Find the unknowns in the following figures. (1  6)
1.
2. ADE is a straight line.
A
A
D
x  10
4x
D
5x  7
B
3x  43
E
B
C
C
3. ABE and CDF are straight lines.
A
4.
D
87
A
x
B
x  16
E
2x  10
F
47
C
B
D
C
5. O is the centre of the circle,
ABE is a straight line.
D
O
6. AB is a diameter.
A
B
64
x
86
y
D
A
B
y
C
x
C
E
2.7
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7. In the figure, O is the centre of the circle and AB is a diameter.
Chords AD and BC are produced to meet at E. Find x, y and z.
A
D
y
42
z
E
x
O
19
C
B
8. In the figure, A, B, C and D are points on the circumferences of
two circles which intersect at E and F. AEGD and CGF are
straight lines. Find CGD.
C
A
E
G
110
D
43
80
B
F
9. In the figure, chords AD and BC are produced to meet at E,
chords AB and DC are produced to meet at F. It is given that
BFC  35 and DAB  2DEC . Find DEC.
E
D
A
C
35
B
F
10. In the figure, chords BC and CE are equal in length. D is a point
on the circumference. A is a point on the circumference such
that BAC  44 . Find CDE.
E
A
44
B
2.8
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D
C
11. In the figure, A, B and D are points on the circumference of the
larger circle. The circumference of the smaller circle passes
through A, D and the centre of the larger circle O. AB produced
intersects the circumference of the smaller circle at C and
DBC  40. Find BCD.
A
B
O
C
40
D
12. In the figure, ABCDE is a semi-circle, where AE is a diameter.
CBF is a straight line and ABF  48.
A
E
(a) Find CDE.
(b) Find CAE.
D
F
48
B
C
13. In the figure, A, B, C and D are points on the circumference,
ABM is a straight line. If CA  CD , prove that BC bisects DBM.
14. In the figure, ABCD is a cyclic quadrilateral. EAD and EBC are
straight lines. EAB  EBA.
(a) Prove that AB // DC .
(b) Prove that EAB ~ EDC .
E
C
D
A
B
B
A
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M
2.9
D
15. In the figure, AEC and BED are straight lines. It is given that
CAB  20, CBE  25 and ABE  45 . Prove that A, B, C
and D are concyclic.
C
E
25
45
20
A
B
16. In the figure, O is the centre of the circle, chords AH and AK are
produced to C and D respectively such that straight line CBD is
perpendicular to diameter AB. Let BHK  b.
A
(a) Find CHB.
O
H
(b) Express ADB in terms of b.
b
K
(c) Prove that C, D, K and H are concyclic.
C
B
17. In the figure, ASCB and PRSQ are straight lines. If RC // AQ,
prove that B, C, R and P are concyclic.
D
P
R
B
C
S
A
Q
18. In the figure, B and D are points on the circumference of the
larger circle. C and F are points on the circumference of the
smaller circle. The circumferences of two circles intersect at E
and G. ADB, AFC and BGC are straight lines. Prove that A, D, E
and F are concyclic.
A
D
F
E
C
B
G
2.10
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F
19. In the figure, the circumferences of two circles intersect at A
and B. C and E are points on the circumference of the smaller
circle. D and F are points on the circumference of the larger
circle. CAD, CEF and BED are straight lines. Let CFD  a ,
DBF  b and BFC  c.
D
c
A
E
b
(a) Prove that BEC  a  c .
(b) Hence prove that a  b .
a
B
C
20. In the figure, ABCD is a quadrilateral, where A  B and
C  D .
A
B
(a) Prove that ABCD is a cyclic quadrilateral.
(b) AB is produced to E and DC is produced to F such that
EF // AD . Prove that BCFE is a cyclic quadrilateral.
D
C
21. In the figure, AB is a diameter of semi-circle ADCB. AC and BD
intersect at K. M is a point on AB such that KM  AB.
D
C
K
(a) Prove that B, C, K and M are concyclic.
(b) Prove that A, D, K and M are concyclic.
(c) Prove that KM bisects DMC.
A
M
22. In the figure, ABC is a triangle, where AB  AC and BAC  36.
AC and BD intersect at E. DB bisects ABC and AB // DC .
B
A
36
(a) Find ABD.
D
(b) Prove that A, B, C and D are concyclic.
(c) (i) Prove that BC  AD.
(ii) Prove that ABC  BAD .
E
B
C
2.11
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23. In the figure, PQR  PST , PQ  PS and QR  ST . X is a point
on ST such that QRX is a straight line.
Q
R
(a) Prove that PRXT is a cyclic quadrilateral.
