11X1 T13 01 definitions & chord theorems (2010)
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Transcript of 11X1 T13 01 definitions & chord theorems (2010)
![Page 1: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/1.jpg)
Circle Geometry
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Circle Geometry Circle Geometry Definitions
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Circle Geometry Circle Geometry Definitions
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Circle Geometry Circle Geometry Definitions
Radius: an interval joining centre to the
circumference
radius
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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
![Page 6: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/6.jpg)
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
![Page 7: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/7.jpg)
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
![Page 8: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/8.jpg)
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
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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
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Sector: a plane figure with two radii and an
arc as boundaries.
sector
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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”
![Page 13: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/13.jpg)
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
![Page 14: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/14.jpg)
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
![Page 15: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/15.jpg)
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.
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Concyclic Points: points that lie on the same circle.
A
B
C D
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Concyclic Points: points that lie on the same circle.
concyclic points
A
B
C D
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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
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A B
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A B
represents the angle
subtended at the centre by
the arc AB
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A B
represents the angle
subtended at the centre by
the arc AB
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A B
represents the angle
subtended at the centre by
the arc AB
represents the angle
subtended at the
circumference by the arc AB
![Page 25: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/25.jpg)
A B
represents the angle
subtended at the centre by
the arc AB
represents the angle
subtended at the
circumference by the arc AB
![Page 26: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/26.jpg)
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.
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Circles touching internally
share a common tangent.
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Circles touching internally
share a common tangent.
Circles touching externally
share a common tangent.
![Page 31: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/31.jpg)
Circles touching internally
share a common tangent.
Circles touching externally
share a common tangent.
![Page 32: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/32.jpg)
Circles touching internally
share a common tangent.
Circles touching externally
share a common tangent.
![Page 33: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/33.jpg)
Chord (Arc) Theorems
![Page 34: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/34.jpg)
Chord (Arc) Theorems Note: = chords cut off = arcs
![Page 35: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/35.jpg)
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.
![Page 36: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/36.jpg)
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
![Page 37: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/37.jpg)
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
![Page 38: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/38.jpg)
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
![Page 39: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/39.jpg)
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
![Page 40: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/40.jpg)
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
![Page 41: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/41.jpg)
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
![Page 42: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/42.jpg)
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
![Page 43: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/43.jpg)
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
![Page 44: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/44.jpg)
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
![Page 45: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/45.jpg)
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
![Page 46: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/46.jpg)
(2) Converse of (1)
The line from the centre of a circle to the midpoint of the chord at
right angles.
![Page 47: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/47.jpg)
(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
![Page 48: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/48.jpg)
(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
![Page 49: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/49.jpg)
(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
![Page 50: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/50.jpg)
(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
![Page 51: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/51.jpg)
(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
![Page 52: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/52.jpg)
(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
![Page 53: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/53.jpg)
(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
![Page 54: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/54.jpg)
(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
![Page 55: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/55.jpg)
(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
![Page 56: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/56.jpg)
(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
![Page 57: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/57.jpg)
(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
![Page 58: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/58.jpg)
(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
![Page 59: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/59.jpg)
(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
![Page 60: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/60.jpg)
(3) Equal chords of a circle are the same distance from the centre and
subtend equal angles at the centre.
![Page 61: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/61.jpg)
(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
![Page 62: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/62.jpg)
(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
![Page 63: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/63.jpg)
(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
![Page 64: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/64.jpg)
(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
![Page 65: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/65.jpg)
(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
![Page 66: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/66.jpg)
(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
![Page 67: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/67.jpg)
(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
![Page 68: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/68.jpg)
(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
![Page 69: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/69.jpg)
(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
![Page 70: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/70.jpg)
(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
![Page 71: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/71.jpg)
(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
![Page 72: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/72.jpg)
(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
![Page 73: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/73.jpg)
(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
![Page 74: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/74.jpg)
(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
![Page 75: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/75.jpg)
A
B
O
C D
![Page 76: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/76.jpg)
A
B
O
C D
CDAB :Data
![Page 77: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/77.jpg)
A
B
O
C D
CDAB :Data
CODAOB :Prove
![Page 78: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/78.jpg)
A
B
O
C D
CDAB :Data
CODAOB :Prove
Proof: given CDAB
![Page 79: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/79.jpg)
A
B
O
C D
CDAB :Data
CODAOB :Prove
Proof: given CDAB
radii DOCOBOAO
![Page 80: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/80.jpg)
A
B
O
C D
CDAB :Data
CODAOB :Prove
Proof: given CDAB
radii DOCOBOAO
SSSCODAOB
![Page 81: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/81.jpg)
A
B
O
C D
CDAB :Data
CODAOB :Prove
Proof: given CDAB
radii DOCOBOAO
matching 's in 's AOB COD
SSSCODAOB
![Page 82: 11X1 T13 01 definitions & chord theorems (2010)](https://reader033.fdocuments.us/reader033/viewer/2022051323/54bbaee24a7959864d8b45d9/html5/thumbnails/82.jpg)
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