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Unit 8: Circle Geometry
Grade 9 Math
Circle Geometry Glossary
Term
Circle
Definition
Illustration
Draw a circle. Measure the radius with
a ruler. Label the centre of the circle
“O”.
Radius
Diameter
Find the length of the diameter of this
circle:_____
Arc
Draw and label arc AC on this circle.
Minor Arc
Chord
Construct 3 chords in this circle. Make
one of the chords a diameter.
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Tangent Line
Draw a tangent line to the circle.
Point of Tangency
Identify a point of tangency in the
following diagram.
B●
C
●
●
A
Central Angle
Identify the inscribed angle:_____
Identify the central angle:______
Inscribed Angle
If the Central angle measures 110°,
what is the measure of the inscribed
angle?
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Section 8.1 Properties of Tangents to a Circle
Tangent-Radius Property
Example Problems
A) Point O is the center of a circle and AB is tangent to the
circle. In
OAB,
AOB = 550. Determine the measure of
OBA.
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Try to find the missing angles in the following diagrams
A)
Application Example
B)
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Using the Pythagorean Theorem in a Circle
1.
Try this one!
2.
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3. An airplane is cruising at an altitude of 9000m. A cross
section of the earth is a circle with a radius approximately
6400km. A passenger wonders how far she is from a point H on
the horizon she sees outside the window. Calculate the distance
to the nearest kilometer.
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Tangent Practice
(Round Answers to the nearest tenth)
1. In the diagram shown, point P is 26cm from the centre O
of the circle, and PT is tangent to the circle at T such that
PT  24cm . What is the diameter of the circle in
centimeters?
T
P
O
A
6
2. In this circle with centre O
and AB tangent to the circle,
what is the length of CB ?
3. If points Y and Z lie on the circle
with centre O as shown, and XY
is tangent to the circle at Y, what
is the length of XZ ?
8
C
O
O
5
Y
Z
12
X
B
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4.
are tangent to the circle with centre O as shown. If
and OC  12 , what is the length of DB ?
BA and BC
AB  35
A
35
D
O
B
12
C
5. In the circle with centre O shown, what is
the length of YZ if YX is tangent to the
circle at X?
O
Z
7
Y
24
X
6. In the circle with centre O shown, TS is tangent to the circle
at S. If the circle has diameter 4, and TS = 4, what is the
length of PT ?
T
S
P
O
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8.2 Properties of Chords in a Circle
In any circle with center O and chord AB:
 If OC bisects AB, then OC ┴ AB
 If OC ┴ AB, then AC = CB
 The perpendicular bisector of AB
goes through the center O.
Example # 1
O is the center of the circle.
Find the length of chord AB.
Example # 2
The diameter of a circle is 18cm.
A chord JK is 5cm from the center.
Find the length of the chord.
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Example # 3
A chord MN is 24cm. The radius of a circle is 20cm. Find the
length of x.
Example # 4:
Finding Angle
Measurements
x , y and z.
Try These
A).
B)
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Chord Practice (Round Answers to the
nearest tenth)
1.
P
O is the centre of this circle and R is the
midpoint of PQ . If the diameter of the circle
is 26 cm, and OR = 5 cm, what is the length
of the chord PQ in cm?
R
O
Q
A
2.
The circle with centre O shown, has diameter 30cm.
If a chord AC is 9cm from the centre, what is the
length of chord AC?
9
C
1
3
O
3.
In this circle with centre 0,
what is the length of chord
O
AB ?
5
B
A
D
14
4.
In this circle with centre O,
what is the length of chord
O
AC ?
2
A
B
C
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5.
In the circle with centre O as shown, the length of chord AB
is 20 cm and it is 5 cm from the centre. What is the diameter
of the circle in cm?
O
5
A
6.
In the circle with centre O shown, PA  4 and
the length of the diameter of the circle?
B
P
2
PO  2 .
4
B
What is
A
O
7.
The circle shown with centre O has a diameter of 20 cm. If
chord AB is 8 cm from the centre, find the length, in
centimeters, of AB .
O
8 cm
A
B
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Section 8.3 Properties of Angles in a Circle
Central Angle and Inscribed Angle Property
The measure of a central angle is twice the measure of an
inscribed angle subtended by the same arc.
Examples
#1
#2
#3
#1
Inscribed Angles Property
Inscribed angles subtended by the same arc are equal.
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Examples
#1
#2
#3
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Angles in a Semicircle Property
Inscribed angles subtended by a
semicircle (half the circle) are right
angles. This means these angles
use the diameter.
Example #1 Find the missing angle measures.
Example # 2 - Try this one!
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Worksheet on Angles in Circles(8.3)
1.
Given the circle shown, what is the
measure of y in degrees?
B
C
o
25
y
A
75o
D
2.
In the circle with centre O shown,
measure of angle C in degrees?
BAC is 50 .
What is the
C
B
O
3.
In the circle with centre O, what is
mDCE ?
A
A
D
E
O
60 °
50 °
B
C
4.
In the circle with centre O shown, C  44 . What is the
measure of minor arc AD in degrees?
C
4 °
O
A
D
B
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5.
In the circle with centre O shown, the measure of angle BOA
is 68°. What is the measure, in degrees, of angle A?
B
O
D
C
A
6.
In the circle with centre O shown, the measure of angle GOJ
is 140°. What is the measure, in degrees, of GHJ ?
H
x°
O
G
J
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7. A student looked at the diagram below and concluded that x =
y. The student justified that conclusion by saying that both
angles are subtended by arc AB.
What is the student’s error?
What are the values of x and y?
8. Point O is the centre of the circle; DB is a diameter.
Determine the values of w, x, y, and z.
Justify your solutions.