mms9 workbook 08 unit8
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MMS9_Prep book_Unit8.1.qxp
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UNIT
8
Circle Geometry
What You’ll Learn How to • Solve problems involving tangents to a circle • Solve problems involving chords of a circle • Solve problems involving the measures of angles in a circle
Why Is It Important? Circle properties are used by • artists, when they create designs and logos
Key Words radius (radii) right angle tangent point of tangency diameter right triangle isosceles triangle
chord perpendicular bisector central angle inscribed angle arc subtended semicircle
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MMS9_Prep book_Unit8.1.qxp
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8.1 Skill Builder Solving for Unknown Measures in Triangles Here are 2 ways to find unknown measures in triangles. Angle Sum Property Pythagorean Theorem In any triangle: In any right 䉭PQR:
P
q
b°
r
c° a°
R
a° b° c° 180°
Q
p
q2 p2 r 2
Here is how to find the unknown measures in right 䉭PQR. In 䉭PQR, the angles add up to 180°. To find x°, start at 180° and subtract the known measures. x° 180° 90° 60° 30°
P q R
7 cm
60°
x°
8 cm
Q
By the Pythagorean Theorem: QR2 PR2 PQ2 82 q2 72 So: q2 82 72 Answer to the same degree of accuracy as the question uses.
q 冪 82 72 ⬟ 3.87 So, x° is 30° and q is about 4 cm.
Check 1. Find each unknown measure. a)
b)
x°
3.0 cm 5.0 cm
x 60° 50°
x° 180° ______ ______ ______ _
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x2 ______ ______ x冪 ⬟
So, x is __________________.
MMS9_Prep book_Unit8.1.qxp
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8.1 Properties of Tangents to a Circle FOCUS Use the relationship between tangents and radii to solve problems. A tangent touches a circle at exactly one point. Tangent Radius O
Point of tangency
Tangent-Radius Property B
A tangent to a circle is perpendicular to the radius drawn to the point of tangency. A So, OP ⬜ AB, ⬔OPA 90° and ⬔OPB 90°
Example 1
P
⬜ means “perpendicular to”.
O
Finding the Measure of an Angle in a Triangle
BP is tangent to the circle at P. O is the centre of the circle. Find the measure of x°.
P
x°
B
50° O
Solution By the tangent-radius property: ⬔OPB 90° Since the sum of the angles in 䉭OPB is 180°: x° 180° 90° 50° 40° So, x° is 40°.
Check 1. Find the value of x°. ⬔_______ 90° x° 180° _____ _____ _____
O
65°
x°
D
C
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MMS9_Prep book_Unit8.1.qxp
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Example 2
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Using the Pythagorean Theorem in a Circle
MB is a tangent to the circle at B. O is the centre. Find the length of radius OB.
B r O 10 cm
Solution By the tangent-radius property: ⬔OBM 90° By the Pythagorean Theorem in right 䉭MOB: OM2 OB2 BM2 102 r 2 82 100 r 2 64 100 64 r 2 36 r 2 冪 36 r r6 Radius OB has length 6 cm.
Check 1. ST is a tangent to the circle at S. O is the centre. Find the length of radius OS. Answer to the nearest millimetre. S r
12 cm
O 16 cm
T
⬔OST ______
By the tangent-radius property
OT2 ______ ______ ______ r 2 ______ ______ r 2 ______ ______ ______ r 2 ______ r 2 冪
r r ⬟ ______
OS is about ______ cm long.
