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Ge
om
etr
y
10.2 Arcs and Chords
Geometry
Mr. Peebles
Spring 2013
Ge
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Bell Ringer: Solve For r .
8 ft.
16 ft.
r
r
A
B
C
Ge
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Bell Ringer
Answer
(r + 8)2 = r2 + 162
Pythagorean Thm.
Substitute values
c2 = a2 + b2
r 2 + 16r + 64 = r2 + 256 Square of binomial
16r + 64 = 256
16r = 192
r = 12
Subtract r2 from each side.
Subtract 64 from each side.
Divide.
The radius of the silo is 12 feet.
8 ft.
16 ft.
r
r
A
B
C
Ge
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Daily Learning Target (DLT)
Monday May 13, 2013
• “I can use properties of arcs and chords
of circles, as applied.”
Ge
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Assignment: • Pgs. 577-580 (1-27 Odds, 41-43)
1. 9π m2 19. 3.3 m2
3. 0.7225π ft2 21. 120.4 cm2
5. 86,394 ft2 23. (54π + 20.25 ) cm2
7. 40.5π yds2 25. (4 – π) ft2
9. 169π/6 m2 27. (784 – 196π) in2
11. 12π ft2 41. A
13. 25π/4 in2 42. G
15. 24π in2 43. 5.2 m2
17. 22.1 cm2
3
Ge
om
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Using Arcs of Circles
• In a plane, an angle
whose vertex is the
center of a circle is
a central angle of
the circle. If the
measure of a
central angle, APB
is less than 180°,
then A and B and the
points of P
central angle
minorarcmajor
arcP
B
A
C
Ge
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Using Arcs of Circles
• in the interior of APB
form a minor arc of the
circle. The points A and
B and the points of P
in the exterior of APB
form a major arc of the
circle. If the endpoints
of an arc are the
endpoints of a diameter,
then the arc is a
semicircle.
central angle
minorarcmajor
arcP
B
A
C
Ge
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Naming Arcs
• For instance, m =
mGHF = 60°.
• m is read “the
measure of arc GF.”
You can write the
measure of an arc next
to the arc. The
measure of a
semicircle is always
180°.
EH F
G
E
60°
60°
180°
GF
GF
Ge
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Naming Arcs
• The measure of a
major arc is defined as
the difference between
360° and the measure
of its associated minor
arc. For example, m
= 360° - 60° = 300°.
The measure of the
whole circle is 360°.
EH F
G
E
60°
60°
180°
GEF
GF
Ge
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Ex. 1: Finding Measures of Arcs
• Find the measure
of each arc of R.
a.
b.
c.
MN
MPN
PMN PR
M
N80°
Ge
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Ex. 1: Finding Measures of Arcs
• Find the measure
of each arc of R.
a.
b.
c.
Solution:
is a minor arc, so
m = mMRN
= 80°
MN
MPN
PMN PR
M
N80°
MN
MN
Ge
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Ex. 1: 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°
MN
MPN
PMN PR
M
N80°
MPN
MPN
Ge
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Ex. 1: Finding Measures of Arcs
• Find the measure
of each arc of R.
a.
b.
c.
Solution:
is a semicircle, so
m = 180°
MN
MPN
PMN PR
M
N80°
PMN
PMN
Ge
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Note:
• Two arcs of the same circle are adjacent if they intersect at exactly one point. You can add the measures of adjacent areas.
• Postulate 26—Arc Addition Postulate. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
ABC
B
A
C
m = m + m ABBC
Ge
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Ex. 2: Finding Measures of Arcs
• Find the measure of
each arc.
a.
b.
c.
m = m + m =
40° + 80° = 120°
GE
R
EF
G
H
GEFGFGE
GH
HE
40°
80°
110°
Ge
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Ex. 2: Finding Measures of Arcs
• Find the measure of
each arc.
a.
b.
c.
m = m + m =
120° + 110° = 230°
GE
R
EF
G
H
GEFGF
EF
40°
80°
110° GEF
GE
Ge
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Ex. 2: Finding Measures of Arcs
• Find the measure of
each arc.
a.
b.
c.
m = 360° - m =
360° - 230° = 130°
GE
R
EF
G
H
GEFGF
40°
80°
110° GF
GEF
Ge
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Ex. 3: Identifying Congruent Arcs
• Find the measures
of the blue arcs.
Are the arcs
congruent?
C
D
A
B45°
45°
Ge
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Ex. 3: 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
AB
DCDC
AB
45°
45°
Ge
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Q
S
P
R
Ex. 3: Identifying Congruent Arcs
• Find the measures
of the blue arcs.
Are the arcs
congruent? 80°
80°
Ge
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Q
S
P
R
Ex. 3: 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
RSRS
PQ
80°
80°
Ge
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X
W
Y
Z
Ex. 3: Identifying Congruent Arcs
• Find the measures of the blue arcs. Are the arcs congruent?
65°
Ge
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X
W
Y
Z
• m = m = 65°, but
and are not arcs of the
same circle or of
congruent circles, so
and are NOT
congruent.
Ex. 3: Identifying Congruent Arcs
• Find the measures of the blue arcs. Are the arcs congruent?
65° XY
ZW
XY
ZW
XY
ZW
Ge
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Using Chords of Circles
• A point Y is called
the midpoint of
if . Any
line, segment, or ray
that contains Y
bisects . XYZ
XYYZ
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Theorem 10.4
• In the same circle, or in
congruent circles, two
minor arcs are
congruent if and only if
their corresponding
chords are congruent.
AB BC if and only if
AB BC
C
B
A
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Theorem 10.5
• If a diameter of a
circle is
perpendicular to a
chord, then the
diameter bisects the
chord and its arc. E
D
G
F
DG
GF
, DE EF
Ge
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Theorem 10.5
• If one chord is a
perpendicular
bisector of another
chord, then the first
chord is a diameter.
JK is a diameter of
the circle.
J
L
K
M
Ge
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Ex. 4: Using Theorem 10.4
• You can use
Theorem 10.4 to
find m . AD
BA
C
2x°
(x + 40)°
Ge
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Ex. 4: Using Theorem 10.4
• You can use
Theorem 10.4 to
find m . AD
• Because AD DC,
and . So,
m = m
ADDC
ADDC
2x = x + 40 Substitute
x = 40 Subtract x from each
side.
BA
C
2x°
(x + 40)°
Ge
om
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or in congruent
circles, two chords
are congruent if and
only if they are
equidistant from the
center.
• AB CD if and only
if EF EG.
Theorem 10.7
F
G
E
B
A
C
D
Ge
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Exit Quiz – 5 Points
• Find the area of the sector shown
below.
4 ft.
P
D
C
Sector CPD intercepts an
arc whose measure is 85°.
The radius is 4 ft.
A m
360° = • r2