geometry circle problems

Math eNrichment and Olympiad Preparation Geometry-Circles

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1. in the figure, there are nine circles, each of radius 1. Find the shaded area.

a. 9 9π− b. 36 9π− c. 40 9π− d. 10 10π− e. 40 10π−

2. In the figure, TQ is the tangent to the tangent to the circle at A. If are AC = are BC and 48 ,PAQ°∠ =

then QAC∠ =

a. 42° b. 48

° c. 66

° d. 71

° e. 84

°

3. In the figure, O is the centre of the circle. If OR PQ� and 42 ,ROQ°∠ = find .RMQ∠

a. 21° b. 42

° c. 63

° d. 84

° e. 126

°




4. In the figure, an equilateral triangle is inscribed in a circle of radius 1. The circumference of the circle is

greater than the perimeter of the triangle by

a. 4 3 3π − b. 3 3

42

π − c. 2 3π − d. 3 3

22

π − e. 2 3 3π −

5. Three equal circles of radii 1 touch each other as shown in the figure, shaded area =

a. 12

π− b. 3

2

π− c. 2 3

2

π− d. 3

6

π− e. 2 3

6

π−

6. In the figure, AB is a diameter and 30BAC°∠ = . If the area of ABC∆ is 3, then the radius of the

circle is

a. 1

2 b. 1 c. 2 d. 3 e. 2

7. In the figure, PA and PC are tangents to the circle ABC. If 48P°∠ = , then ABC∠ =

a. 84° b. 96

° c. 106

° d. 114

° e. 132

°




8. In the figure, TA and TB are tangents to the circle ABC. If TA TB⊥ and BD AC⊥ , find .CBD∠

a. 30° b. 40

° c. 45

° d. 50

° e. 60

°

9. In the figure AB, AC and BC are three tangents touching the circle at D, E and F respectively.

If AC = 24, BC = 18 and 90 ,ACB°∠ = find the radius of the circle.

a. 3 b. 4 c. 5 d. 6 e. 7

10. In the figure, TB touches the semi-circle at B. TA cuts the semi-circle at P such that TP = PA.

If the radius of the semi-circle is 2, find the area of the shaded region.

a. 12 π− b. 8 π− c. 6 π− d. 4 π− e. ( )2 4 π−

11. In the figure, XPY and YQZ are semi-circles with areas A1 and A2 respectively. 60XYZ°∠ = and

45 .YZX°∠ = The ratio A1 : A2 =




a. 2 : 3 b. 2 :3 c. 2 : 3 d. 2 : 3 e. 3 : 2

12. In the figure, O is the centre of the circle. Find a + c.

a. b b. 2b c. 180 b° − d. 360 b

° − e. 360 2b° −

13. In the figure, O is the centre of the circle BCD. ABC and EDC are straight lines. BC = DC and

70 .AED°∠ = Find .BOD∠

a. 40° b. 70

° c. 80

° d. 90

° e. 140

°

14. In the figure, TPA and TQB are tangents to the circle at P and Q respectively. If PQ = PR, which of the

following must be true?

I. APR QRP∠ = ∠

II. QTP QPR∠ = ∠

III. QPR APR∠ = ∠ _QPR = _APR

a. I only b. II only c. III only d. I and II only e. I and III only

15. From a rectangular metal sheet of width 3 cm and length 40 cm, at most how many circles each of

radius 1 cm can be cut?

a. 20 b. 21 c. 22 d. 23 e.. 24




16. In the figure, arc AB : arc BC : arc CD : arc DE : arc EA = 1 : 2 : 3 : 4 : 5. Find.

a. 30° b. 36

° c. 60

° d. 72

° e. 120

°

17. In the figure, TP and TQ are tangent to the circle of radius 3cm. Find the length of the minor arc PQ.

a. 3 cmπ b. 2 cmπ c. 3

2cm

π d. cmπ e.

