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
10.
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.
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.
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
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
2AXO 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
2AXO 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
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