S
(b) Hence prove that PX bisects QXT.
X
P
24. In the figure, diameters AB and CD are perpendicular to each
other and intersect at the centre of the circle O. Chord CP and
AB intersect at E. Chord BQ intersects CD and CP at F and K
respectively. OE  OF .
T
A
Q
D
F
O
K
E
C
(a) Prove that BOF  COE .
(b) (i) Prove that B, C, O and K are concyclic.
P
(ii) Prove that CP  BQ .
B
(c) (i) Prove that DCQ  BCP .
(ii) Prove that ACQ  DCP.
[ In this exercise, O is the centre of the circle in each of the figures. ]
In each of the following figures, TA and TB are tangents to the circle. Find the unknowns. (25  30)
26. EOT is a straight line.
25.
E
D
b
O
24
a
B
O
2.12
66
C
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T
C
A
27. OCT is a straight line.
28. OCT is a straight line.
T
T
z
y
C
A
B
x cm
12 cm
y cm
50 x
C
O
A
B
5 cm
O
29. OMA is a straight line.
30.
A
M
48 cm
O
T
x cm
B
In each of the following figures, TA and TB are tangents to the circle. Find the unknowns. (31  34)
31. DOCA is a straight line.
32. COT is a straight line.
A
C
39
34 C
O
T
y
x
D
O
y
z
B
x
D
A
T
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A
T
64
x
2.13
O
y
33. TDC and CEOA are straight lines.
34. AEC and BED are straight lines.
C
T
y
E
O
26 x
B
D
x
E
A
y
T
C
62
A
D
In each of the following figures, the inscribed circle of ABC touches AC, BC and AB at D, E and F
respectively. Find the unknowns. (35  36)
35.
C
36.
C
(5z  8) cm
3z cm
18 cm
2x cm
E
D
D
(3y  4) cm
x cm
E
A
A
x cm
10 cm
F
F
B
14 cm
B
37. In the figure, CAF, DBG and AB are tangents to the circle at
C, D and E respectively. OA and OB intersect the circumference
at R and S respectively. If CF // DG and BAF  128  , find x
and y.
F
A
128
G
x
2.14
 2010 Chung Tai Educational Press. All rights reserved.
R
E
y
B
S
38. In the figure, the inscribed circle of ABC touches AB, AC
and BC at P, Q and R respectively. Given that OCR  38
and AOC  118 , find x and y.
A
x
Q
P
118
y
O
C
R
38
x
67
F
B
39. In the figure, the inscribed circle of ABC touches AB, BC
and AC at D, E and F respectively. Given that ECF  48
and DFE  67 , find x, y and z.
A
D
z
y
48
B
40. In the figure, the inscribed circle of ABC touches AB,
AC and BC at P, Q and R respectively. It is given that
AB  10 cm, BC  20 cm and BR  x cm .
E
C
A
Q
P
10 cm
(a) Express the length of AP in terms of x.
(b) Express the length of CQ in terms of x.
(c) If AC  15 cm, find the value of x.
B
R
C
x cm
20 cm
41. In the figure, AD and BC are two common tangents to two
circles, where A, B, C and D are points of contact. AD and
BC intersect at E. Prove that AD  BC .
A
[ Hint: A straight line that is a tangent to two circles is called a common
tangent to the two circles. ]
C
E
B
D
2.15
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42. In the figure, AD is a common tangent to two circles at D.
AB and AC are tangents, where B and C are their respective
points of contact. If BAC  60, prove that ABC is an
equilateral triangle.
43. In the figure, A is a point on the circumference, OB
intersects the circumference at C. Given that OA  8 cm ,
AB  15 cm and BC  9 cm, prove that AB is the tangent to
the circle at A.
44. In the figure, PA and PB are tangents to the circle at A and
B respectively. C is a point on the circumference and APQ
is a straight line. Prove that BPQ  2ACB.
45. In the figure, OD and OE are radii. AB touches the circle
at F. AD and BE are tangents to the circle at D and E
respectively. AD // BE . Let AOF  x and BOF  y .
A
60
(a) Express DAF in terms of x.
(b) Express EBF in terms of y.
B
C
(c) Prove that DE is a diameter.
D
BA
D
9 cm
2.16
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15Pcm
Q
C
A
x
C y
O
B
F
46. In the figure, the circumferences of two circles C1 and C 2
intersect at W and Y. PX and PY are the tangents to C1 . QY
and QZ are the tangents to C 2 . PX // QY and PY // QZ . Prove
that XYZ is a straight line.
47. In the figure, AB and BC are the tangents to the circle, where
AB  4 and BC  12 . AOC is a straight line. Find the area of
the shaded region. (Express your answer in terms of .)