2cm

π

18. In the figure, the equilateral triangle ACE of side 4 cm is inscribed in the circle. Find the area of the

inscribed regular hexagon ABCDEF.

a. 2

8 3 cm b. 2

8 2 cm c. 2

4 3 cm d. 2

4 2 cm e. 2

16cm

19. In the figure, O is the center of the circle. find θ .

a. 42º b. 36º c. 24º d. 21º e. 18º




20. In the figure, the circle is inscribed in a regular pentagon. P, Q and R are points of contact. Find θ .

a. 30º b. 32º c. 35º d. 36º e. 45º

21. In the figure, ST is a tangent to the smaller circle. ABC is a straight line. If 2TAD x∠ = and

2 ,DPC x∠ = find x .

a. 30º b. 36º c. 40º d. 42º e. 45º

22. In the figure, the two circles touch each other at C. The diameter AB of the bigger circle is tangent to

the smaller circle at D. If DE bisects ,ADC∠ findθ .

a. 24º b. 38º c. 45º d. 52º e. 66º

23. In the figure, O is the center of the circle. If the diameter AOB rotates about O, which of the following

is/are constant?

I. θ φ+ II. AC + BD III. AC × BD

a. I only b. II only c. III only d. I and II only e. I and III only




24. In the figure, AB is a diameter. Find .ADC∠

a. 100º b. 110º c. 120º d. 135º e. 140º

25. In the figure, if arcBC : arcCA : arcAB = 1: 2 : 3, which of the following is/are true?

I. 1: 2 : 3A B C∠ + ∠ + ∠ = II. a : b : c = 1: 2 : 3 III. sin A : sin B : sinC = 1: 2 : 3

a. I only b. II only c. III only d. I and II only e. I, II and III

26. In the figure, TP and TQ are tangents to the circle at P and Q respectively. If M is a point on the minor

arc PQ and ,PMQ θ∠ = PTQ∠ =

a. 2

θ b. 90θ °− c. 180 θ° − d. 180 2θ° − e. 2 180θ °−

27. In the figure, O is the center of the circle. AB touches the circle at N. Which of the following is/are

correct?

I. M, N, K, O are concyclic. II. HNB NKB∆ ∆∼ III. ONAN NOB∠ = ∠

a. I only b. II only c. III only d. I and II only e. I, II and III




28. In the figure, the three circles touch one another. XY is their common tangent. The two larger circles

are equal. If the radius of the smaller circle is 4 cm, find the radii of the larger circles.

a. 8 cm b. 10 cm c. 12 cm d. 14 cm e. 16 cm

29. In the figure, ABCD is a cyclic quadrilateral with AB = 5, BC = 2 and 120 .ADC°∠ = Find AC.

a. 19 b. 21 c. 2 6 d. 34 e. 39

30. In the figure, O is the center of the circle. If AC = 3 and ,6

BACπ

∠ = find the diameter AB.

a. 3

2 b. 6 c.

3 3

2 d. 2 3 e. 3 3

31. In the figure, PA is tangent to the circle at A, 28CAP°∠ = and BA = BC. Find x.

a. 28º b. 48º c. 56º d. 62º e. 76º




32. In the figure, O is the center of the inscribed circle of .ABC∆ If 30OAC°∠ = and 25OAC

°∠ = , find

.ABC∠

a. 50º b. 55º c. 60º d. 62.5º e. 70º

33. In the figure, CDEF is a sector of a circle which touched AB at E. If AB = 25 and BC = 15, find the

radius of the sector.

a. 9 b. 10 c. 11.25 d. 12 e. 12.5

34. In the figure, ABCD is a semi-circle, CDE and BAE are straight lines. If 30CBD°∠ = and

22 ,DEA°∠ = find x.

a. 38º b. 41º c. 44º d. 52º e. 60º

35. In the figure, OABCD is a sector of a circle. If �AB = �BC = �,CD then x =

a. 105º b. 120º c. 135º d. 144º e. 150º

36. In the figure, ABCD is a semicircle. CAD∠ =

a. 25º b. 40º c. 45º d. 50º e. 65º

37. In the figure, O is the center of the circle, POQR is a straight line. TR is the tangent to the circle at T.

PRT∠ =




a. 20º b. 35º c. 45º d. 50º e. 70º

38. In the figure, O is the center of the circle. Find the area of the major segment ABC.

a. 2

4r

π b.

23

4r

π c.

21

4 2r

π −

d.

23 1

4 2r

π −

e.

