10.2 and 10.3 Arcs of Circles

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Sect. 10.2 Arcs and Chords
Goal 1
Using Arcs of Circles
Goal 2
Using chords of Circles.
Using Arcs of Circles
The Central Angle
of a Circle –
A CENTRAL ANGLE is
an angle whose vertex is
at the center of a circle.
Sum of Central Angles - The sum of
the measures of the central angles of a
circle with no interior points in
common is 360°.
Using Arcs of Circles
Every central angle cuts the circle into two arcs.
The smaller arc is called the
Minor Arc. The MINOR ARC
is always less than 180°. It is
named by only two letters with
an arc over them as in our
example,
.
The Minor Arc
The larger arc is called the
Major Arc. The MAJOR ARC
is always more than 180°. It is
named by three letters with an
arc over them as in our
example,
.
The Major Arc
Using Arcs of Circles
The Semicircle (Major Arc = Minor Arc) :
The measure of the semicircle is
180°. SEMICIRCLES are congruent arcs
formed when the diameter of a circle separates
the circles into two arcs.
Using Arcs of Circles
Definition of Arc Measure
• The measure of a minor arc is the
measure of its central angle.
Central Angle = Minor Arc
The measure of a
major arc is 360°
minus the measure
of its central angle.
Using Arcs of Circles
Example 1:
Find the
measure of
each arc.
1.
2.
3.


X
A
148°
G
XB
GXB
B
Using Arcs of Circles
Postulate: Arc Addition Postulate
The measures of an arc formed by two
adjacent arcs is the sum of the measures of
the two arcs. That is, if B is a point on
,
then
+
=
.
Using Arcs of Circles
Example 2:
Find the
measure of
each arc
J
K



60°
82°
1. JKB
100°
2. BGJ
3.
JG
A
G
B
Using Arcs of Circles
Example 3:


Find the measures of KJ
and GB . Are the arcs
congruent? Why?
J
K
60°
A
60°
G
B
Using Chords of Circles
If two arcs of one circle have the same
measure, then they are congruent
arcs. Congruent arcs also have the
same length.
Using Chords of Circles
When a minor arc and a chord
share the same endpoints, we call
the arc the ARC OF THE CHORD.
Using Chords of Circles
Theorems about Chords
Theorem congruent arcs
In a circle or in congruent circles, two
minor arcs are congruent if and only if
their corresponding chords are
congruent.

EF  AB
Using Chords of Circles
Example 4:

G
Find the
measure of GJ
(2x+48)°
(3x+11)°
A
J
K
B
Using Chords of Circles
Theorem diameter bisector
In a circle, if a diameter
is perpendicular to a
chord, then it bisects the
chord and its
arc. (Hint): This
diagram creates right
triangles if you add
radius OA or OB.

AC  BC
AN  B N
Using Chords of Circles
Example 5:
In the diagram,
FK = 40, AC = 40,
AE = 25. Find
EG, GH, and EF.
K
H
G
F
E
A
D
B
C
Using Chords of Circles
Theorem perpendicular chords
If one chord is a
perpendicular bisector of
another chord, then the
first chord is a diameter
F
A
FB is a diameter of circle E
E
D
B
C
Using Chords of Circles
Theorem Equidistant Chords
In a circle or
congruent circles, two
chords are congruent if
and only if they are
equidistant from the
center.
Chords are congruent if they are equidistant
from the center, they are also congruent if
there arcs are the same size.
Using Chords of Circles
Example 7:
Find the length of the radius of
a circle if a chord is 10” long
and 12” from the center.
Using Chords of Circles
Example 8:
 
C
Find the measure of:
BC , DC , and BDC
D
(3x + 11)°
A
(2x + 48)°
B
Homework

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