48. In the figure, PABQ is a common tangent to two circles with
centres O1 and O2 , where A and B are their respective points
of contact. If the radii of two circles are 10 cm and 6 cm, and
O1O2  18 cm , find the length of AB. (Leave your answer as
surd in its simplest form.)
In each of the following figures, TA and TB are tangents to the circle. Find the unknowns. (49  54)
C
49.
C
50.
D
C
3x  6
72
C1
D
C2
W
O
Y
74  x
45
x
y
A
T
B
X
A
T
A
O1
P
P
 2010 Chung Tai Educational Press. All rights reserved.
A
Z
B
B
Q
O2
B
2.17
Q
51. O is the centre of the circle,
COD is a straight line.
52.
53.
54. O is the centre of the circle.
C
24
30
D
38
O
y
x
A
B
T
In each of the following figures, TA and TB are tangents to the circle. Find the unknowns. (55  60)
55. ACD is a straight line.
56. O is the centre of the circle.
D
A
38
C
C
y
x
y
124
O
T
78
72
A
x
O
C
B
D
D
39
x
134
T
A
68
x
A
2.18
B
C B
T
B
T
x
A
 2010 Chung Tai Educational Press. All rights reserved.
T
D
B
y
57. TBD is a straight line.
58. TBE is a straight line.
D
A
y
x
C
C
x
34
53
y
T
B
E
A
D
B
70
T
59. ADE and BCD are straight lines.
60. O is the centre of the circle, COD, AED and
CET are straight lines.
B
14
D
O
C
x
C
E
52
y
T
A
y
T
B
103 x
E
D
71
A
61. In the figure, C, D, E and T are points on the circumference. AB is the tangent to the circle at T.
Given that BTD  75 and CTD  60 , find x and y.
D
x
E
y
C
60
75
B
T
A
2.19
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62. In the figure, the inscribed circle of ABC touches AB, AC
and BC at D, E and F respectively. Given that ADE  58
and EDF  52, find x and y.
A
D
58
52
x
E
y
B
F
63. In the figure, O is the centre of the circle. TC and SD are
tangents to the circle at A and B respectively. E is a point
on the circumference such that EA // DS. AF is a diameter.
Given that EAF  42, find x and y.
C
C
A
E
42
x
y
O
T
F
D
S
B
64. In the figure, C, D and E are points on the circumference.
AB and DF touch the circle at C and D respectively. Given
that CDF  BCE , prove that AB // DE.
E
D
F
B
C
A
65. In the figure, A, B and D are points on the circumference.
ADC is a straight line and BD  AC. It is given that
ABD  BCD  x.
A
D
(a) Express BAD and CBD in terms of x.
(b) Prove that BC is the tangent to the circle at B.
x
x
B
2.20
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C
66. In the figure, TC and TD are tangents to the circle at A and
B respectively. E is a point on the circumference. G is a
point on TD. EG and AB intersect at F and BF  BG . Given
that EAC  29 and ATB  56 , find x, y and z.
D
B
y
z
E
29
C
F
G
56
A
T
C
67. In the figure, AC and BC are tangents to the circle at A and
B respectively. D is a point on the circumference. It is given
that ACB  x .
x
(a) Prove that ADB  90  .
2
(b) If ADB  2ACB , find x.
68. In the figure, chords BE and CD intersect at F. AD is the
tangent to the circle at D and AD // BE. It is given that
BC  CE , ADB  x and BEC  y .
x
A
x
D
B
D
x
E
A
y
F
(a) Prove that FC is the angle bisector of BCE.
(b) Prove that x  y  90 .
B
C
P
69. In the figure, PQ is the tangent to semi-circle RTKS at T. RS
is a diameter, PKS is a straight line and QP  PS . Prove that
 
R T  TK .
T
K
Q
R
S
2.21
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70. In the figure, O1 and O2 are the centres of circles DAC and
CBE respectively. AB is a common tangent. DAP, PBE and
DO1CO 2 E are straight lines.
P
A
B
(a) Express b in terms of a.
(b) Prove that DP  PE .
71. In the figure, O is the centre of the circle. Chord PQ is
produced to T such that TB and TD are tangents to the circle
at B and D respectively. Chords PQ and BC intersect at K,
PQ // CD .
(a) Prove that T, B, K and D are concyclic.
(b) Prove that T, B, O and D are concyclic.
(c) Prove that BDT  CDK .
2.22
 2010 Chung Tai Educational Press. All rights reserved.
b
D
a
C
O1
O2
B
Q
P
K
T
O
C
D
E