Circle Geometry
Definitions
secant
radius
diameter
centre
chord
tangent
concentric circles
Definitions
secant
radius
diameter
centre
chord
tangent
concentric circles
Definitions
minor arc
major arc
Definitions
minor sector
major sector
Definitions
major
segment
minor
segment
Definitions
minor sector
minor arc
major arc
major sector
major
segment
minor
segment
Theorem 1
Equal chords subtend equal angles at
the centre
A
Proof :
O
B
In  AOB,  COD
C
D
Theorem 1
Equal chords subtend equal angles at
the centre
A
Proof :
O
B
In  AOB,  COD
OA  OC (radii )
OB  OD (radii )
AB  CD ( given)
C
D
Theorem 1
Equal chords subtend equal angles at
the centre
A
Proof :
O
B
In  AOB,  COD
OA  OC (radii )
OB  OD (radii )
C
D
AB  CD ( given)
 AOB   COD ( S .S .S )
 AOB   COD (corresponding angles of congruent triangles)
Theorem 1
Equal chords subtend equal angles at
the centre
A
Proof :
O
B
In  AOB,  COD
OA  OC (radii )
OB  OD (radii )
C
D
AB  CD ( given)
 AOB   COD ( S .S .S )
 AOB   COD (corresponding angles of congruent triangles)
Theorem 2
If the angles subtended at the centre of a
circle are equal, then the chords are equal
in length. (Converse of Theorem 1)
A
Proof :
O
B
Your turn
C
D
Theorem 2
If the angles subtended at the centre of a
circle are equal, then the chords are equal
in length. (Converse of Theorem 1)
A
Proof :
O
B
In  AOB,  COD
OA  OC (radii )
C
OB  OD (radii )
 AOB   COD ( given)
 AOB   COD ( S . A.S )
AB  CD (corresponding sides of congruent triangles)
D
Theorem 3
Equal chords are equidistant from the centre.
A
Proof :
In  AOB,  COD
O
B
C
D
Theorem 3
Equal chords are equidistant from the centre.
A
Proof :
In  AOB,  COD
OA  OC (radii )
OB  OD (radii )
AB  CD ( given)
 AOB   COD ( S .S .S )
O
B
C
D
Theorem 3
Equal chords are equidistant from the centre.
A
Proof :
In  AOB,  COD
OA  OC (radii )
OB  OD (radii )
AB  CD ( given)
C
 AOB   COD ( S . A.S )
 perpendicular heights are equal
(corresponding heights of congruent triangles)
O
B
D
Theorem 3
Equal chords are equidistant from the centre.
A
Proof :
In  AOB,  COD
OA  OC (radii )
OB  OD (radii )
AB  CD ( given)
C
 AOB   COD ( S .S .S )
 perpendicular heights are equal
(corresponding heights of congruent triangles)
O
B
D
Theorem 4
The line from the centre of a circle to a chord
bisects the chord.
Proof :
In  AOC ,  AOD
O
C
A
D
Theorem 4
The line from the centre of a circle which
bisects the chord, is perpendicular to it.
Proof :
In  AOC ,  AOD
OC  OD (radii )
O
OAis common
AC  AD ( given)
 AOB   COD ( S .S .S )
C
A
D
Theorem 4
The line from the centre of a circle which
bisects the chord, is perpendicular to it.
Proof :
In  AOC ,  AOD
O
OC  OD (radii )
OAis common
AC  AD ( given)
C
A
D
 AOB   COD ( S .S .S )
OAC  OAD (corresponding angles of congruent triangles)
OAC  OAD  180 (straight angle)
OAC  OAD  90
Theorem 4
The line from the centre of a circle which
bisects the chord, is perpendicular to it.
Proof :
In  AOC ,  AOD
O
OC  OD (radii )
OAis common
AC  AD ( given)
C
A
D
 AOC   AOD ( S .S .S )
OAC  OAD (corresponding angles of congruent triangles)
OAC  OAD  180 (straight angle)
OAC  OAD  90
Theorem 5
The perpendicular from the centre of a circle to
a chord bisects the chord.
Proof :
In  AOC ,  AOD
O
Your turn
C
A
D
Theorem 5
The perpendicular from the centre of a circle to
a chord bisects the chord.
Proof :
In  AOC ,  AOD
O
OC  OD (radii )
OAis common
OAC  OAD ( given)
C
A
D
Theorem 5
The perpendicular from the centre of a circle to
a chord bisects the chord.
Proof :
In  AOC ,  AOD
O
OC  OD (radii )
OAis common
OAC  OAD ( given)
C
 OAC   OAD ( R.H .S )
AC  AD (corresponding sides of congruent triangles)
A
D
Theorem 5
The perpendicular from the centre of a circle to
a chord bisects the chord.
Proof :
In  AOC ,  AOD
O
OC  OD (radii )
OAis common
OAC  OAD ( given)
C
 AOB   COD ( R.H .S )
AC  AD (corresponding sides of congruent triangles)
A
D