450, prove that OA ⊥BOC.

 

81. Two chords AB and AC of a circle subtends angles equal to 90؛ and 150؛, respectively at the centre. Find ∠BAC, if AB and AC lie on the opposite sides of the centre.

 

82. If BM and CN are the perpendiculars drawn on the sides AC and AB of the triangle ABC, prove that the points B, C, M and N are concyclic.

 

83. If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides, prove that the quadrilateral so formed is cyclic.

 

84. If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are also equal.

 

85. The circumcentre of the triangle ABC is O. Prove that ∠OBC + ∠BAC = 900.

 

86. A chord of a circle is equal to its radius. Find the angle subtended by this chord at a point in major segment.

 

87. In the below figure, ∠ADC = 130° and chord BC = chord BE. Find ∠CBE.

 

88. In the above right sided figure, ∠ACB = 400. Find ∠OAB.

 

89. A quadrilateral ABCD is inscribed in a circle such that AB is a diameter and ∠ADC = 1300. Find ∠BAC.

 

90. Two circles with centres O and O’ intersect at two points A and B. A line PQ is drawn parallel to OO’ through A(or B) intersecting the circles at P and Q. Prove that PQ = 2 OO’

 

91. In the below figure, AOB is a diameter of the circle and C, D, E are any three points on the semicircle. Find the value of ∠ACD + ∠BED.

 

92. In the above right sided figure, ∠OAB = 300 and ∠OCB = 570. Find ∠BOC and ∠AOC.

 

93. In the below figure, O is the centre of the circle, ∠BCO = 300, find x and y.

94. In the above right sided figure, O is the centre of the circle BD = OD and CD ⊥ AB. Find ∠CAB.

 

95. Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre.

 

Please click the link below to download CBSE Class 9 Circles Sure Shot Questions.