Transcript
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10.1Properties of Tangents
Circle
Secant
Tangent
Radius
Diameter
Chord
Radius
Chord
Diameter
Secant
Tangent
Example
Tell whether the line or segment is best descrbed as a radius, chord, diameter, secant, or tangent ofC.
A
D
C
E
B
Fa.
b.
c.
d.
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In K, J is a point of tangency. Find the radius of K.
36 cm
48 cm
r
rK
J L
Theorem 10.2
Find x.
6x - 8
25
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10.1 Homework
3-10 13.
19. 23.
24. 25.
26. 29.
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34.
37.
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10.2Find Arc Measures
Central angle
central angle
Minor arc
Major arc
major arc ADB
minor arc AB
B
A
C
D
Semicircle
65
G
E
F
Measuring Arcs
Measure of a major arc
Find the measure of each arc of K where HJ is a diameter.
80
H
K
J
Ia.b.c.d.
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Examplea result of a survey about the ages of people in a town are shown. Find the indicated arcmeasures.
17-44
15-17
45-64
>65
Ages of People (in years)
8060
10090
QU
TS
RV
a. mRUb. mRSTc. mRVTd. mUST
Congruent circles
Congruent arcs
Tell whether arcs CD and EF are congruent. Why?
45 45CP
F
EDa. b. c.
F
E
CP
D
110
D
C
Q
F
E
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10.2 HW PICTURES:
3 10
17. 20.
21.
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10.3Properties of Chords
Theorem 10.3
Theorem 10.4
Theorem 10.5
108
A
B
C
D
Examples
1. In R mAB = _______ . Find the mCD.
2. Use the diagram of C to 3. In the diagram of P, PV = PW,
find the length of BF . QR = 2x + 6, and ST = 3x 1. Find QR.
W
V
PQ
R
S
T
F15 B
G
C
A
D
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17. 21.
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23.
30.
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10.4Inscribed angles and Polygons
Inscribed angle
D
C
B
A
Intercepted arc
Theorem 10.7Measure of an Inscribed Angle Theorem
E
D
C
B
A
Theorem 10.8
Theorem 10.9
Q
T
S
R
--Conversely,
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Theorem 10.10
Examples
1. Find the indicated measures in X
104
33
X
U
V
W
Y
a. mUW b. mVWY
2. Find mWX and m WYX.
44A
Z
Y
X
W
Find the measure of each angle in the quadrilateral.
3.
(5y - 5)
(4y + 5)
5x
7x
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10.5Other Angle Relationships in Circles
D
2
1
B
C
A
Theorem 10.11
Find the indicated measures.
mAB = 124
D
2
1
B
C
A
Intersecting lines and circlesif two lines intersect a circle, there are three places where thelines can intersect
Theorem 10.12Angles Inside the Circle
2
1
E
G
D
F
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Theorem 10.13Angles Outside the Circle
Examples
Line m is tangent to the circle. Find x or y.
y
118
1. 2.
63
89x
(17x + 6)(7x - 2)39
228
x
3. 4.
x
44
30
5.
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1
E
D
C
B
A
0.6Finding Segment Lengths in Circles
ind x.
gment
egment
Theorem 10.14Segments of Chords
F
Secant se
External s
Theorem 10.15Segments of Secants Theorem
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F
x
21
x27R
T
V
ind x.
ind RT.
ind x.
F
10.16Segments of Secants and Tangents Theorem
F
x
x - 4
x
24
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1
-10 -5 5 10
6
4
2
-2
-4
-6
-8
0.7Write and Graph Equations of Circles
ircle Centered at the Origin:
tandard Equation of a Circle Centered at (h, k):
rite the equations of the circles using the given information.
1. Center (0, 0), radius 5
2. Center (-3, 8), radius 5/3
3. Center (1, 2) and a point on the circle is (4, 2)
Standard Equation of aC
S
3
2
1
-1
-4 -2 2 4
h k
-
2
W
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Graph each equation.
. x2+ y2= 25 5. (x - 2)2+ (y + 1)2= 44
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