chapter 9 circles
DESCRIPTION
Chapter 9 Circles. Define a circle and a sphere . Apply the theorems that relate tangent s, chords and radii . Define and apply the properties of central angles and arcs. Bring a Compass Tomorrow. 9.1 Basic Terms. Objectives Define and apply the terms that describe a circle. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/1.jpg)
Chapter 9Circles
• Define a circle and a sphere.
• Apply the theorems that relate tangents, chords and radii.
• Define and apply the properties of central angles and arcs.
![Page 2: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/2.jpg)
Bring a Compass Tomorrow
![Page 3: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/3.jpg)
9.1 Basic Terms
Objectives
• Define and apply the terms that describe a circle.
![Page 4: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/4.jpg)
The Circle
is a set of points in a plane equidistant from a given point.
A
B
![Page 5: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/5.jpg)
The Circle
The given distance is a radius (plural radii)
A
B
radius
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The Circle
The given point is the center
A
B
radius
center
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The Circle
A
BPoint on circle
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Chord
any segment whose endpoints are on the circle.
A
BC
chord
![Page 9: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/9.jpg)
Diameter
A chord that contains the center of the circle
A
BC
diameter
![Page 10: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/10.jpg)
any line that contains a chord of a circle.
Secant
A
BC
secant
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Tangent
any line that contains exactly one point on the circle.
A
B
tangent
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Point of Tangency
A
BPoint of tangency
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Sphere
is the set of all points equidistant from a given point.
AB
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Sphere
Radii
Diameter
Chord
Secant
TangentA
B
D
C
E
F
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Congruent Circles (or Spheres)
have equal radii.
A D
BE
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Concentric Circles (or Spheres)
share the same center.
O
G
Q
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Inscribed/Circumscribed
A polygon is inscribed in a circle and the circle is circumscribed about the polygon if each vertex of the polygon lies on the circle.
![Page 18: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/18.jpg)
P
M
Q
O
N
R
L
Name each segment
![Page 19: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/19.jpg)
P
M
Q
O
N
R
L
OM
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P
M
Q
O
N
R
L
MN
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P
M
Q
O
N
R
L
MN
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P
M
Q
O
N
R
L
MQ
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P
M
Q
O
N
R
L
ML
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P
M
Q
O
N
R
L
ML
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P
M
Q
O
N
R
L
Point M
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9.2 Tangents
Objectives
• Apply the theorems that relate tangents and radii
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TheoremIf a line is tangent to a circle, then the line is perpendicular to the radius
drawn to the point of tangency.
A
B
tangent
C
90m ABC Sketch
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Corollary
Tangents to a circle from a common point are congruent.
A
X
Y
ZXY XZSketch
tangent
tangent
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Theorem
If a line in the plane of a circle is perpendicular to a radius at its endpoint, then the line is a tangent to the circle.
AX
B
tangent
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Inscribed/Circumscribed
When each side of a polygon is tangent to a circle, the polygon is said to be circumscribed about the circle and the circle is inscribed in the polygon.
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White Board Practice
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Common Tangents
are lines tangent to more than one coplanar circle.
A
X
B
tangentR
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Common External Tangents
A
XB
R
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Common External Tangents
A
X
B
R
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Common Internal Tangents
A
X
B
R
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Common Internal Tangents
A
X
B
R
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Construction 8Given a point on a circle, construct the tangent to the circle through the point.
Given:
Construct:
Steps:
with point A Btangent line l to through A B
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Remote Time
• How many common external tangents can be drawn?
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Remote Time
• How many common external tangents can be drawn?
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Remote Time
• How many common external tangents can be drawn?
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Remote Time
• How many common external tangents can be drawn?
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Remote Time
• How many common external tangents can be drawn?
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Remote Time
• How many common external tangents can be drawn?
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Remote Time
• How many common internal tangents can be drawn?
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Remote Time
• How many common internal tangents can be drawn?
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Remote Time
• How many common internal tangents can be drawn?
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Remote Time
• How many common internal tangents can be drawn?
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Remote Time
• How many common internal tangents can be drawn?
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Remote Time
• How many common internal tangents can be drawn?
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Tangent Circles
are circles that are tangent to each other.
A
B
R
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Externally Tangent Circles
A
B
R
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Internally Tangent Circles
A
B
R
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Remote Time
• Are the circlesA. Externally Tangent
B. Internally Tangent
C. None
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Remote Time
• Are the circlesA.Externally Tangent
B.Internally Tangent
C.None
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Remote Time
• Are the circlesA.Externally Tangent
B.Internally Tangent
C.None
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Remote Time
• Are the circlesA.Externally Tangent
B.Internally Tangent
C.None
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Remote Time
• Are the circlesA.Externally Tangent
B.Internally Tangent
C.None
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Remote Time
• Are the circlesA.Externally Tangent
B.Internally Tangent
C.None
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9.3 Arcs and Central Angles
Objectives
• Define and apply the properties of arcs and central angles.
![Page 60: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/60.jpg)
Central Angle
is formed by two radii, with the center of the circle as the vertex.
B
A C
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Arc
an arc is part of a circle like a segment is part of a line.
B
AC
AC
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Arc Measure
the measure of an arc is given by the measure of its central angle.
B
AC
80
80
AC
80mAC
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Minor Arc
an unbroken part of a circle with a measure less than 180°.
B
AC
AC
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Semicircle
an unbroken part of a circle that shares endpoints with a diameter.
