Transcript 10.2 Arcs and Chords

Geometry
10.2 Arcs and Chords
Geometry
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
A
major
arc
minor
arc
P
B
C
Geometry
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
A
major
arc
minor
arc
P
B
C
Naming Arcs
Geometry
G
• Arcs are named by
their endpoints. For
example, the minor
arc associated with
APB above is AB.
Major arcs and
semicircles are
named by their
endpoints and by a
point on the arc.

60°
60°
E
H
F
E
180°
Naming Arcs
Geometry
G
• For example, the
major arc
associated with
APB is ACB .
E GFhere on the
right is a semicircle.
The measure of a
minor arc is defined
to be the measure of
its central angle.


60°
60°
E
H
F
E
180°
Geometry
Naming Arcs

• For instance, m GF=
mGHF = 60°.
• m GF 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°.

G
60°
60°
E
H
F
E
180°
Geometry
Naming Arcs

G
• The measure of a GF
minor arc is defined as
the difference between E
360° and the measure
of its associated major
arc. For example, m GEF
= 360° - 60° = 300°.
The measure of the
whole circle is 360°.
60°
60°

H
F
E
180°
Geometry
Ex. 1: Finding Measures of Arcs
•
Find the measure
of each arc of R.

a. MN
b. MPN
c. PMN

N
80°
R
M
P
Geometry
Ex. 1: Finding Measures of Arcs
•
Find the measure
of each arc of R.

a. MN
b. MPN
c. PMN
Solution:
MN is a minor arc, so
m MN = mMRN
= 80°

 
N
80°
R
M
P
Geometry
Ex. 1: Finding Measures of Arcs
•
Find the measure
of each arc of R.

a. MN
b. MPN
c. PMN
Solution:
MPN is a major arc, so
m MPN = 360° – 80°
= 280°

 
N
80°
R
M
P
Geometry
Ex. 1: Finding Measures of Arcs
•
Find the measure
of each arc of R.

a. MN
b. MPN
c. PMN
Solution:
PMN is a semicircle, so
m PMN = 180°

 
N
80°
R
M
P
Note:
C
Geometry
A
• 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.

 
B
m ABC = m AB + m BC
Geometry
Ex. 2: Finding Measures of Arcs
•
G
Find the measure of
each arc.

  
a. GE
b. GEF
c. GF
m GE = m GH + m HE =
40° + 80° = 120°
H
40°
80°
R
110°
F
E
Geometry
Ex. 2: Finding Measures of Arcs
•
G
Find the measure of
each arc.

  
a. GE
b. GEF
c. GF
m GEF = m GE + m EF
120° + 110° = 230°
H
40°
80°
R
110°
=
F
E
Geometry
Ex. 2: Finding Measures of Arcs
•
G
Find the measure of
each arc.


a. GE
b. GEF
c. GF
m GF = 360° - m GEF =
360° - 230° = 130°
H
40°

80°
R
110°
F
E
Geometry
Ex. 3: Identifying Congruent Arcs
• Find the measures
of the blue arcs.
Are the arcs
congruent?
•
 
 
AB and DC are in
the same circle and
m AB = m DC = 45°.
So, AB  DC
A
D
B
45°
45°
C
Geometry
Ex. 3: Identifying Congruent Arcs
• Find the measures
of the blue arcs.
Are the arcs
congruent?
•
 
 
P Q and RS are in
congruent circles and
m P Q = m RS = 80°.
So, P Q  RS
80°
Q
P
80°
S
R
Geometry
Ex. 3: Identifying Congruent Arcs
• Find the measures of
the blue arcs. Are the
arcs congruent?


 

Z
X
• m XY = m ZW = 65°, but
XYand ZW are not arcs of the
same circle or of
congruent circles, so XY
and ZW are NOT
congruent.

65°
Y
W
Geometry
Using Chords of Circles
• A point Y is called
the midpoint of
if XY  YZ . Any
line, segment, or ray
that contains Y
bisects X YZ .
 

Geometry
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
A
C
B
Geometry
Theorem 10.5
• If a diameter of a
circle is
perpendicular to a
chord, then the
diameter bisects the
chord and its arc.
F
E
 
G
DE  EF ,
DG  GF
D
Geometry
Theorem 10.5
• If one chord is a
perpendicular
bisector of another
chord, then the first
chord is a diameter.
J
M
JK is a diameter of
the circle.
K
L
Ex. 4: Using Theorem 10.4
Geometry
(x + 40)°
• You can use
Theorem 10.4 to
find m AD .



 
C
A
• Because AD  DC,
and AD  DC . So,
m AD = m DC
2x = x + 40
x = 40
2x°
B
Substitute
Subtract x from each
side.
Geometry
Ex. 5: Finding the Center of a
Circle
• Theorem 10.6 can
be used to locate a
circle’s center as
shown in the next
few slides.
• Step 1: Draw any
two chords that are
not parallel to each
other.
Geometry
Ex. 5: Finding the Center of a
Circle
• Step 2: Draw the
perpendicular
bisector of each
chord. These are
the diameters.
Geometry
Ex. 5: Finding the Center of a
Circle
• Step 3: The
perpendicular
bisectors intersect
at the circle’s
center.
center
Geometry
Theorem 10.7
• In the same circle,
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.
C
G
D
E
B
F
A
Geometry
Ex. 7: Using Theorem 10.7
AB = 8; DE = 8, and
CD = 5. Find CF.
A
8 F
B
C
E
5
8
G
D
Geometry
Ex. 7: Using Theorem 10.7
Because AB and DE
are congruent
chords, they are
equidistant from the
center. So CF 
CG. To find CG,
first find DG.
CG  DE, so CG
bisects DE.
Because
DE = 8,
8
DG = 2 =4.
A
8 F
B
C
E
5
8
G
D
Geometry
Ex. 7: Using Theorem 10.7
Then use DG to find
CG. DG = 4 and
CD = 5, so ∆CGD is
a 3-4-5 right
triangle. So CG = 3.
Finally, use CG to
find CF. Because
CF  CG, CF = CG
=3
A
8 F
B
C
E
5
8
G
D