CBSE Class 9 Circles Sure Shot Questions

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Study Material for Class 9 Mathematics Chapter 10 Circles

Class 9 Mathematics students should refer to the following Pdf for Chapter 10 Circles in Class 9. These notes and test paper with questions and answers for Class 9 Mathematics will be very useful for exams and help you to score good marks

Class 9 Mathematics Chapter 10 Circles

 

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.”

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 the remaining two sides are equal and the diagonals are also equal.”
 
23. Prove that “If two opposite sides of cyclic quadrilateral are equal, then the other two sides are parallel.”
 
24. Prove that “If two non parallel sides of a trapezium are equal, it is cyclic.”
 
25. Prove that “The sum of the angles in the four segments exterior to a cyclic quadrilateral is equal to 6 right angles.”
 
26. Two circles with centres A and B intersect at C and D. Prove that ∠ACB = ∠ADB.
 
27. Bisector AD of AC of ∠ABC passes through the centre of the circumcircle of ∠ABC. Prove that AB = AC.
 
28. In the below figure A, B and C are three points on a circle such that angles subtended by the chords AB and AC at the centre O are 800 and 1200 respectively. Determine ∠BAC.
 CBSE Class 9 Circles Sure Shot Questions
29. In the above right-sided figure, P is the centre of the circle. Prove that ∠XPZ = 2 (∠XZY + ∠YXZ).
 
30. Prove that the midpoint of the hypotenuse of a right triangle is equidistant from its vertices.
 
31. In the below figure ABCD is a cyclic quadrilateral, O is the centre of the circle. If ∠BOD = 1600, find ∠BPD.
 CBSE Class 9 Circles Sure Shot Questions
32. Prove that in a triangle if the bisector of any angle and the perpendicular bisector of its opposite side intersect, they will intersect on the circumcircle of the triangle.
 
33. The diagonals of a cyclic quadrilateral are at right angles. Prove that perpendiculars from the point of their intersection on any side when produced backward bisect the opposite side.
 
34. If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.
 
35. If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.
 
36. 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.
 
37. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
 
38. 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.
 
39. In the below figure, AB is a diameter of the circle, CD is a chord equal to the radius of the circle. AC and BD when extended intersect at a point E. Prove that ∠AEB = 60°.
 CBSE Class 9 Circles Sure Shot Questions
40. In the above right-sided figure, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If ∠DBC = 55° and ∠BAC = 45°, find ∠BCD.
 
41. Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic.
 
42. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD.
 
43. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
 
44. Two circles intersect at two points A and B. AD and AC are diameters to the two circles. Prove that B lies on the line segment DC.
 
45. Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic.
 
46. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
 
47. Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively. Prove that ∠ACP = ∠QCD.
 CBSE Class 9 Circles Sure Shot Questions
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.
 CBSE Class 9 Circles Sure Shot Questions
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.
 CBSE Class 9 Circles Sure Shot Questions
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.
 CBSE Class 9 Circles Sure Shot Questions
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
 CBSE Class 9 Circles Sure Shot Questions
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
 CBSE Class 9 Circles Sure Shot Questions
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
 CBSE Class 9 Circles Sure Shot Questions
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.
 CBSE Class 9 Circles Sure Shot Questions
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.
 CBSE Class 9 Circles Sure Shot Questions
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.
CBSE Class 9 Circles Sure Shot Questions
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.

Chapter 01 Number Systems
CBSE Class 9 Number Systems Sure Shot Questions
Chapter 02 Polynomials
CBSE Class 9 Polynomials Sure Shot Questions
Chapter 03 Coordinate Geometry
CBSE Class 9 Coordinate Geometry Sure Shot Questions
Chapter 04 Linear Equations In Two Variables
CBSE Class 9 Linear Equations in Two Variables Sure Shot Questions
Chapter 05 Introduction to Euclids Geometry
CBSE Class 9 Introduction to Euclids Geometry Sure Shot Questions
Chapter 06 Lines and Angles
CBSE Class 9 Lines and Angles Sure Shot Questions
Chapter 07 Triangles
CBSE Class 9 Triangles Sure Shot Questions
Chapter 08 Quadrilaterals
CBSE Class 9 Quadrilaterals Sure Shot Questions
Chapter 09 Areas of Parallelograms and Triangles
CBSE Class 9 Areas of Parallelogram and Triangle Sure Shot Questions
Chapter 10 Circles
CBSE Class 9 Circles Sure Shot Questions
Chapter 11 Constructions
CBSE Class 9 Constructions Sure Shot Questions
Chapter 12 Herons Formula
CBSE Class 9 Herons Formula Sure Shot Questions
Chapter 13 Surface Areas and Volumes
CBSE Class 9 Surface Areas and Volumes Sure Shot Questions
Chapter 14 Statistics
CBSE Class 9 Statistics Sure Shot Questions
Chapter 15 Probability
CBSE Class 9 Probability Sure Shot Questions

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CBSE Class 9 Mathematics Chapter 10 Circles Study Material

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