Upload
ngophuc
View
223
Download
5
Embed Size (px)
Citation preview
Circles and Spheres Key
Standards
MM2G3. Students will understand the
properties of circles.
b. Understand and use properties of central,
inscribed, and related angles.
Locus of Points
Look at the investigation on page 460 –
461 of the Geometry book.
Investigate: Given a point A, what is the
locus of points in a plane that are 4
inches from point A?
Method: Locate (fold paper hotdog and
hamburger) and mark the center of a
piece of paper. Use a ruler, a piece of
paper, and a radius of 4 inches.
Circle What is the definition of a circle?
A circle is the locus of points, in a plane,
that are a constant distance from a given
point, called the center.
The circle is named for its center, ex P
What is that constant distance called?
A radius is a segment whose endpoints are
the center and any point on the circle.
How many radii does circle have?
An infinite number
Central Angle
Two radii form a central angle
A central angle of a circle is an angle
whose vertex is the center of the circle.
Chords
A chord is a segment whose endpoints
are on a circle
A diameter is a chord that contains the
center of the circle.
Secants
A secant is a line that intersects the
circle at two points
A secant that included the center also
includes the diameter
Tangent
A tangent is a line that intersects the
circle only once
A tangent is always perpendicular to a
radius at the point of tangency
Arcs An arc is an unbroken part of a circle.
Minor Arcs are named for their end points.
The measure of a minor arc is defined to
be the measure of its central angle.
Minor arc: Central angle < 180
Arcs The measure of a major arc is defined as
the difference between 360 and the
measure of its associated minor arc.
Semicircle: Central angle = 180
Major arc: Central angle > 180
Major arcs and semicircles are named by
their end points and a point on the arc
Nomenclature
Pay particular attention to the
nomenclature as shown in the following
slide.
The arc AB is designated:
This same nomenclature will be used to
designate the length of the arc later.
The measure of the arc in degrees is
designated:
AB
ABm
Warm-Up
Draw, name and label a:
• Circle
• Chord
• Secant
• Tangent
• Radius
• Diameter
• Central Angle
• Minor arc
• Major Arc
• Point of
Tangency
•Example 1:
60 60
Central Angle = APB
Minor arc = AB mAB = mAPB = 60
Major arc = ACB mACB = mACB =
360 - 60 = 300
Minor arcMajor
arc
C
P
B
A
Ex. 2: Finding Measures of Arcs
Find the measure
of each arc of R.
a.
b.
c.
MNMPN
PMN PR
M
N80°
Ex. 2: Finding Measures of Arcs
Find the measure
of each arc of R.
a.
b.
c.
Solution:
is a minor arc, so
m = mMRN
= 80°
MNMPN
PMN PR
M
N80°
MN
MN
Ex. 2: Finding Measures of Arcs
Find the measure
of each arc of R.
a.
b.
c.
Solution:
is a major arc, so
m = 360° – 80°
= 280°
MNMPN
PMN PR
M
N80°
MPN
MPN
Ex. 2: Finding Measures of Arcs
Find the measure
of each arc of R.
a.
b.
c.
Solution:
is a semicircle, so
m = 180°
MNMPN
PMN PR
M
N80°
PMN
PMN
Arc Addition Postulate Adjacent arcs have exactly one point in
common.
The measure of an arc formed by two
adjacent arcs is the sum
of the measures
of the two arcs
m ABC = m AB + m BC
B
C
A
Ex. 3: Finding Measures of Arcs
Find the measure of
each arc.
a.
b.
c.
m = m + m =
40° + 80° = 120°
GE
GEFR
EF
G
H
GFGE
GH
HE
40°
80°
110°
Ex. 3: Finding Measures of Arcs
Find the measure of
each arc.
a.
b.
c.
m = m + m =
120° + 110° = 230°
GE
GEFR
EF
G
H
GF
EF
40°
80°
110° GEF
GE
Ex. 3: Finding Measures of Arcs
Find the measure of
each arc.
a.
b.
c.
m = 360° - m =
360° - 230° = 130°
GE
GEFR
EF
G
H
GF
40°
80°
110° GF
GEF
W X
40
Q
40
Z Y
Congruent Arcs
In a circle or in congruent circles, two
minor arcs are congruent iff their
corresponding central angles are
congruent.
