1. Circle Geometry

2. Circle Geometry
Circle Geometry Definitions

3. Circle Geometry
Circle Geometry Definitions

4. Circle Geometry
Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
radius

5. Circle Geometry
Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
Diameter: an interval passing through the
radius centre, joining any two points
on the circumference
diameter

6. Circle Geometry
Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
Diameter: an interval passing through the
radius centre, joining any two points
on the circumference
diameter Chord:an interval joining two points on
the circumference
chord

7. Circle Geometry
Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
Diameter: an interval passing through the
radius centre, joining any two points
on the circumference
diameter Chord:an interval joining two points on
secant the circumference
Secant: a line that cuts the circle
chord

8. Circle Geometry
Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
Diameter: an interval passing through the
radius centre, joining any two points
on the circumference
diameter Chord:an interval joining two points on
secant the circumference
Secant: a line that cuts the circle
chord Tangent: a line that touches the circle
tangent

9. Circle Geometry
Circle Geometry Definitions
Radius: an interval joining centre to the
arc
circumference
Diameter: an interval passing through the
radius centre, joining any two points
on the circumference
diameter Chord:an interval joining two points on
secant the circumference
Secant: a line that cuts the circle
chord Tangent: a line that touches the circle
tangent Arc: a piece of the circumference

11. Sector: a plane figure with two radii and an
arc as boundaries.
sector

12. Sector: a plane figure with two radii and an
arc as boundaries.
The minor sector is the small “piece of the
pie”, the major sector is the large “piece”
sector

13. Sector: a plane figure with two radii and an
arc as boundaries.
The minor sector is the small “piece of the
pie”, the major sector is the large “piece”
A quadrant is a sector where the angle at
sector the centre is 90 degrees

14. Sector: a plane figure with two radii and an
arc as boundaries.
The minor sector is the small “piece of the
pie”, the major sector is the large “piece”
A quadrant is a sector where the angle at
sector the centre is 90 degrees
segment
Segment: a plane figure with a chord and an
arc as boundaries.

15. Sector: a plane figure with two radii and an
arc as boundaries.
The minor sector is the small “piece of the
pie”, the major sector is the large “piece”
A quadrant is a sector where the angle at
sector the centre is 90 degrees
segment
Segment: a plane figure with a chord and an
arc as boundaries.
A semicircle is a segment where the chord is
the diameter, it is also a sector as the diameter
is two radii.

17. Concyclic Points: points that lie on the same circle.
A
B
D C

18. Concyclic Points: points that lie on the same circle.
A
B
concyclic points
D C

19. Concyclic Points: points that lie on the same circle.
A
B
cyclic quadrilateral concyclic points
D C
Cyclic Quadrilateral: a four sided shape with all vertices on the
same circle.

21. 
A
B

22. 
A
B
 represents the angle
subtended at the centre by
the arc AB

23. 

A
B
 represents the angle
subtended at the centre by
the arc AB

24. 

A
B
 represents the angle
subtended at the centre by
the arc AB
 represents the angle
subtended at the
circumference by the arc AB

25. 

A
B
 represents the angle
subtended at the centre by
the arc AB
 represents the angle
subtended at the
circumference by the arc AB

26. 

A
B
 represents the angle Concentric circles have the
subtended at the centre by same centre.
the arc AB
 represents the angle
subtended at the
circumference by the arc AB

29. Circles touching internally
share a common tangent.

30. Circles touching internally
share a common tangent.
Circles touching externally
share a common tangent.

31. Circles touching internally
share a common tangent.
Circles touching externally
share a common tangent.

32. Circles touching internally
share a common tangent.
Circles touching externally
share a common tangent.

33. Chord (Arc) Theorems

34. Chord (Arc) Theorems
Note: = chords cut off = arcs

35. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.

36. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.
A
X
B
O

37. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.
AX  BX  from centre, bisects chord
A
X
B
O

38. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.
AX  BX  from centre, bisects chord
A Data : AB  OX
X
B
O

39. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.
AX  BX  from centre, bisects chord
A Data : AB  OX
X Prove : AX  BX
B
O

40. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.
AX  BX  from centre, bisects chord
A Data : AB  OX
X Prove : AX  BX
B Proof: Join OA, OB
O

41. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.
AX  BX  from centre, bisects chord
A Data : AB  OX
X Prove : AX  BX
B Proof: Join OA, OB
O
AXO  BXO  90 given R 

42. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.
AX  BX  from centre, bisects chord
A Data : AB  OX
X Prove : AX  BX
B Proof: Join OA, OB
O
AXO  BXO  90 given R 
AO  BO  radii H 

43. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.
AX  BX  from centre, bisects chord
A Data : AB  OX
X Prove : AX  BX
B Proof: Join OA, OB
O
AXO  BXO  90 given R 
AO  BO  radii H 
OX is common S 

44. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.
AX  BX  from centre, bisects chord
A Data : AB  OX
X Prove : AX  BX
B Proof: Join OA, OB
O
AXO  BXO  90 given R 
AO  BO  radii H 
OX is common S 
 AXO  BXO RHS 

45. Chord (Arc) Theorems
Note: = chords cut off = arcs
(1) A perpendicular drawn to a chord from the centre of a circle bisects
the chord, and the perpendicular bisector of a chord passes through
the centre.
AX  BX  from centre, bisects chord
A Data : AB  OX
X Prove : AX  BX
B Proof: Join OA, OB
O
AXO  BXO  90 given R 
AO  BO  radii H 
OX is common S 
 AXO  BXO RHS 
 AX  BX  matching sides in  's 

46. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.

47. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
A
X
B
O

48. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A
X
B
O

49. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
B
O

50. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
Prove : AB  OX
B
O

51. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
Prove : AB  OX
B
O Proof: Join OA, OB

52. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
Prove : AB  OX
B
O Proof: Join OA, OB
AX  BX given S 

53. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
Prove : AB  OX
B
O Proof: Join OA, OB
AX  BX given S 
AO  BO  radii S 

54. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
Prove : AB  OX
B
O Proof: Join OA, OB
AX  BX given S 
AO  BO  radii S 
OX is common S 

55. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
Prove : AB  OX
B
O Proof: Join OA, OB
AX  BX given S 
AO  BO  radii S 
OX is common S 
 AXO  BXO SSS 

56. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
Prove : AB  OX
B
O Proof: Join OA, OB
AX  BX given S 
AO  BO  radii S 
OX is common S 
 AXO  BXO SSS 
AXO  BXO  matching 's in  's 

57. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
Prove : AB  OX
B
O Proof: Join OA, OB
AX  BX given S 
AO  BO  radii S 
OX is common S 
 AXO  BXO SSS 
AXO  BXO  matching 's in  's 
AXO  BXO  180 straight  AXB 

58. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
Prove : AB  OX
B
O Proof: Join OA, OB
AX  BX given S 
AO  BO  radii S 
OX is common S 
 AXO  BXO SSS 
AXO  BXO  matching 's in  's 
AXO  BXO  180 straight  AXB 
2AXO  180
AXO  90

59. (2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
OX  AB line joining centre to midpoint,  to chord
A Data : AX  BX
X
Prove : AB  OX
B
O Proof: Join OA, OB
AX  BX given S 
AO  BO  radii S 
OX is common S 
 AXO  BXO SSS 
AXO  BXO  matching 's in  's 
AXO  BXO  180 straight  AXB 
2AXO  180
AXO  90
 AB  OX

60. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.

61. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
A
O X
B
C Y D

62. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
A
O X
B
C Y D

63. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
A
O X
B
C Y D

64. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
O X
B
C Y D

65. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
Prove : OX  OY
O X
B
C Y D

66. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
Prove : OX  OY
O Proof: Join OA, OC
X
B
C Y D

67. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
Prove : OX  OY
O Proof: Join OA, OC
X
AB  CD given 
B
C Y D

68. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
Prove : OX  OY
O Proof: Join OA, OC
X
AB  CD given 
1
C Y D
B AX  AB  bisects chord
2

69. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
Prove : OX  OY
O Proof: Join OA, OC
X
AB  CD given 
1
C Y D
B AX  AB  bisects chord
2
1
CY  CD  bisects chord
2

70. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
Prove : OX  OY
O Proof: Join OA, OC
X
AB  CD given 
1
C Y D
B AX  AB  bisects chord
2
1
CY  CD  bisects chord
2
 AX  CY S 

71. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
Prove : OX  OY
O Proof: Join OA, OC
X
AB  CD given 
1
C Y D
B AX  AB  bisects chord
2
1
CY  CD  bisects chord
2
 AX  CY S 
AXO  CYO  90 given R 

72. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
Prove : OX  OY
O Proof: Join OA, OC
X
AB  CD given 
1
C Y D
B AX  AB  bisects chord
2
1
CY  CD  bisects chord
2
 AX  CY S 
AXO  CYO  90 given R 
OA  OC  radii H 

73. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
Prove : OX  OY
O Proof: Join OA, OC
X
AB  CD given 
1
C Y D
B AX  AB  bisects chord
2
1
CY  CD  bisects chord
2
 AX  CY S 
AXO  CYO  90 given R 
OA  OC  radii H 
 AXO  CYO RHS 

74. (3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
OX  OY  chords, equidistant from centre
AOB  COD  chords subtend  ' s at centre
Data: AB  CD, OX  AB, OY  CD
A
Prove : OX  OY
O Proof: Join OA, OC
X
AB  CD given 
1
C Y D
B AX  AB  bisects chord
2
1
CY  CD  bisects chord
2
 AX  CY S 
AXO  CYO  90 given R 
OA  OC  radii H 
 AXO  CYO RHS 
 OX  OY  matching sides in  's 

75. A
O
B
C D

76. Data : AB  CD
A
O
B
C D

77. Data : AB  CD
A Prove : AOB  COD
O
B
C D

78. Data : AB  CD
A Prove : AOB  COD
O
Proof: AB  CD given 
B
C D

79. Data : AB  CD
A Prove : AOB  COD
O
Proof: AB  CD given 
AO  BO  CO  DO  radii 
B
C D

80. Data : AB  CD
A Prove : AOB  COD
O
Proof: AB  CD given 
AO  BO  CO  DO  radii 
C D
B  AOB  COD SSS 

81. Data : AB  CD
A Prove : AOB  COD
O
Proof: AB  CD given 
AO  BO  CO  DO  radii 
C D
B  AOB  COD SSS 
AOB  COD  matching 's in  's 

82. Data : AB  CD
A Prove : AOB  COD
O
Proof: AB  CD given 
AO  BO  CO  DO  radii 
C D
B  AOB  COD SSS 
AOB  COD  matching 's in  's 
Exercise 9A; 1 ce, 2 aceg, 3, 4, 9, 10 ac, 11 ac, 13, 15, 16, 18