11 x1 t13 01 definitions & chord theorems (2013)
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Circle Geometry
Circle Geometry Circle Geometry Definitions
Circle Geometry Circle Geometry Definitions
Circle Geometry Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
radius
Circle Geometry Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
radius
Diameter: an interval passing through the
centre, joining any two points
on the circumference
diameter
Circle Geometry Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
radius
Diameter: an interval passing through the
centre, joining any two points
on the circumference
diameter Chord: an interval joining two points on
the circumference
chord
Circle Geometry Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
radius
Diameter: an interval passing through the
centre, joining any two points
on the circumference
diameter Chord: an interval joining two points on
the circumference
chord
Secant: a line that cuts the circle secant
Circle Geometry Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
radius
Diameter: an interval passing through the
centre, joining any two points
on the circumference
diameter Chord: an interval joining two points on
the circumference
chord
Secant: a line that cuts the circle secant
Tangent: a line that touches the circle
tangent
Circle Geometry Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
radius
Diameter: an interval passing through the
centre, joining any two points
on the circumference
diameter Chord: an interval joining two points on
the circumference
chord
Secant: a line that cuts the circle secant
Tangent: a line that touches the circle
tangent Arc: a piece of the circumference
arc
Sector: a plane figure with two radii and an
arc as boundaries.
sector
Sector: a plane figure with two radii and an
arc as boundaries.
sector
The minor sector is the small “piece of the
pie”, the major sector is the large “piece”
Sector: a plane figure with two radii and an
arc as boundaries.
sector
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
the centre is 90 degrees
Sector: a plane figure with two radii and an
arc as boundaries.
sector
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
the centre is 90 degrees
Segment: a plane figure with a chord and an
arc as boundaries.
segment
Sector: a plane figure with two radii and an
arc as boundaries.
sector
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
the centre is 90 degrees
Segment: a plane figure with a chord and an
arc as boundaries.
segment
A semicircle is a segment where the chord is
the diameter, it is also a sector as the diameter
is two radii.
Concyclic Points: points that lie on the same circle.
A
B
C D
Concyclic Points: points that lie on the same circle.
concyclic points
A
B
C D
Concyclic Points: points that lie on the same circle.
concyclic points
Cyclic Quadrilateral: a four sided shape with all vertices on the
same circle.
cyclic quadrilateral
A
B
C D
A B
A B
represents the angle
subtended at the centre by
the arc AB
A B
represents the angle
subtended at the centre by
the arc AB
A B
represents the angle
subtended at the centre by
the arc AB
represents the angle
subtended at the
circumference by the arc AB
A B
represents the angle
subtended at the centre by
the arc AB
represents the angle
subtended at the
circumference by the arc AB
A B
represents the angle
subtended at the centre by
the arc AB
represents the angle
subtended at the
circumference by the arc AB
Concentric circles have the
same centre.
Circles touching internally
share a common tangent.
Circles touching internally
share a common tangent.
Circles touching externally
share a common tangent.
Circles touching internally
share a common tangent.
Circles touching externally
share a common tangent.
Circles touching internally
share a common tangent.
Circles touching externally
share a common tangent.
Chord (Arc) Theorems
Chord (Arc) Theorems Note: = chords cut off = arcs
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.
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
B O
X
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.
chord bisects centre, from BXAX
A
B O
X
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.
chord bisects centre, from BXAX
A
B O
X OXAB :Data
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.
chord bisects centre, from BXAX
A
B O
X OXAB :Data
BXAX :Prove
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.
chord bisects centre, from BXAX
A
B O
X OXAB :Data
BXAX :Prove
Proof: Join OA, OB
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.
chord bisects centre, from BXAX
A
B O
X OXAB :Data
BXAX :Prove
Proof: Join OA, OB
RBXOAXO given 90
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.
chord bisects centre, from BXAX
A
B O
X OXAB :Data
BXAX :Prove
Proof: Join OA, OB
RBXOAXO given 90
HBOAO radii
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.
chord bisects centre, from BXAX
A
B O
X OXAB :Data
BXAX :Prove
Proof: Join OA, OB
RBXOAXO given 90
HBOAO radii
SOX common is
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.
chord bisects centre, from BXAX
A
B O
X OXAB :Data
BXAX :Prove
Proof: Join OA, OB
RBXOAXO given 90
HBOAO radii
SOX common is
RHSBXOAXO
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.
