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1.

In the diagram, ABC is a tangent to the circle centre O. Given that

55BFE ,

70FED and

72GOB , calculate

(i)

OBG ,[1]

(ii)

GDB [1]

(iii)

EDG [1]

(iv)

GBA[2]

(v) the sum of interior angles of the pentagonBDEFG. [2]

2. In the diagram, the points A, B, Cand D lies on a circle of centre O. AC is a

diameter which intersects BD atEand the tangent to the circle at Cmeets AD

produced atF.

Given that BCA = 56 and DAC = 36, find

(i) CAB , [2]

(ii) DEA , [2]

(iii) FCD . [2]

5570

72

A B C

D

E

G

F

O

A

B

O

E

D

F

36

56C

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3.

O is the centre of the circlePQRS. The straight lineAR cuts the circle at Q and C.

APis a tangent to the circle atP.

Given that POQ = 74and PAQ = ,36 calculate, stating your reasons clearly

(a) QRP, [1]

(b) QSP, [1]

(c) QPA, [2]

(d) PSR, [2]

(e) RPO. [2]

O

A

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4. In the figure, O is the centre of the circle.AB and CB are tangents to the circle atFand G. Given

that ABC= 40, FEA = 66 andDE=EF, calculate, giving reasons

(i) GFB, [1]

(ii) FGD, [1]

(iii) obtuse angle

FOD, [1](iv) GDO, [2]

(v) EFA [3]

5. In the diagram below,ACis the diameter of the circle, centre O. The chordsAEandBC

meet atD when produced. Given that CDE= 22 o and BAC= 39 o, calculate

(i) ACB, [2]

(ii) AOB, [1]

(iii) CBE. [3]

Diagram is not drawn to scale

66

40G

F

E

D

C

B

A

O

B

D

A

E

O

C

39 o

22 o

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6. In the diagram below,A,B, C,D andEare points on the circle with centerO.DCis parallel to the diameter

EB of the circle with centerO.

(a) Given that OAC = 45, ACB = 30, DEF is a straight line and FAH is a tangent to the circle,

calculate,

(i) AOE, [2]

(ii) BAC, [2]

(iii) AEF. [2]

(b) Prove that AEFis similar to CAE. [2]

C

B

A

D

E

F

O

30

45

G

H

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7. The pointsP, Q,R, Sand Tlie on a circle, centre O.PSis a diameter of the circle.

QVis a tangent to the circle and 24PTQ .

Calculate, stating your reasons clearly, the following angles,

(a) (i) QSP , [1]

(ii) QTS , [2]

(iii) QRS , [1]

(iv) QVP , [3]

(b) Given that 114QPT , determine whether the lines QO andPTare

parallel. Give a reason for your answer. [2]

O

P

S

T

RQ

V

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8. In the diagram, O is the centre of the circlePQRS. SR is parallel toPO. TPA is a tangent to the

circle and SPT= 37 .

Find

(i) OPS, [1]

(ii) PSR, [1]

(iii) obtuse POR, [2]

(iv) PQR, [1]

(v) ORP, [1]

(vi) SQR. [2]

T

A

O

R

P

S

Q

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9. It is given that O is the centre of the circle.PCQ and RBQ are tangents to the circle at CandB

respectively.

It is also given that o64BCQ .

(a) Explain why OCB = 26 [1]

(b) Calculate, stating your reasons clearly,

(i) CAB, [2]

(ii) CQB, [2]

(i) COB, [1]

(ii) ABR, [1]

(v) DBC. [1]

(c) Prove that OBQCis a cyclic quadrilateral. [2]

B

O

CP

64o

Q

A

R

D

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10. In the figure, O is the centre of the circle. The tangentPATmeets the line CD produced at T. It is

given that ABC= 82 and TAD = 44.

(a) Name one triangle in the diagram which is isosceles (Give reasons for your answers.

[2]

(b) Calculate

(i) OAD, [1]

(ii) AOD, [1]

(iii) ACD, [1]

(iv) ODC, [2]

(v) ATD. [2]

A

B

C

D

P

TO82

44

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11. In the diagram, the pointsA,B, C,FandD are on the circle with centre O.

CED is a straight line, AEC= 100o, ADC= 2y

o, ABC=xo and

DAE= 3yox

o .