23 1

4 2r

π +

39. In the figure, C1 and C2 are two circles. If area of region I : area of region II : area of region III = 2 :

1 : 3, then radius of C1 : radius C2 =

a. 9 : 16 b. 2 : 3 c. 3 : 4 d. 2 : 3 e. 3: 2




40. In the figure, O is the centre of the circle. AB and AC are tangents to the circle at B and C

respectively. Area of the shaded region =

a. 2

26

cmπ

−

b. 2

23

cmπ

−

c. 2

36

cmπ

−

d. 2

33

cmπ

−

e.

23

2 6cm

π −

41. In the figure, O is the centre of the circle. Find x.

a. 20° b. 27.5° c. 35° d. 37.5° e. 40°

42. In the figure, O is the centre of the circle. PA is the tangent to the circle at A and CB // PA. Find x.

a. 21° b. 24° c. 42° d. 45° e. 48°

43. In the figure, O is the centre of the circle. AP, AB and BR are tangents to the circle at P, Q and R

respectively. Which of the following must be true?

I. AP + BR = AB

II. OQ bisects AOB∠

III. 1

2AOB POR∠ = ∠

a. I only b. II only c. I and II only d. I and III only e. I, II and III




44. In the figure, BEA is a semicircle. ABCD is a rectangle and DC touches the semicircle at E. Find the

area of the shaded region.

a. 9π b. 18π c. 36π d. 36 9π− e. 36 9π+

45. In the figure, BCA is a semicircle. If AC = 6 and CB = 4, find the area of the semicircle.

a. 5

2π b.

13

2π c. 10π d. 13π e. 26π

46. In the figure, EC is the tangent to the circle at C. Find .CBD∠

a. 40° b. 50° c. 65° d. 70° e. 75°

47. In the figure, OXY is a sector with centre O. If Z is the mid point of YO, find area of triangle OXZ∆ :

area of sector .OXY∆

a. 1 : 2 b. 2 : 3π c. 2 :3π d. 3: 2π e. 3 3 : 2π




48. In the figure, AC is the angle bisector of .BAD∠ Which of the following statements must be true?

I. BCE ADE∆ ∆∼ II. ABC AED∆ ∆∼ III. ABC BDA∆ ∆∼

a. I only b. I and II only c. I and III only d. II and III only e. I, II and III

49. In the figure, � � �2, 3, 4AB BC CD= = = and � 6,DA = Find .BCD∠

a. 72 ° b. 84° c. 90° d. 96° e. 144°

50. In the figure, the radii of the two circles are 3 cm and 1 cm respectively. Find the ratio of the area of

the shaded part to that of the smaller circle.

a. 2 : 1 b. 3 : 1 c. 4 : 1 d. 8 : 1 e. 9 : 1

51. In the figure, OABC is a sector. Find the area of the shaded region.

a. ( ) 22 cmπ − b. ( ) 2

2 4 cmπ − c. ( ) 24 8 cmπ − d. ( ) 2

8 8 cmπ − e. ( ) 28 16 cmπ −




52. In the figure, AB is a diameter of the circle and ABD is a straight line. CBD∠ =

a. 2θ b. 4θ c. 90 θ° + d. 180 θ° − e. 180 2θ° −

53. In the figure, AD is a diameter of the circle. If � � �: : 3:5 :7,AB BC CD = ADC∆ =

a. 36° b. 45° c. 48° d. 49° e. 72°

54. In the figure, ABC is a semicircle. Find the area of the shaded part.

a. 2

6 cmπ b. 2

15 cmπ c. ( ) 26 9 3 cmπ − d. ( ) 2

6 9 3 cmπ + e. ( ) 212 9 3 cmπ −

55. In the figure, CE is tangent to the circle at C. Find .DCE∠

a. 40° b. 42° c. 49° d. 54° e. 78°

56. In the figure, a square is inscribed in a circle with radius 1 cm. Find the area of the shaded region.

a. ( ) 22 cmπ − b. ( ) 2

2 cmπ − c. ( ) 21 cmπ − d. ( ) 2

2 2 cmπ − e.