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By the Pythagorean Theorem
8 cm M
MMS9_Prep book_Unit8.1.qxp
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Practice In each question, O is the centre of the circle. 1. From the diagram, identify: B C
J
A H
O E F D G
a) 3 radii
_______, _______, _______
b) 2 tangents
_______, _______
c) 2 points of tangency
_______, _______
d) 4 right angles
⬔_______, ⬔_______, ⬔_______, ⬔_______
2. What is the measure of each angle? a)
b)
B
Q
P
O
O
P
R
⬔OBP _____
⬔PQO _____
⬔PRO _____
3. Find each value of x°. a)
b)
W
x°
35°
O
T
T
x°
O
25°
M
⬔OTW ______ x° 180° ______ ______ ______
____________________________ ____________________________ ______
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MMS9_Prep book_Unit8.1.qxp
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4. Find each value of x. Answer to the nearest tenth of a unit. a) x
C
O
10 km 12 km T
⬔OCT 90°
By the tangent-radius property By the Pythagorean Theorem in 䉭OCT
______ x 2 ______ ______ x 2 ______ ______ ______ x 2 _____ x 2 冪
x x ⬟ _____
So, OC is about _____ km. b)
c)
P 6 cm
4 cm O
x
O
Q
x
P
15 cm
15 cm M
⬔OPQ ______, and:
x 2 ______ ______
x 2 ______ ______
x 2 ______ ______
x 2 ______ ______
x 2 ______
x 2 ______ x冪 x ⬟ ______ So, OQ is about ______ cm.
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x冪 x ⬟ ______ So, OP is about ______ cm.
MMS9_Prep book_Unit8.2.qxp
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8.2 Properties of Chords in a Circle FOCUS Use chords and related radii to solve problems. A chord of a circle joins 2 points on the circle. Radius
Chord O
Diameter
Chord Properties In any circle with centre 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 centre O.
Example 1
A
C
B O
Finding the Measure of Angles in a Triangle
Find x°, y°, and z°.
z°
O 30°
y° x°
B C
A
Solution OC bisects chord AB, so OC ⬜ AB Therefore, x° 90° By the angle sum property in 䉭OAC: y° 180° 90° 30° 60° Since radii are equal, OA OB, and 䉭OAB is isosceles. ⬔OBA ⬔OAB So, z° 30°
z°
O 30°
y° x°
B C
A
In an isosceles triangle, 2 base angles are equal.
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Check 1. Find the values of x° and y°.
O y°
B x°
40°
C
A
_____ ⬜ _____
So, x° _____
y° _____ _____ _____
By the chord properties By the angle sum property
_____ 2. Find the values of x°, y°, and z°. A
° C x
55°
y°
O
z° B
_____ ⬜ _____
So, x° _____
y° _____ _____ _____ _____ Since OA _____, 䉭_____ is isosceles and ⬔_____ ⬔_____ So, z° _____
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By the chord properties By the angle sum property
MMS9_Prep book_Unit8.2.qxp
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Example 2
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Using the Pythagorean Theorem in a Circle
O is the centre of the circle. Find the length of chord AB.
O 6 cm A
10 cm B
C
Solution
O 6 cm A
10 cm
C
102 62 BC2 100 36 BC2 100 36 BC2 64 BC2 BC 冪 64 8 So, BC 8 cm
B
By the Pythagorean Theorem in right 䉭OCB
Since OC ⬜ AB, OC bisects AB. By the chord properties So, AC BC 8 cm The length of chord AB is: 2 8 cm 16 cm
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Check 1. Find the values of a and b.
