Circle Properties
Part I
A circle is a set of all points in a plane that
are the same distance from a fixed point in
a plane
The set of points form the Circumference .
The line joining the centre of a circle and a
point on the circumference is called
Radius
the……………….
A chord is a straight line segment joining
two points on the circle
A chord that passes through the centre is a
diameter
……………………….
secant
A………………………
is a straight line that
cuts the circle in two points
An arc is part of the circumference of a circle
Major arc
Minor arc
sector
A ……………………is
part of the circle
bounded by two radii and an arc
major
sector
Minor
sector
segment
A ……………………is
part of the circle
bounded by a chord and an arc
major
segment
Minor
segment
The arc AB subtends an angle of  at the
centre of the circle.
Subtends means “to extend under” or “ to be
opposite to”
O

A
B
Instructions:
• Draw a circle
• Draw two chords of equal length
• Measure angles  AOB and  DOC
B
O
A
C
D
What do you notice?
Equal chords subtend equal angles at the centre


Conversely
Equal angles at the centre of a circle stand on equal arcs


Instructions:
• select an arc AB
• subtend the arc AB to the centre O
• subtend the arc AB to a point C on the circumference
C
• Measure angles  AOB and  ACB
O
What do you notice?
A
B
Instructions:
• select an arc AB
• subtend the arc AB to the centre O
• subtend the arc AB to a point C on the circumference
C
• Measure angles  AOB and  ACB
O
What do you notice?
A
B
The angle that an arc of a circle subtends
at the centre is twice the angle it subtends
at the circumference

2
Instructions:
• select an arc AB
• select two points C, D on the circumference
• subtend the arc AB to a point C on the circumference
• subtend the arc AB to a point D on the circumference
D
• Measure angles  ACB and  ADB
C
O
A
B
Instructions:
• select an arc AB
• select two points C, D on the circumference
• subtend the arc AB to a point C on the circumference
• subtend the arc AB to a point D on the circumference
D
• Measure angles  ACB and  ADB
C
O
What do you notice?
A
B
Angles subtended at the circumference by the
same arc are equal




Instructions:
• Draw a circle and its diameter
• subtend the diameter to a point on the circumference
•Measure

ACB
C
A
What do you notice?
B
An angle in a semicircle is
a right angle
Instructions:
•Draw a cyclic quadrilateral (the vertices of the
quadrilateral lie on the circumference
•Measure all four angles


γ
β
What do you notice?
The opposite angles of a cyclic quadrilateral
are supplementary
180-

180-

If the opposite angles of a quadrilateral are
supplementary the quadrilateral is cyclic
180-

Instructions:
• Draw a cyclic quadrilateral
• Produce a side of the quadrilateral
•Measure angles  and β
β

If a side of a cyclic quadrilateral is produced,
the exterior angle is equal to the interior
opposite angle


Circle Properties
Part II
tangent properties
A tangent to a circle is a straight line that touches the
circle in one point only
Tangent to a circle is perpendicular to the
radius drawn from the point of contact.
Tangents to a circle from an exterior point are equal
When two circles touch, the line through their centres
passes through their point of contact
External Contact
Point of contact
When two circles touch, the line through their centres
passes through their point of contact
Internal Contact
Point of contact
The angle between a tangent
and a chord through the point of contact
is equal to the angle in the alternate segment


The square of the length of the tangent
from an external point is equal to
the product of the intercepts of the secant
passing through this point
A
B
B=external point
C
D
2
BA =BC.BD
The square of the length of the tangent
from an external point is equal to
the product of the intercepts of the secant
passing through this point
A
Note: B is the crucial point in the formula
B
C
D
2
BA =BC.BD
Circle Properties
Chord properties
Triangle AXD is similar to triangle CXB
hence
C
A
X
D
B
AX.XB=CX.XD
Note: X is the crucial point in the formula
C
A
X
D
B
AX.XB=CX.XD
Chord AB and CD intersect at X
Prove AX.XB=CX.XD
In AXD and CXB
C
A
AXD =  CXB
(Vertically Opposite Angles)
DAX =  BCX
(Angles standing on same arc)
X
ADX =  CBX
D
(Angles standing on same arc)
B
 AXD    CXB
Hence
AX CX

XD XB
AX .XB  CX .XD
(Equiangular )
AAA test for similar triangles
A perpendicular line from the centre off a circle to a
chord bisects the chord
A
C
B
Conversley: A line from the centre of a circle that
bisects a chord is perpendicular to the chord
A
C
B
Equal chords are equidistant from the centre of the circle
A
C
B
Conversley: Chords that are equidistant from the centre
are equal
A
C
B
Quick
Quiz
a
a= 40
40
40
b= 80
C
b
d
60
C
d= 120
f
f= 55
55
C
m= 62
62
m
C
e
e= 90
C
12 cm
x= 12
102
C
102
x cm
k
k= 35
C
70
a
a= 50
120
10
x= 50
C
100
x
y= 55
y
C
35 
Quick
Quiz
Which quadrilateral is concyclic?
A
105
75
answer= A
110
B
C
100
20
140
c = 60
60
C
c
g
g= 90
C
4cm
h cm
h= 4
Tangent
C
y
m =50
y = 50
C
40
m
Q
P
50 
a= 65
C
a
R
PQ, RQ are tangents
10
n= 5
n
C
nx8=4x10
4
8
8n =40
n =5
q= 25
C
2
q
4q=10
4q=100
4
10
q=25
2
BA =BC.BD
x= 12
4(4+x)=8
C
2
x
4(4+x)=64
4
8
4+x=16
x=12
k= 5
C
K2=32+42
k
K =5