B
A C
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Major Arc
an unbroken part of a circle with a measure greater than 180°.
BA C
D
ACD
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Adjacent Arcs
arcs that have exactly one point in common.
B
A C
D
AD DC
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Arc Addition Postulate
The measure of the arc formed by two adjacent arcs is the sum of the arcs.
B
A C
D
Sketch
mADCmDCmAD
![Page 68: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/68.jpg)
Congruent Arcs
arcs in the same circle or in congruent circles that have the same measure.
B
A C
D90
90
DCAD
mDCmAD
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White Board Practice
Name two minor arcs
R
C
SA
O
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White Board Practice
AR, RC, RS, AS, SC
R
C
SA
O
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White Board Practice
Name two major arcs
R
C
SA
O
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White Board Practice
ARS, ACR, RCS, RSA, RSC, CRS, CSR
R
C
SA
O
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White Board Practice
Name two semicircles
R
C
SA
O
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White Board Practice
ARC, ASC
R
C
SA
O
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White Board Practice
Name an acute central angle
R
C
SA
O
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White Board Practice
AOR
R
C
SA
O
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Theorem
In the same circle or in congruent circles, two minor arcs are congruent only if their central angles are congruent.
B
A C
D
90 90DCAD
DBCABD
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White Board Practice
Name two congruent arcs
R
C
SA
O
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White Board Practice
ARC, ASC
R
C
SA
O
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Group Practice
• Give the measure of each arc.
4x
3x 3x + 10
2x
2x-1
4
A
B
C
D
E
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Group Practice
m AB = 88
m BC = 52
m CD = 38
m DE = 104
m EA = 784x
3x 3x + 10
2x
2x-1
4
A
B
C
D
E
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The radius of the Earth is about 6400 km.
6400
6400
O
BA
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The latitude of the Arctic Circle is 66.6º North. That means the m BE 66.6º.
6400
6400
O
BA
EW
66.6º
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Find the radius of the Arctic Circle
6400
O
BA
EW
66.6º
xº
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Find the radius of the Arctic Circle
6400
O
BA
EW
66.6º
23.4º
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Lecture 4 (9-4)
Objectives
• Define the relationships between arcs and chords.
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Chord of the ArcThe minor arc between the endpoints of a
chord is called the arc of the chord, and the chord between the endpoints of an arc is the chord of the arc.
BA
D
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Theorem 9-4
Sketch
In the same circle or in congruent circles, congruent arc have congruent chords and congruent chords have congruent arcs.
B AC
D
BD DC
BD DC
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Theorem 9-5
Sketch
A diameter that is perpendicular to a chord bisects the chord and its arc.
B
AC
DX
Y
DC BY
DX XC
DY YC
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Theorem 9-6
Sketch
In the same circle or in congruent circles, chords are equally distant from the center only if they are congruent.
B
AC
D
X
YA XA
BD EC
Y
E
![Page 91: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/91.jpg)
9.5 Inscribed Angles
Objectives• Solve problems and
prove statements about inscribed angles.
• Solve problems and prove statements about angles formed by chords, secants and tangents.
![Page 92: Chapter 9 Circles](https://reader035.vdocuments.net/reader035/viewer/2022081504/56812ac6550346895d8e9f36/html5/thumbnails/92.jpg)
Inscribed Angle
B
A
C
An angle formed by two chords or secant lines whose vertex lies on the circle.
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Theorem
B
A
C
The measure of an inscribed angle is half the measure of the intercepted arc.
mACABCm2
1
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Corollary
B
A
C
If two inscribed angles intercept the same arc, then they are congruent.
ABC ADC
Sketch
D
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Corollary
C
A
An angle inscribed in a semicircle is a right angle.
90m ABC
B
O
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Corollary
C
A
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
B
O D180
180
m A m C
m B m D
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An angle formed by a chord and a tangent has a measure equal to half of the intercepted arc.
Theorem
C
A
B
O
D
mADBABCm2
1
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Construction 9Given a point outside a circle, construct the tangent to the circle through the point.
Given:
Construct:
Steps:
with point A Btangent line l to through A B
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9.6 Other Angles
Objectives
• Solve problems and prove statements involving angles formed by chords, secants and tangents.
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TheoremThe angle formed by two intersecting chords
is equal to half the sum of the intercepted arcs.
A
D
B
C
E
1)(
2
11 mDEmCBm
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TheoremThe angle formed by secants or tangents with the
vertex outside the circle has a measure equal to half the difference of the intercepted arcs.
A
D
B
CE
1
F
)(2
11 mEFmBDm
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AO
G
F
D
E
CB
123
45
6
7
8
AB is tangent to circle O.AF is a diameterm AG = 100m CE = 30m EF = 25
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9.7 Circles and Lengths of Segments
Objectives
• Solve problems about the lengths of chords, secants and tangents.
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TheoremWhen two chords intersect, the product of
their segments is equal.
A
D
B
XE
F
XBFXXDEX
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TheoremWhen two secant segments are drawn to a circle
from a common point, the product of their length times their external segments is equal.
A
D
B
CE
1
F
CFCDCECB
Whole Piece Outside Piece = Whole Piece Outside Piece
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TheoremWhen a secant and a tangent are drawn from a
common point, the product of the secant and its external segment is equal to the tangent squared.
A
D
C
E
FCECECFCD
Whole Piece Outside Piece = Whole Piece Outside Piece