circle properties part i. a circle is a set of all points in a plane that are the same distance from...
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![Page 1: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/1.jpg)
Circle Circle PropertiesProperties
Part I
![Page 2: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/2.jpg)
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
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The line joining the centre of a circle and a point on the circumference is called
the……………….Radius
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A is a straight line segment joining two points on the circle
chord
![Page 5: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/5.jpg)
A chord that passes through the centre is a
……………………….
diameter
![Page 6: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/6.jpg)
A……………………… is a straight line that cuts the circle in two points
secant
![Page 7: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/7.jpg)
An arc is part of the circumference of a circle
Major arc
Minor arc
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A ……………………is part of the circle bounded by two radii and an arc
sector
Minor sector
major sector
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A ……………………is part of the circle bounded by a chord and an arc
segment
Minor segment
major segment
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The arc AB subtends an angle of at the centre of the circle.
AB
O
Subtends means “to extend under” or “ to be opposite to”
![Page 11: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/11.jpg)
Instructions:
• Draw a circle
• Draw two chords of equal length
• Measure angles AOB and DOC
A
B
C
D
O
What do you notice?
![Page 12: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/12.jpg)
Equal chords subtend equal angles at the centre
![Page 13: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/13.jpg)
Conversely
Equal angles at the centre of a circle stand on equal arcs
![Page 14: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/14.jpg)
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?
![Page 15: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/15.jpg)
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?
![Page 16: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/16.jpg)
2
The angle that an arc of a circle subtends at the centre is twice the
angle it subtends at the circumference
![Page 17: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/18.jpg)
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?
![Page 19: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/19.jpg)
Angles subtended at the circumference by the same arc are equal
![Page 20: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/20.jpg)
Instructions:
• Draw a circle and its diameter
• subtend the diameter to a point on the circumference
•Measure ACB C
B
What do you notice?
A
![Page 21: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/21.jpg)
An angle in a semicircle is a right
angle
![Page 22: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/22.jpg)
γ
Instructions:
•Draw a cyclic quadrilateral (the vertices of the quadrilateral lie on the circumference
•Measure all four angles
β
What do you notice?
![Page 23: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/23.jpg)
180-
The opposite angles of a cyclic quadrilateral are supplementary
180-
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180-
If the opposite angles of a quadrilateral are supplementary the quadrilateral is cyclic
![Page 25: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/25.jpg)
β
Instructions:
• Draw a cyclic quadrilateral
• Produce a side of the quadrilateral
•Measure angles and β
![Page 26: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/26.jpg)
If a side of a cyclic quadrilateral is produced, the exterior angle is equal to
the interior opposite angle
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Circle Circle PropertiesProperties
Part II tangent properties
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A tangent to a circle is a straight line that touches A tangent to a circle is a straight line that touches the circle in one point onlythe circle in one point only
![Page 29: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/29.jpg)
Tangent to a circleTangent to a circleis perpendicular to is perpendicular to the the radius drawn from the point of contact.radius drawn from the point of contact.
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Tangents to a circleTangents to a circlefrom an exterior from an exterior pointpoint
are equalare equal
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When two circles touch,When two circles touch,the line through their the line through their centrescentres
passes through their point of contactpasses through their point of contact
Point of contact
External Contact
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When two circles touch,When two circles touch,the line through their the line through their centrescentres
passes through their point of contactpasses through their point of contact
Point of contact
Internal Contact
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The angle between a The angle between a tangent tangent and a chord through the point of and a chord through the point of
contact contact is equal to the angle in the alternate is equal to the angle in the alternate segmentsegment
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The square of the length of the tangentThe square of the length of the tangent
from an external point is equal tofrom an external point is equal to
the product of the intercepts of the secantthe product of the intercepts of the secant
passing through this pointpassing through this point
AA
BB
BA2=BC.BD
CC
DD
B=external point
![Page 35: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/35.jpg)
The square of the length of the tangentThe square of the length of the tangent
from an external point is equal tofrom an external point is equal to
the product of the intercepts of the secantthe product of the intercepts of the secant
passing through this pointpassing through this point
AA
BB
BA2=BC.BD
CC
DD
Note: B is the crucial point in the formula
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Circle Circle PropertiesProperties
Chord properties
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A
B
C
D
X
AX.XB=CX.XD
Triangle AXD is similar to triangle CXB hence
![Page 38: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/38.jpg)
A
B
C
D
X
AX.XB=CX.XD
Note: X is the crucial point in the formula
![Page 39: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/39.jpg)
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
![Page 40: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/40.jpg)
A
B
C
A perpendicular line from the centre off a circle A perpendicular line from the centre off a circle to a chord bisects the chord to a chord bisects the chord
![Page 41: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/41.jpg)
A
B
C
Conversley: A line from the centre of a circle Conversley: A line from the centre of a circle that bisects a chord is perpendicular to the that bisects a chord is perpendicular to the chord chord
![Page 42: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/42.jpg)
A
B
C
Equal chords are equidistant from the centre of Equal chords are equidistant from the centre of the circle the circle
![Page 43: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/43.jpg)
A
B
C
Conversley: Chords that are equidistant from the Conversley: Chords that are equidistant from the centre are equalcentre are equal
![Page 44: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/44.jpg)
Quick Quick QuizQuiz
![Page 45: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/45.jpg)
a
40
a= 40
![Page 46: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/46.jpg)
b
40
b= 80C
![Page 47: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/47.jpg)
d
60 d= 120C
![Page 48: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/48.jpg)
f
55 f= 55C
![Page 49: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/49.jpg)
m=62
C62
m
![Page 50: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/50.jpg)
e
e= 90C
![Page 51: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/51.jpg)
x= 12
C102
102
12 cm
x cm
![Page 52: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/52.jpg)
k
70
k= 35C
![Page 53: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/53.jpg)
a
120
a= 50
10
![Page 54: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/54.jpg)
x100
x= 50C
![Page 55: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/55.jpg)
y y= 55C
35
![Page 56: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/56.jpg)
![Page 57: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/57.jpg)
Quick Quick QuizQuiz
![Page 58: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/58.jpg)
answer=A
10575
Which quadrilateral is concyclic?
A
B
C
100
110
20
140
![Page 59: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/59.jpg)
c
60 c =60C
Tangent
![Page 60: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/60.jpg)
g
g= 90C
Tangent
![Page 61: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/61.jpg)
h= 4C
4cm
h cm
Tangent
Tangent
![Page 62: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/62.jpg)
m
40
m =50C
Tangent
y =50
y
![Page 63: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/63.jpg)
a= 65C
50
Q
a
P
R
PQ, RQ are tangents
![Page 64: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/64.jpg)
n= 5
C
10
4
8
n
nx8=4x10
8n =40
n =5
![Page 65: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/65.jpg)
q= 25
C
10
4
q4q=10
2
4q=100
q=25
![Page 66: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/66.jpg)
x= 12
C
8
4
x
4(4+x)=82
4(4+x)=64
4+x=16
x=12
BA2=BC.BD
![Page 67: 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](https://reader038.vdocuments.site/reader038/viewer/2022110402/56649e445503460f94b38f7d/html5/thumbnails/67.jpg)
k= 5
C
8m
k
3m
K2=32+42
K =5
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