O 5 cm E
b
G
13 cm F
a
By the Pythagorean Theorem in right 䉭OFG
_____ _____ a2 ___________________ ___________________ ___________________ ___________________ ___________________ So, a _____ cm _____ _____
By the chord properties
So, b _____ cm
Practice In each diagram, O is the centre of the circle. 1. Name all radii, chords, and diameters. a)
A
D
O
B
R
P
C
Radii: _________________________ Chords: _______________________ Diameters: ____________________
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b)
T
O
S
Q
N
Radii: _________________________ Chords: _______________________ Diameters: ____________________
MMS9_Prep book_Unit8.2.qxp
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2. On each diagram, mark line segments with equal lengths. Then find each value of a. a)
A
b)
a
M
P
3 cm
N
4 cm O
C
O
a B
AC CB ______ cm So, a ______ cm
c)
MN 2 ______ 2 ______ cm ______ cm So, a ______ d)
K
O 18 cm a
1.5 cm
O T
a S
M L
OL 12 ______ 12 ______ cm
OS ______ ______ cm So, a _______
______ cm So, a ______
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MMS9_Prep book_Unit8.2.qxp
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3. Find each value of x° and y°. a)
b)
O y° 45°
A
P
x° C
35°
O y° x°
B
Q
x° ______
______ ______ 䉭OPQ is ________________
y° 180° ______ ______
⬔______ ⬔______
______
So, x° ______ y° 180° ______ ______ ______
4. Find the length of chord BC. ______ ______ DB2
By the Pythagorean Theorem
______ ______ DB2
___________________ O
___________________ 15 cm
___________________
12 cm B D
C
___________________ So, DB ______ cm ______ ______ ______ cm By the chord properties
So, chord BC has length: 2 ______ cm ______ cm 1 AN 2 ______ 12 ______ cm
5. Find ON.
By the chord properties
______ cm ______ ______ ON2 17 cm
______ ______ ON2
O
A
____________________
N 30 cm B
____________________ ____________________ ____________________ ____________________ So, ON is ______ cm.
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By the Pythagorean Theorem
MMS9_Prep book_Check Point.qxp
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Can you …
CHECKPOINT
• Solve problems using tangent properties? • Solve problems using chord properties?
8.1
In each diagram, O is the centre of the circle. Assume that lines that appear to be tangent are tangent. 1. Name the angles that measure 90°. a)
b)
G
M E
O
O M
N
2. Find the unknown angle measures. a)
b)
O
O
75° p° M
q°
s°
A
N
p° ⫽ ______
Tangent-radius property
q° ⫽ 180° ⫺ ______ ⫺ ______
Angle sum property
21°
q° ⫽ ______
B
_______ ⫽ 90° s° ⫽ ______ ⫺ ______ ⫺ ______ s° ⫽ ______
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3. Find the values of a and b to the nearest tenth. P
25 cm
a) Q
b)
20 cm
a
b
B
O
O 15 cm
20 cm
C
⬔OPQ ⫽ ______
⬔OBC ⫽ ______
By the tangent-radius property
OQ is _________________ of 䉭OPQ.
OB is _______ of 䉭OBC.
a2 ⫽ ______ ⫹ ______
______ ⫽ b2 ⫹ ______
_______________________
By the Pythagorean Theorem
_______________________
_______________________
_______________________
_______________________
_______________________
So, a ⬟ ______ cm
_______________________ So, b ⬟ ______ cm
8.2
4. Find the unknown measures. a)
b)
X
d M
b° O
L
6 cm
N
O
35°
Y
OX ⫽ ______
MN ⫽ 2 ⫻ ______
䉭OXY is ___________.
MN ⫽ ______ ⫻ ______ cm
So, b° ⫽ ______
⫽ ______ cm So, d ⫽ ______ cm
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MMS9_Prep book_Check Point.qxp
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5. Find each value of x°, y°, and z°. B z°
x°
C
O
y°
65°
x° ⫽ ______ By the chord properties y° ⫽ ______ ⫺ ______ ⫺ ______ By the angle sum property ⫽ ______ ______ ⫽ ______, so ______ is isosceles. ⬔______ ⫽ ⬔______ So, z° ⫽ ______
A
6. Find the length of OP.
O
10 cm Q P 16 cm
R
QP ⫽ 1 ⫻ QR ⫽
2 1 2
By the chord properties
⫻ ______ cm
⫽ ______ cm OQ2 ⫽ ______ ⫹ OP2
By the Pythagorean Theorem
______ ⫽ ______ ⫹ OP2 ____________________ ____________________ ____________________ ____________________ ⫽ ______ So, the length of OP is ______ cm.
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MMS9_Prep book_Unit8.3.qxp
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8.3 Properties of Angles in a Circle FOCUS Use inscribed angles and central angles to solve problems. Inscribed angle ∠ACB
C
In a circle: • A central angle has its vertex at the centre.
