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10.3 Arcs Of A Circle
After studying this section, you will be able to identify the different types of arcs, determine the measure of an arc, recognize congruent
arcs, apply the relationships between congruent arcs, chords, and central angles.
Definition An arc consists of two points on a circle and all points on the circle needed to connect the points by a single path.
Types of Arcs
Definition The center of an arc is the center of the circle of which the arc is a part.
Definition A central angle is an angle whose vertex is at the center of a circle.
A
B
O
Definition A minor arc is an arc whose points are on or between the sides of a central angle.
A
B
O
Definition A major arc is an arc whose points are on or outside the sides of a central angle.
Definition A semi circle is an arc whose endpoints are the endpoints of a diameter.
A
B
O
E
The symbol is used to label arcs. The minor arc is named with 2 letters. The major arc is named with 3 letters. The middle letter helps indicate the direction of a major arc. Can you name a major and a minor arc?
X
F
The measure of an arc
Definition The measure of a minor arc or a semicircle is the same as the measure of the central angle that intercepts the arc.
Definition The measure of a major arc is 360 minus the measure of the minor arc with the same endpoints.
Example
AB
P
C
: 20
:
Given AB
Find ACB
Example
XY
QD
: 110
:
Given XQY
Find XDY
Congruent Arcs
A
B
O
Two arcs that have the same measure are not necessarily congruent arcs.
D
E
In concentric circles above, arcs AB and DE have the same measure but they are not congruent.
Why do you think they are not congruent?
Definition Two arcs are congruent whenever they have the same measure and are parts of the same circle or congruent circles.
We may conclude
A
BO
If circle P is congruent to circle Q, we may conclude that
C
D
AB CD
80
80
Q P
98 98
EF GH
E
F
G
H
Relating Congruent Arcs, Chords, and Central Angles
A
BIn the diagram, points A and B determine one central angle, one chord, and two arcs (one major and one minor).
minor arc
major arc
chord
central angle
Theorem If two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent.
Theorem If two arcs of a circle (or of congruent circles) are congruent, then their corresponding central angles are
congruent.
Theorem If two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent.
Theorem If two chords of a circle (or of congruent circles) are congruent, then their corresponding central angles are congruent.
Theorem If two arcs of a circle (or of congruent circles) are congruent, then their corresponding chords are congruent.
Theorem If two chords of a circle (or of congruent circles) are congruent, then their corresponding arcs are congruent.
Given: Circle B
Conclusion
Example 1 A
B
C
D
is the midpoint of D AC
bisects BD ABC��������������
Example 2
If in circle O,
A
B
O102
102mAB
find and in circle Om A m B
Example 3
Given: Circles P and Q
Prove AB CDA D
QP B
R
C
;P Q AR RD
Example 4
A. What fractional part of a circle is 36 ? Of 200 ?B. Find the measure of an arc that is of its circle.
7
12
Summary
Summarize what you have learned about arcs of a circle.
Homework: worksheet