( ) 22 1 cmπ −




57. In the figure, ABCD is a semicircle. Find the area of the shaded region correct to the nearest 0.01

2cm .

a. 5.332

cm b. 2.872

cm c. 2.672

cm d. 1.332

cm e. 0.172

cm

58. In the figure, O is the centre of the circle. Find x.

a. 12° b. 20° c. 24° d. 40° e. 60°

59. In the figure, AB is a diameter of the circle. Find x.

a. 26° b. 32° c. 38° d. 52° e. 64°

60. In the figure, AT is tangent to the circle at T and ABC is a straight line. Find AT.

a. 9 cm b. 12 cm c. 15 cm d. 16 cm e. 20 cm

61. In the figure, O is the center of the circle. EAOB and EDC are straight lines. Find x.




a. 40° b. 46° c. 57° d. 66° e. 68°

62. In the figure, PXQ, QYR and RZP are semicircles with areas A1 2

cm , A2 2

cm and A3 2

cm respectively.

If 1

12A = and 2

5,A = find 3

A

a. 13 b. 17 c. 169 d. 13π e. 169

8π

63. In the figure, CAB is a semicircle and ABCD is a parallelogram. Find the area of ABCD.

a. 65 2

cm b. 60 2

cm c. 52 2

cm d. 32.52

cm e. 30 2

cm

64. In the figure, AB is tangent to the circle at B. Find .DCE∠

a. 70° b. 75° c. 90° d. 95° e. 105°




65. In the figure, � � �: : 2 :1:3.AB BC CD = Find .ADC∠

a. 56° b. 60° c. 63° d. 72° e. 84°

66. In the figure, AEC is a diameter and DEB is a straight line. Find x.

a. 54° b. 70° c. 74° d. 92° e. 94°

67. In the figure, OABC is a sector. Find the length of the arc ABC.

a. 2

3cm

π b. 4 cmπ c. 5 cmπ d. 6 cmπ e. 12 cmπ

68. In the figure, A, B and C are the centres of three equal circles, each of radius 1 cm. Find the area of

the shaded region.

a. 23

2 2cm

π −

b.

23 3

2 4cm

π −

c.

23

2 4cm

π +

d.

2

2cm

π e.

23

2 4cm

π −




69. In the figure, ABCD is a semicircle, AB : BD = 4 : 3 . Find AB correct to the nearest 0.1 cm.

a. 5.7 cm b. 7.6 cm c. 10.7 cm d. 13.0 cm e. 14.3 cm

70. In the figure, O is the center of the circle, AOB is a straight line and BCD is the tangent to the circle at

C. Find x.

a.50° b. 53° c. 56° d. 59° e. 62°

71. In the figure, � � �1.

2AB BC CD= = Find .ABC∠

a.100° b. 105° c. 112.5° d. 130° e. 150°

72. The figure shows a rectangular inscribed in a circle. Find the area of the shaded region correct to the

nearest 0.1 2.cm

a. 60.0 2

cm b. 72.7 2

cm c. 132.7 2

cm d. 470.9 2

cm




73. In the figure, OCD and OAB are two sectors. The length of �AB is

a. 8

3cmπ b.

10

3cmπ c. ( )2 2 cmπ + d. 4 cmπ

74. In the figure, O is the center of the semicircle ABCD and BC // AD. If 42 ,COD°∠ = then x =

a. 48° b. 63° c. 84° d. 90°

75. In the figure, AED =1� 1AED = and � 4.CFD = If 100 ,ABC°∠ = then ABD∠ =

a. 18° b. 20° c. 24° d. 25°

76. In the figure, EAF is a common tangent to the circles at the point A. Chords AC and BC of the smaller

circle are produced to meet the larger circle at G and D respectively. Which of the following must be

true?




I. ADG EAG∠ = ∠

II. ABD AGD∠ = ∠

III. BAE ADB∠ = ∠

a. I only b. II only c. I and III only d. II and III only

77. In the figure, OAB is a sector and � .AB cmπ= Find the area of the sector.

a. 23

2cmπ b.

23 cmπ c.

29

2cmπ d.

26 cmπ

78. In the figure, ABC is a semicircle with � 7BC = and 55 ,ACB°∠ = Find �.AB

a. 9 b. 10 c. 11 d. 14

79. The figure shows a circle with diameter AD. If AB = BC = CD, find x + y + z.

a. 315° b. 324° c. 330° d. 360°

80. In the figure, XAB and XDC are straight lines. If DX = 5, AX = 6 and AB = 4, find CD.

a. 5 b. 7 c. 10

3 d.