Central angle ∠AOB
O
• An inscribed angle has its vertex on the circle. Both angles in the diagram are subtended by arc AB.
B
A
Arc AB
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. B x° O C 2x°
A
So, ⬔AOC 2⬔ABC, or ⬔ABC 1 ⬔AOC 2
Inscribed Angles Property Inscribed angles subtended by the same arc are equal. So, ⬔ACB ⬔ADB ⬔AEB
C x° D
E
x°
B
O
x° A
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MMS9_Prep book_Unit8.3.qxp
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Example 1
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Using Inscribed and Central Angles
Find the values of x° and y°. D y° C
22°
O x°
B
A
Solution D
y° C
22°
O x°
A
B
Central ⬔AOB and inscribed ⬔ACB are both subtended by arc AB. So, ⬔AOB 2⬔ACB x° 2 22° 44° ⬔ACB and ⬔ADB are inscribed angles subtended by the same arc AB. So, ⬔ADB ⬔ACB y° 22°
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Check 1. Find each value of x°. a)
b)
C
c)
C
25°
D C
x°
O
x° O
O
x°
28°
62° B
A
B
B A
A
⬔AOB 2 ⬔ACB
⬔ACB 1 ________ 2
x° 2 ______
x° 2 ______
⬔ADB ________
1
______
x° ______
______
2. Find the values of x° and y°. a)
T
b)
y°
C x°
P
D S
30°
O
x°
O
y°
70°
Q
B
⬔QOP 2 ⬔QSP
A
x° 2 ______
⬔ACB 1 ________ 2
x° ______
x° 2 ______
⬔QTP ________ y° ______
1
x° ______ ⬔ADB ________ y° ______
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MMS9_Prep book_Unit8.3.qxp
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Angles in a Semicircle Property Inscribed angles subtended by a semicircle are right angles. ⬔AFB ⬔AGB ⬔AHB 90°
H G F
B O
A
Example 2
Finding Angles in an Inscribed Triangle
Find x° and y °. N 40°
I
O
x°
y° M
Solution ⬔MIN is an inscribed angle subtended by a semicircle. So, x° 90° By the angle sum property in 䉭MIN
y° 180° 90° 40° 50°
Check 1. Find the values of x° and y °. x°
60°
O
x° ______ y ° 180° ______ ______ y ° ______
y°
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MMS9_Prep book_Unit8.3.qxp
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Practice 1. Name the following from the diagram. C
a) the central angle subtended by arc CB: ⬔_______ b) the central angle and inscribed angle subtended by arc AD: ⬔_______ and ⬔_______
A
O
c) the inscribed angle subtended by a semicircle: ⬔_______ D
d) the right angle: ⬔_______ 2. In each circle, name a central angle and an inscribed angle subtended by the same arc. Shade the arc. a)
b)
C
Q
R
O A
O
P
B
Central angle: ⬔_______ Inscribed angle: ⬔_______
Central angle: ⬔_______ Inscribed angle: ⬔_______
3. Determine each indicated measure. a)
G
b)
F x°
O 84°
O
x°
S
T
28°
R H
⬔TSR 1 ⬔_______
⬔GOF 2 ⬔GHF
2
x° 2 _______
x°
_______ E
O
x°
F x° G
⬔DEG _______
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L
d)
72°
x° _______
_______
x° _______
c) D
1 2
J O
K
x° _______
B
MMS9_Prep book_Unit8.3.qxp
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4. Determine each value of x° and y°. x°
a)
34° O y°
x° _______ y° _______ b) y° O 15°
x°
x° _______ _______ _______ y° _______ 5. Find the value of x° and y°. B
⬔ACB _______
x° O 25°
x° 180° _______ _______ C
y°
By the angle sum property
_______ y° _______ _______
A
_______ 6. Find the value of x°, y °, and z °.
z° B
A
⬔AOB 2 _______
y°
x° 2 _______ _______ In 䉭OAB, _______ _______
x° O
䉭OAB is ________________.
