The document defines various terms related to circle geometry, including radius, diameter, chord, secant, tangent, arc, sector, segment, concyclic points, cyclic quadrilateral, concentric circles, and theorems about chords and arcs. Specifically, it states that (1) a perpendicular from the circle center to a chord bisects the chord, and the perpendicular bisector of a chord passes through the center, and (2) conversely, the line from the center to the midpoint of a chord is perpendicular to the chord.

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