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Circle Properties
Part I
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A circle is a set of all points in a plane thatare the same distance from a fixed point in
a plane
The set of points form the .Circumference
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The line joining the centre of a circle and apoint on the circumference is called
the.Radius
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A is a straight line segment joiningtwo points on the circle
chord
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A chord that passes through the centre is a
.diameter
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A is a straight line thatcuts the circle in two points
secant
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An arc is part of the circumference of a circle
Major arc
Minor arc
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A is part of the circlebounded by two radii and an arc
sector
Minorsector
majorsector
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A is part of the circlebounded by a chord and an arc
segment
Minor
segment
majorsegment
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The arc AB subtends an angle of at thecentre of the circle.
AB
O
Subtendsmeans to extend under or to beopposite to
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Instructions:
Draw a circle
Draw two chords of equal length
Measure angles AOB and DOC
A
B
C
D
O
What do you notice?
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Equal chords subtend equal angles at the centre
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Conversely
Equal angles at the centre of a circle stand on equal arcs
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Instructions:
select an arc AB
subtend the arc AB to the centre O
subtend the arc AB to a point C on the circumference
Measure angles AOB and ACB
B
O
A
C
What do you notice?
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Instructions:
select an arc AB
subtend the arc AB to the centre O
subtend the arc AB to a point C on the circumference
Measure angles AOB and ACB
B
O
A
C
What do you notice?
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2
The angle that an arc of a circle subtendsat the centre is twice the angle it subtends
at the circumference
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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
Measure angles ACB and ADB
B
O
A
C
D
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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
Measure angles ACB and ADB
B
O
A
C
D
What do you notice?
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Angles subtended at the circumference by thesame arc are equal
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Instructions:
Draw a circle and its diameter
subtend the diameter to a point on the circumferenceMeasure ACB
C
B
What do you notice?
A
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An angle in a semicircle isa right angle
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Instructions:
Draw a cyclic quadrilateral (the vertices of thequadrilateral lie on the circumference
Measure all four angles
What do you notice?
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180-
The opposite angles of a cyclic quadrilateralare supplementary
180-
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180-
If the opposite angles of a quadrilateral aresupplementary the quadrilateral is cyclic
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Instructions:
Draw a cyclic quadrilateral
Produce a side of the quadrilateral
Measure angles and
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If a side of a cyclic quadrilateral is produced,the exterior angle is equal to the interior
opposite angle
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Circle Properties
Part II tangent properties
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A tangent to a circle is a straight line that touches thecircle in one point only
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Tangent to a circle is perpendicular to the
radius drawn from the point of contact.
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Tangents to a circle from an exterior point are equal
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When two circles touch, the line through their centres
passes through their point of contact
Point of contact
External Contact
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When two circles touch, the line through their centres
passes through their point of contact
Point of contact
Internal Contact
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The angle between a tangent
and a chord through the point of contact
is equal to the angle in the alternate segment
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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
BA2
=BC.BD
C
D
B=external point
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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
BA2
=BC.BD
C
D
Note: B is the crucial point in the formula
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Circle Properties
Chord properties
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A
B
C
D
X
AX.XB=CX.XD
Triangle AXD is similar to triangle CXBhence
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A
B
C
D
X
AX.XB=CX.XD
Note: X is the crucial point in the formula
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Chord AB and CD intersect at X
Prove AX.XB=CX.XD
A
B
C
D
X
In AXD and CXB
AXD = CXB (Vertically Opposite Angles)
DAX = BCX (Angles standing on same arc)
ADX = CBX (Angles standing on same arc)
AXD CXB
Hence (Equiangular )XB
CX
XD
AX
XDCXXBAX .. AAA test for similar triangles
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A
B
C
A perpendicular line from the centre off a circle to achord bisects the chord
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A
B
C
Conversley: A line from the centre of a circle thatbisects a chord is perpendicular to the chord
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A
B
C
Equal chords are equidistant from the centre of the circle
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A
B
C
Conversley: Chords that are equidistant from the centreare equal
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Quick
Quiz
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a
40a= 40
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b
40
b= 80
C
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d
60 d= 120
C
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f
55 f= 55
C
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m=62
C
62m
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e
e= 90
C
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x= 12
C
102102
12 cm
x cm
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k
70
k= 35
C
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a
120a= 50
10
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x100
x= 50
C
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y y= 55
C
35
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QuickQuiz
Which quadrilateral is concyclic?
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answer= A
10575
Which quadrilateral is concyclic?A
B
C
100
110
20
140
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c
60 c= 60
C
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g
g= 90
C
4cm
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h= 4
C
4cm
h cm
Tan
gent
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m
40
m=50
C
y=50
y
Q
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a= 65
C
50 Q
a
P
R
PQ, RQ are tangents
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n= 5
C
10
4
8
n
nx8=4x10
8n =40
n =5
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q= 25
C
10
4
q 4q=102
4q=100
q=25
BA2
BC BD
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x= 12
C
8
4
x
4(4+x)=82
4(4+x)=64
4+x=16
x=12
BA2=BC.BD
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k= 5
C
k
K2=32+42
K =5
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