50° C
In 䉭OAB: y° z ° y° y° _______ _______
By the angle sum property
2y° _______ y°
2
So, y° _______ and z° _______
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MMS9_Prep book_UnitBM.qxp
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Unit 8 Puzzle Circle Geometry Word Search
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Find these words in the puzzle above. You can move in any direction to find the entire word. A letter may be used in more than one word. degrees
radius
perpendicular
centre
inscribed angle
bisect
diameter
central angle
circle
tangent
chord
triangle
circumference
point
angle
arc
isosceles
equal
point of tangency
semicircle
subtended
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Unit 8 Study Guide Skill
Description
Recognize and apply tangent properties
Example x°
6 cm
O B
53°
O
8 cm y
P A
⬔APO ⬔BPO 90° Recognize and apply chord properties in circles
O y° 10 5 x° M
O
A
K
C
B
If OB ⬜ AC, then AB CB. If AB CB, then OB ⬜ AC. Recognize and apply angle properties in a circle
x ° 90°
L
x ° 90° and y ° 60° ML2 102 52
• Inscribed and central angles B
30°
50°
O
x° y°
z°
O
A
C
⬔BOC 2⬔BAC, or ⬔BAC 1 ⬔BOC
x ° 90° y ° 50° z° 100°
2
• Inscribed angles C D O
B
E A
⬔ACB ⬔ADB ⬔AEB • Angles on a semicircle E D A
C B
O
⬔ACB ⬔ADB ⬔AEB 90°
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Unit 8 Review 8.1
1. Find each value of x ° and y °. Segments RS and MN are tangents. R
a)
b)
x°
M y°
x°
O
O 75°
20° S
N
x ° ______ y ° 180° ______ ______ ______
⬔ONM ______ x ° 180° ______ ______ ______
2. Find each value of x to the nearest tenth. Segments GH and ST are tangents. a) x
4 cm
13 cm
O 15 cm
O
G
12 cm
x
⬔OHG ______
⬔OST ______
______ x 2 ______
______________________
______________________
______________________
______________________
______________________
______________________
______________________
______________________
So, x ⬟ ________ cm
So, x ⬟ ________ cm
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S
b)
H
T
MMS9_Prep book_UnitBM.qxp
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3. Find the values of x ° and y °.
O 25° A
x ° ______
By the chord properties
y ° ______ ______ ______
By the angle sum property
__________________________
y° x°
y ° ______
B C
4. Find the values of x °, y °, and z °. x ° ______
By the ____________________
OM ON, so 䉭______ is isosceles. ⬔ONP ⬔OMP
O z° M
27° x° P
y°
So, y ° ______
N
z ° ______ ______ ______
__________________________
z ° ______ 5. Find the length of the radius of the circle to the nearest tenth. XY 1 ______ 2 X
12 cm Y 2 cm O
Z
1 ______ cm 2 ______ cm Draw radius OX. OX2 ______ XY2 OX2 ______ ______ ___________________________ ___________________________ ___________________________ OX ⬟ ______ The radius is about ______ cm.
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MMS9_Prep book_UnitBM.qxp
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6. Find each value of x °. a)
b)
c) x°
14°
O
43°
O
O
x°
x ° 2 ______
x°
x ° ______
x ° ______
x ° ______ 7. Find each value of x ° and y °. a)
b) 25° y°
y°
O 35°
O x°
x° 35°
x ° ______
x ° 2 ______
y ° ______
______ y ° ______
8. Find the value of w°, x °, y °, and z °. A
x ° y ° ______
25°
z ° ______ ______ ______
w° O C
y°
x° B z° D
________________________ z° ______ 䉭ACD is ____________. So, ⬔CDA ⬔CAD w ° w ° w ° ______ ______ 2w ° ______ w°
2
w ° ______
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By the angle sum property
By the angle sum in 䉭ACD
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