Post on 30-Dec-2015
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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.
Bring a Compass Tomorrow
9.1 Basic Terms
Objectives
• Define and apply the terms that describe a circle.
The Circle
is a set of points in a plane equidistant from a given point.
A
B
The Circle
The given distance is a radius (plural radii)
A
B
radius
The Circle
The given point is the center
A
B
radius
center
The Circle
A
BPoint on circle
Chord
any segment whose endpoints are on the circle.
A
BC
chord
Diameter
A chord that contains the center of the circle
A
BC
diameter
any line that contains a chord of a circle.
Secant
A
BC
secant
Tangent
any line that contains exactly one point on the circle.
A
B
tangent
Point of Tangency
A
BPoint of tangency
Sphere
is the set of all points equidistant from a given point.
AB
Sphere
Radii
Diameter
Chord
Secant
TangentA
B
D
C
E
F
Congruent Circles (or Spheres)
have equal radii.
A D
BE
Concentric Circles (or Spheres)
share the same center.
O
G
Q
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.
P
M
Q
O
N
R
L
Name each segment
P
M
Q
O
N
R
L
OM
P
M
Q
O
N
R
L
MN
P
M
Q
O
N
R
L
MN
P
M
Q
O
N
R
L
MQ
P
M
Q
O
N
R
L
ML
P
M
Q
O
N
R
L
ML
P
M
Q
O
N
R
L
Point M
9.2 Tangents
Objectives
• Apply the theorems that relate tangents and radii
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
Corollary
Tangents to a circle from a common point are congruent.
A
X
Y
ZXY XZSketch
tangent
tangent
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
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.
White Board Practice
Common Tangents
are lines tangent to more than one coplanar circle.
A
X
B
tangentR
Common External Tangents
A
XB
R
Common External Tangents
A
X
B
R
Common Internal Tangents
A
X
B
R
Common Internal Tangents
A
X
B
R
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
Remote Time
• How many common external tangents can be drawn?
Remote Time
• How many common external tangents can be drawn?
Remote Time
• How many common external tangents can be drawn?
Remote Time
• How many common external tangents can be drawn?
Remote Time
• How many common external tangents can be drawn?
Remote Time
• How many common external tangents can be drawn?
Remote Time
• How many common internal tangents can be drawn?
Remote Time
• How many common internal tangents can be drawn?
Remote Time
• How many common internal tangents can be drawn?
Remote Time
• How many common internal tangents can be drawn?
Remote Time
• How many common internal tangents can be drawn?
Remote Time
• How many common internal tangents can be drawn?
Tangent Circles
are circles that are tangent to each other.
A
B
R
Externally Tangent Circles
A
B
R
Internally Tangent Circles
A
B
R
Remote Time
• Are the circlesA. Externally Tangent
B. Internally Tangent
C. None
Remote Time
• Are the circlesA.Externally Tangent
B.Internally Tangent
C.None
Remote Time
• Are the circlesA.Externally Tangent
B.Internally Tangent
C.None
Remote Time
• Are the circlesA.Externally Tangent
B.Internally Tangent
C.None
Remote Time
• Are the circlesA.Externally Tangent
B.Internally Tangent
C.None
Remote Time
• Are the circlesA.Externally Tangent
B.Internally Tangent
C.None
9.3 Arcs and Central Angles
Objectives
• Define and apply the properties of arcs and central angles.
Central Angle
is formed by two radii, with the center of the circle as the vertex.
B
A C
Arc
an arc is part of a circle like a segment is part of a line.
B
AC
AC
Arc Measure
the measure of an arc is given by the measure of its central angle.
B
AC
80
80
AC
80mAC
Minor Arc
an unbroken part of a circle with a measure less than 180°.
B
AC
AC
Semicircle
an unbroken part of a circle that shares endpoints with a diameter.
B
A C
Major Arc
an unbroken part of a circle with a measure greater than 180°.
BA C
D
ACD
Adjacent Arcs
arcs that have exactly one point in common.
B
A C
D
AD DC
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
Congruent Arcs
arcs in the same circle or in congruent circles that have the same measure.
B
A C
D90
90
DCAD
mDCmAD
White Board Practice
Name two minor arcs
R
C
SA
O
White Board Practice
AR, RC, RS, AS, SC
R
C
SA
O
White Board Practice
Name two major arcs
R
C
SA
O
White Board Practice
ARS, ACR, RCS, RSA, RSC, CRS, CSR
R
C
SA
O
White Board Practice
Name two semicircles
R
C
SA
O
White Board Practice
ARC, ASC
R
C
SA
O
White Board Practice
Name an acute central angle
R
C
SA
O
White Board Practice
AOR
R
C
SA
O
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
White Board Practice
Name two congruent arcs
R
C
SA
O
White Board Practice
ARC, ASC
R
C
SA
O
Group Practice
• Give the measure of each arc.
4x
3x 3x + 10
2x
2x-1
4
A
B
C
D
E
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
The radius of the Earth is about 6400 km.
6400
6400
O
BA
The latitude of the Arctic Circle is 66.6º North. That means the m BE 66.6º.
6400
6400
O
BA
EW
66.6º
Find the radius of the Arctic Circle
6400
O
BA
EW
66.6º
xº
Find the radius of the Arctic Circle
6400
O
BA
EW
66.6º
23.4º
Lecture 4 (9-4)
Objectives
• Define the relationships between arcs and chords.
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
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
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
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
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.
Inscribed Angle
B
A
C
An angle formed by two chords or secant lines whose vertex lies on the circle.
Theorem
B
A
C
The measure of an inscribed angle is half the measure of the intercepted arc.
mACABCm2
1
Corollary
B
A
C
If two inscribed angles intercept the same arc, then they are congruent.
ABC ADC
Sketch
D
Corollary
C
A
An angle inscribed in a semicircle is a right angle.
90m ABC
B
O
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
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
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
9.6 Other Angles
Objectives
• Solve problems and prove statements involving angles formed by chords, secants and tangents.
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
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
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
9.7 Circles and Lengths of Segments
Objectives
• Solve problems about the lengths of chords, secants and tangents.
TheoremWhen two chords intersect, the product of
their segments is equal.
A
D
B
XE
F
XBFXXDEX
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
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