Need Congruent:
Central angles
Radii.
Ex. 4: Identifying Congruent Arcs
Find the measures
of the blue arcs.
Are the arcs
congruent?
C
D
A
B
AB and are in the
same circle and
m = m = 45°.
So, =
DC
AB
DCDC
AB
45°
45°
Q
S
P
R
Ex. 4: Identifying Congruent Arcs
Find the measures
of the blue arcs.
Are the arcs
congruent? RS
PQ and are in
congruent circles and
m = m = 80°.
So, =
PQ
RS
RS
PQ
80°
80°
X
W
Y
Z
Ex. 4: Identifying Congruent Arcs Find the measures of
the blue arcs. Are the arcs congruent?
65°
m = m = 65°, but
and are not arcs of the
same circle or of
congruent circles, so
and are NOT
congruent.
XY
ZW
XY
ZW
XY
ZW
Practice
Page 193, # 3 – 39 by 3’s and 19
(14 problems)
Warm-Up: Identify Diameter
Chord
3 Radii
m AB
m CAB
Name of circle
3 minor arcs
3 major arcs
Semicircle
mBSC
mBSA = 40.00
mCSD = 25.00
B
A
C
S
D
F E
Application:
Determine each central angles to make
a pie chart from the following data:
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 25
Orange 15
Green 10
Application:
Determine each central angles to make
a pie chart from the following data:
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 25
Orange 15
Green 10
Total 50
Application:
Determine each central angles to make
a pie chart from the following data:
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 25 50
Orange 15 30
Green 10 20
Total 50
Application:
Determine each central angles to make
a pie chart from the following data:
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 25 50
Orange 15 30
Green 10 20
Total 50 100
Application:
Determine each central angles to make
a pie chart from the following data:
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 25 50 180
Orange 15 30 108
Green 10 20 72
Total 50 100
Application:
Determine each central angles to make
a pie chart from the following data:
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 25 50 180
Orange 15 30 108
Green 10 20 72
Total 50 100 360
Application:
What is the central angles if we wanted
to combine Blue and Green?
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 25 50 180
Orange 15 30 108
Green 10 20 72
Total 50 100 360
252°
Class Work – In Groups
Determine each central angles to make a
pie chart from the following data:
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 8
Orange 7
Green 5
Class Work – In Groups
Determine each central angles to make a
pie chart from the following data:
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 8
Orange 7
Green 5
20
Class Work – In Groups
Determine each central angles to make a
pie chart from the following data:
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 8 40
Orange 7 35
Green 5 25
Total 20 100
Class Work – In Groups
Determine each central angles to make a
pie chart from the following data:
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 8 40 144
Orange 7 35 126
Green 5 25 90
Total 20 100 360
Class Work – In Groups
What would the central angle be if we
combined the Blue and Green?
Category Number of
each color
% Number of
Degrees in the
Central Angle
Blue 8 40 144
Orange 7 35 126
Green 5 25 90
Total 20 100 360
234°
40%
35%
25%
According to the 2007-2008
Pet Owners survey:[
Animal # Households
with a Pet
(millions)
# Pets
(millions)
Bird 6.0 15.0
Cat 38.2 93.6
Dog 45.6 77.5
Equine 3.9 13.3
Freshwater fish 13.3 171.7
Saltwater fish 0.7 11.2
Reptile 4.7 13.6
Small pets 5.3 15.9
Homework
Problem18 of 11 – 2 Practice
Problems 19 – 27 of Exercises Handout
page 456
Homework:
11 – 3 Study Guide