Circle Geometry

Circle GeometryCircle Geometry Definitions

Circle GeometryCircle Geometry Definitions

Circle GeometryCircle Geometry Definitions

Radius:an interval joining centre to the circumference

radius

Circle GeometryCircle 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 GeometryCircle 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 GeometryCircle 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

chordSecant: a line that cuts the circle

secant

Circle GeometryCircle 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

chordSecant: a line that cuts the circle

secant

Tangent: a line that touches the circle

tangent

Circle GeometryCircle 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

chordSecant: 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

CD

Concyclic Points:points that lie on the same circle.

concyclic points

A

B

CD

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

CD

AB

AB

represents the angle subtended at the centre by the arc AB

AB

represents the angle subtended at the centre by the arc AB

AB

represents the angle subtended at the centre by the arc AB

represents the angle subtended at the circumference by the arc AB

AB

represents the angle subtended at the centre by the arc AB

represents the angle subtended at the circumference by the arc AB

AB

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) TheoremsNote: = chords cut off = arcs

Chord (Arc) TheoremsNote: = 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) TheoremsNote: = 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

BO

X

Chord (Arc) TheoremsNote: = 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 BXAXA

BO

X

Chord (Arc) TheoremsNote: = 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 BXAXA

BO

XOXAB :Data

Chord (Arc) TheoremsNote: = 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 BXAXA

BO

XOXAB :DataBXAX :Prove

Chord (Arc) TheoremsNote: = 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 BXAXA

BO

XOXAB :DataBXAX :Prove

Proof: Join OA, OB

Chord (Arc) TheoremsNote: = 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 BXAXA

BO

XOXAB :DataBXAX :Prove

Proof: Join OA, OB RBXOAXO given 90

Chord (Arc) TheoremsNote: = 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 BXAXA

BO

XOXAB :DataBXAX :Prove

Proof: Join OA, OB RBXOAXO given 90

HBOAO radii

Chord (Arc) TheoremsNote: = 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 BXAXA

BO

XOXAB :DataBXAX :Prove

Proof: Join OA, OB RBXOAXO given 90

HBOAO radii SOX common is

Chord (Arc) TheoremsNote: = 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 BXAXA

BO

XOXAB :DataBXAX :Prove

Proof: Join OA, OB RBXOAXO given 90

HBOAO radii SOX common is RHSBXOAXO

Chord (Arc) TheoremsNote: = 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 BXAXA

BO

XOXAB :DataBXAX :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

BO

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

BO

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

BO

XBXAX :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

BO

XBXAX :DataOXAB :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

BO

XBXAX :DataOXAB :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

BO

XBXAX :DataOXAB :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

BO

XBXAX :DataOXAB :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

BO

XBXAX :DataOXAB :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

BO

XBXAX :DataOXAB :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

BO

XBXAX :DataOXAB :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

BO

XBXAX :DataOXAB :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

BO

XBXAX :DataOXAB :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

BO

XBXAX :DataOXAB :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 DY

(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 DY

(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 DY

(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 DY

(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 DY

(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 DY

(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 DY

(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

21 ABAX

centreat s' subtend chords CODAOB

A

B

O X

C DY

(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

21 ABAX

centreat s' subtend chords CODAOB

chord bisects 21 CDCY

A

B

O X

C DY

(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

21 ABAX

SCYAX

centreat s' subtend chords CODAOB

chord bisects 21 CDCY

A

B

O X

C DY

(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

21 ABAX

SCYAX RCYOAXO given 90

centreat s' subtend chords CODAOB

chord bisects 21 CDCY

A

B

O X

C DY

(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

21 ABAX

SCYAX RCYOAXO given 90

centreat s' subtend chords CODAOB

chord bisects 21 CDCY

HOCOA radii

A

B

O X

C DY

(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

21 ABAX

SCYAX RCYOAXO given 90

centreat s' subtend chords CODAOB

chord bisects 21 CDCY

HOCOA radii RHSCYOAXO

A

B

O X

C DY

(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

21 ABAX

SCYAX RCYOAXO given 90

matching sides in 's OX OY

centreat s' subtend chords CODAOB

chord bisects 21 CDCY

HOCOA radii RHSCYOAXO

A

B

O X

C DY

A

B

O

C D

A

B

O

C D

CDAB :Data

A

B

O

C D

CDAB :DataCODAOB :Prove

A

B

O

C D

CDAB :DataCODAOB :Prove

Proof: given CDAB

A

B

O

C D

CDAB :DataCODAOB :Prove

Proof: given CDAB radii DOCOBOAO

A

B

O

C D

CDAB :DataCODAOB :Prove

Proof: given CDAB radii DOCOBOAO

SSSCODAOB

A

B

O

C D

CDAB :DataCODAOB :Prove

Proof: given CDAB radii DOCOBOAO

matching 's in 's AOB COD SSSCODAOB

A

B

O

C D

CDAB :DataCODAOB :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