24

5

81. In the figure, BE and BF are tangents to the circle at A and C respectively. If 100 ,ADC°∠ = then

ABC∠ =




a. 20° b. 30° c. 40° d. 50°

82. In the figure, O is the center of the circle ABCD. If EAB and EDOC are straight lines and EA = AO, find

.AEO∠

a. 18° b. 24° c. 27° d. 36°

83. In the figure, O is the center of the circle ABC. Find x.

a. 17.5° b. 27.5° c. 35° d. 55°

84. In the figure, ABCD is a circle. AC and BD meet at E. If AD = 4, AE = 2, EC = 5 and BE = 4, then BC=

a. 6 b. 7 c. 8 d. 10

85. In the figure, ABC is a circle. If 30ABC°∠ = and � 4,AC = then the circumference of the circle is




a. 24 b. 48 c. 8π d. 16π

86. In the figure, ABCD is a circle. If � � � �2 2 2 ,CD DA AB BC= = = then x =

a. 108° b. 112° c. 120° d. 144°

87. In the figure, TS, SQ and QP are tangents to the circle at T, R and P respectively. If TS // PQ, TS = 3

and QP =12, then the radius of the circle is

a. 4.5 b. 6 c. 7.5 d. 9

88. In the figure, OAB is a sector of radius 2 cm. If the length of �AB is 3π cm, then the area of the sector

OAB is

a. 23.

2cm

π b.

23 .cmπ c.

24 .cmπ d.

26 .cmπ

89. In the figure, ABCD is a circle. AB produced and DC produced meet at E. If AC and BD intersect at F,

then ABD∠ =




a. 41° b. 52° c. 56° d. 60°

90. In the figure, ABCD is a circle. If AC is a diameter of the circle and AB = BD, then CAD∠ =

a. 18° b. 21° c. 27° d. 36°

91. In the figure, AB and AC are tangents to the circle at X and Y respectively. Z is a point lying on the

circle. If 100 ,BAC°∠ = then XYZ∠ =

a. 40° b. 45° c. 50° d. 55°

92. In the figure, O is the centre of the circle and AOC is a straight line. If AB and BC are tangents to the

circle such that AB = 3, and BC = 4, then the radius of the circle is




a. 3

2 b.

12

7 c. 2 d.

5

2

93. In the figure, ABCD is a circle. If � � � �: : : 1: 2 : 3 : 3AB BC CD DA = and E is a point lying on BD, then

CAE∠ =

a. 45° b. 50° c. 55° d. 60°

94. In the figure, O is the center of the circle. B and C are points lying on the circle. If OC = 2 cm and OA

= 1 cm, then the area of the shaded region OABC is

a. 2

2cm

π b.

22

3cm

π c.

23

2 3cm

π +

d.

223

3cm

π +

95. In the figure, O is the centre of the circle ABC. If 50OBC°∠ = and 20ACO

°∠ = , then BOA∠ =

a. 50° b. 60° c. 70° d. 80°




96. In the figure, O is the centre of the circle. A and B are points lying on the circle. If AOC is a straight

line and BC is a tangent to the circle, then the radius of the circle is

a. 3

2 b. 3 c. 2 3 d. 3 3

97. In the figure, A, B, C and D are points lying on the circle. If AB = 5, AD = 3 and BD = 7, then

BCD∠ =

a. 60° b. 85° c. 95° d. 120°

98. In the figure, A, B and C are points lying on the circle. AB is a diameter of the circle. DB is the tangent

to the circle at B. If ACD is a straight line with AC = 4 and CD = 2, then AB =

a. 2 6 b. 4 3 c. 4 6 d. 8 3

99. In the figure, OAB and OCD are sectors with centre O. It is given that the area of the shaded region

ABCD is 2

54 .cmπ If AC = 6 cm, then OA =

a. 15 cm b. 21 cm c. 24 cm d. 30 cm




100. In the figure, O is the centre of the circle ABCD. If 84ADC°∠ = and 38 ,CBO

°∠ = then AOB∠ =

a. 64° b. 88° c. 104° d. 168°

101. In the figure, AB is the tangent to the circle at B and ADC is a straight line. If AB : AD = 2 : 1 , then

the area of ABD∆ : the area of BCD∆ =

a. 1 : 2 b. 1 : 3 c. 1 : 4 d. 2 : 3

geometry circle problems

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