(a) Give a reason why

(i) x + 2y = 180, [1]

(ii) 3yx + 2y = 100. [1]

(b) Find the value ofx and ofy. [2]

(c) Hence, calculate, giving the reasons,

(i) AOC, [1]

(ii) AFC, [1]

(iii) ACD. [2]

D

A

B

C

FO

E

x

2y

100

3yx

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12 In the figure, O is the centre of the circle through A,B, CandD. TA is a tangent to the circle

atA andACintersectsBD atX. ACB = 620 and DAC= 320.

(a) Explain why OAT= 90. [1]

(b) Find

(i) BAO, [2]

(ii) AOD, [1]

(iii) BXC, [1]

(iv) TAD, [1]

(v) DCB. [1]

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13. The points A, B, C, D and F lie on a circle. The line GE meets the circle at a tangent at point F.

Given that BAC = 26o, CAD = 35

o, AFC = 55o and CFE = 61

o,

a) Calculate

i) ABC [1]

ii) ADC [1]

iii) DAF [2]

iv) AHC [2]

b) Show that AD is the diameter of the circle [2]

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14.

The pointsA,B, C,D andElie on a circle, centre O. AB is a diameterof the circle. TG is a tangent to the circle atE. 30EDA and 62DEO .

(a) Find, giving your reasons,

(i) EOA , [1]

(ii) EAD , [2]

(iii) GED , [2]

(iv) DCB , [2]

(b) Given that the radius of the circle is 6 cm, find

(i) the length ofET

, [2](ii) the area of the shaded region. [3]

T

B

D

C

62

30

E

OG

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15. In the diagram, O is the centre of the small circleABD.ABCandADPare straight lines andBP

and CD intersect at point Q.

(a) Given that DAB 78, showing all the necessary reasons, find

(i) BOD, [1]

(i) ACD, [1]

(ii) ADC, [1]

(iii) PQC. [3]

(b) Prove, stating your reasons clearly, that BCQ and DPQ are similar. [2]

16. In the diagram, RS is the diameter of the circle RSTQ, PQR and PTS are straightlines. US is the tangent to the circle at S. Given that SRT = 21 and QRT = 43, find

(i) STQ, [1]

(ii) PSU, [1]

(iii) QPT. [2]

All reasons must be clearly stated.

O

B

C

D

P

Q

78

P

Q

R

T

SU

43

21

O

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17. In the diagram,BOHis the diameter of the circle with centre O.

ABCandEDCare tangents to the circle atB andD respectively.

It is given that FDG = 21, BFD = 64 and GFD = 78.

Stating your reasons clearly, find

(a)

BOD, [2](b) FDB, [2]

(c) HDG, [2]

(d) BCD. [2]

A

B

C

D

E

H

O

F

G

21

64

78

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18. The pointsA,B, C,D andElie on a circle with centre O where EDAE .ACis the diameter of the circle and GF is tangent to the circle at C.

AngleAED = 140 and angleBCG = 10

Find

(a) OCD, [1]

(b) ABC, [1]

(c) FCD, [1]

(d) BAC, [1]

(e) AOD, [1]

(f) CDE. [2]

19. In the diagram, O is the centre of the circle throughA, B, C, D andE.MNis the tangent atA.

The diameterEB and the chord DA intersect at X. Given that 050

BOA and , 052BED

calculate, giving your reasons clearly, the following angles:

(i) BEA

, [2]

(ii) BCA [2]

(iii) EAO , [2]

(iv) BCD [2]

A

E

C

B

DF

G

O

O

DE

A

C

520

50X

B

MN

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20. Given thatAB andAFare tangents to the circle atB andFrespectively. O is the centre of the

circle. EOB is the diameter of the circle, OGA is a straight line. o60 FOE and 15BEC .

(a) Calculate, stating your reasons clearly,

(i) FDE , [1]

(ii) EBC , [2]

(iii) FDC , [2]

(b) Show that triangle ABFis equilateral. [3]

F

O

B

A

C

D

E

1560

G

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21.

The pointsA,B, C, Sand Q lie on a circle, centre O.

ASis a diameter of the circle and BO is parallel toRT.

RTis a tangent to the circle andAQTis a straight line.

Angle ORS= 45 .

(a) Find the angles, showing your reasons clearly,

(i) BOC [1]

(ii) BAC [1]

(iii) OCA [2]

(b) Given also that angle QST= 42 , find the angles

(i) QAS [1]

(ii) QSC [2]

(c) Show that triangles ABO and ORSare similar. [1]

R

A

B

C

S

T

Q

O