chord bisects centre, from BXAX
A
B O
X OXAB :Data
BXAX :Prove
Proof: Join OA, OB
RBXOAXO given 90
HBOAO radii
SOX common is
RHSBXOAXO
matching sides in 'sAX BX
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
A
B O
X
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
OXAB :Prove
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
OXAB :Prove
Proof: Join OA, OB
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
OXAB :Prove
Proof: Join OA, OB
SBXAX given
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
OXAB :Prove
Proof: Join OA, OB
SBXAX given
SBOAO radii
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
OXAB :Prove
Proof: Join OA, OB
SBXAX given
SBOAO radii
SOX common is
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
OXAB :Prove
Proof: Join OA, OB
SBXAX given
SBOAO radii
SOX common is
SSSBXOAXO
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
OXAB :Prove
Proof: Join OA, OB
SBXAX given
SBOAO radii
SOX common is
SSSBXOAXO
matching 's in 'sAXO BXO
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
OXAB :Prove
Proof: Join OA, OB
SBXAX given
SBOAO radii
SOX common is
SSSBXOAXO
matching 's in 'sAXO BXO
AXBBXOAXO straight 180
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
OXAB :Prove
Proof: Join OA, OB
SBXAX given
SBOAO radii
SOX common is
SSSBXOAXO
matching 's in 'sAXO BXO
AXBBXOAXO straight 180
90
1802
AXO
AXO
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
chord tomidpoint, tocentre joining line ABOXA
B O
X BXAX :Data
OXAB :Prove
Proof: Join OA, OB
SBXAX given
SBOAO radii
SOX common is
SSSBXOAXO
matching 's in 'sAXO BXO
AXBBXOAXO straight 180
90
1802
AXO
AXO
OX AB
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
centreat s' subtend chords CODAOB
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
centreat s' subtend chords CODAOB
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
OYOX :Prove
centreat s' subtend chords CODAOB
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
OYOX :Prove
Proof: Join OA, OC
centreat s' subtend chords CODAOB
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
OYOX :Prove
Proof: Join OA, OC
given CDAB
centreat s' subtend chords CODAOB
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
OYOX :Prove
Proof: Join OA, OC
given CDAB
chord bisects 2
1 ABAX
centreat s' subtend chords CODAOB
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
OYOX :Prove
Proof: Join OA, OC
given CDAB
chord bisects 2
1 ABAX
centreat s' subtend chords CODAOB
chord bisects 2
1 CDCY
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
OYOX :Prove
Proof: Join OA, OC
given CDAB
chord bisects 2
1 ABAX
SCYAX
centreat s' subtend chords CODAOB
chord bisects 2
1 CDCY
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
OYOX :Prove
Proof: Join OA, OC
given CDAB
chord bisects 2
1 ABAX
SCYAX
RCYOAXO given 90
centreat s' subtend chords CODAOB
chord bisects 2
1 CDCY
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
OYOX :Prove
Proof: Join OA, OC
given CDAB
chord bisects 2
1 ABAX
SCYAX
RCYOAXO given 90
centreat s' subtend chords CODAOB
chord bisects 2
1 CDCY
HOCOA radii
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
OYOX :Prove
Proof: Join OA, OC
given CDAB
chord bisects 2
1 ABAX
SCYAX
RCYOAXO given 90
centreat s' subtend chords CODAOB
chord bisects 2
1 CDCY
HOCOA radii RHSCYOAXO
A
B
O X
C D Y
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
centre fromt equidistan chords, OYOX
Data: , ,AB CD OX AB OY CD
OYOX :Prove
Proof: Join OA, OC
given CDAB
chord bisects 2
1 ABAX
SCYAX
RCYOAXO given 90
matching sides in 's OX OY
centreat s' subtend chords CODAOB
chord bisects 2
1 CDCY
HOCOA radii RHSCYOAXO
A
B
O X
C D Y
A
B
O
C D
A
B
O
C D
CDAB :Data
A
B
O
C D
CDAB :Data
CODAOB :Prove
A
B
O
C D
CDAB :Data
CODAOB :Prove
Proof: given CDAB
A
B
O
C D
CDAB :Data
CODAOB :Prove
Proof: given CDAB
radii DOCOBOAO
A
B
O
C D
CDAB :Data
CODAOB :Prove
Proof: given CDAB
radii DOCOBOAO
SSSCODAOB
A
B
O
C D
CDAB :Data
CODAOB :Prove
Proof: given CDAB
radii DOCOBOAO
matching 's in 's AOB COD
SSSCODAOB
A
B
O
C D
CDAB :Data
CODAOB :Prove
Proof: given CDAB
radii DOCOBOAO
matching 's in 's AOB COD
SSSCODAOB
Exercise 9A; 1 ce, 2 aceg, 3, 4, 9, 10 ac, 11 ac, 13, 15, 16, 18