warm-up 1. calculate the exact value of sine and cosine of 30° 2. calculate the sum of the square...
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Warm-Up1. Calculate the exact value of sine and
cosine of 30°
2. Calculate the sum of the square of the sine and cosine of 30°
3. Explain what you think you would get if you did the same thing (find the sine and cosine of the angle, square them, and add them together) with 60°, 45°, or any other angle
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Circles and Spheres Key Standards
MM2G3. Students will understand the properties of circles.
b. Understand and use properties of central, inscribed, and related angles.
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CircleWhat is the definition of a circle?
A circle is the locus of points that are a constant distance from a given point, called the center.
The circle is named for its center, ex PWhat 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
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Locus of PointsLook at the investigation on page 460 –
461 of the Geometry book.
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Central AngleTwo radii form a central angleA central angle of a circle is an angle
whose vertex is the center of the circle.
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ChordsA chord is a segment whose endpoints
are on a circleA diameter is a chord what contains the
center of the circle.
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ArcsAn 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
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ArcsThe measure of a major arc is defined as
the difference between 360 and the measure of its associated minor arc.
Major arcs and semicircles are named by their end points and a point on the arc
Major arc: Central angle > 180
Semicircle: Central angle = 180
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NomenclaturePay 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
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•Example 1:
60 60
Central Angle = APB
Minor arc = AB mAB = mAPB = 60
Major arc = ACB mACB = mACB = 360 - 60 = 300
Minor arcMajorarc
C
P
B
A
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Ex. 2: Finding Measures of Arcs
Find the measure of each arc of R.
a.
b.
c.
MNMPN
PMN PR
M
N80°
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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
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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
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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
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Arc Addition PostulateAdjacent 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 = mAB + mBC
B
C
A
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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°
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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
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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
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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.
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Ex. 4: Identifying Congruent Arcs
Find the measures of the blue arcs. Are the arcs congruent?
C
D
A
BAB and are in the
same circle and m = m = 45°. So, =
DC
ABDCDC
AB
45°
45°
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Q
S
P
R
Ex. 4: Identifying Congruent Arcs
Find the measures of the blue arcs. Are the arcs congruent?
RSPQ and are in
congruent circles and m = m = 80°.
So, =
PQ
RSRS
PQ
80°
80°
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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
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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
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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
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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
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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
![Page 27: Warm-Up 1. Calculate the exact value of sine and cosine of 30° 2. Calculate the sum of the square of the sine and cosine of 30° 3. Explain what you think](https://reader030.vdocuments.net/reader030/viewer/2022032605/56649e6f5503460f94b6c36f/html5/thumbnails/27.jpg)
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
![Page 28: Warm-Up 1. Calculate the exact value of sine and cosine of 30° 2. Calculate the sum of the square of the sine and cosine of 30° 3. Explain what you think](https://reader030.vdocuments.net/reader030/viewer/2022032605/56649e6f5503460f94b6c36f/html5/thumbnails/28.jpg)
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
![Page 29: Warm-Up 1. Calculate the exact value of sine and cosine of 30° 2. Calculate the sum of the square of the sine and cosine of 30° 3. Explain what you think](https://reader030.vdocuments.net/reader030/viewer/2022032605/56649e6f5503460f94b6c36f/html5/thumbnails/29.jpg)
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°
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PracticePage 193, # 3 – 39 by 3’s and 19
(14 problems)