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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:
148°
XA
B
G
Find the measure of each arc.
1. XBGXB2.
3.GBX
Using Arcs of Circles
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 + = .
Postulate 26 Arc Addition Postulate
60°
82°100°
A
B
G
J
K
Example 2:
Find the measure of each arc1.
JKB
2.BGJ
3.JG
Using Arcs of Circles
60°
60°A
B
G
J
K
Example 3:
Find the measures of and . Are the arcs congruent? Why?
KJ
GB
Using Arcs of Circles
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.
When a minor arc and a chord share the same endpoints, we call the arc the ARC OF THE CHORD.
Using Chords of Circles
Using Chords of Circles
Theorems about Chords
In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
Theorem 10.4
ABEF
Using Chords of Circles
(2x+48)°(3x+11)°
A
B
G
JK
Example 4:
Find the measure of
GJ
Using Chords of Circles
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.
Theorem 10.5
BCAC NAN B
Using Chords of Circles
Example 5:
In the diagram, FK = 40, AC = 40, AE = 25. Find EG, GH, and EF.
H
G
D
B
E
A C
F
K
Using Chords of Circles
Theorem 10.6
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter
D
B
E
A C
F
EcircleofdiameteraisFB
Using Chords of Circles
In a circle or congruent circles, two chords are congruent if and only if they are equidistant from the center.
Theorem 10.7
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.
(2x + 48)°
(3x + 11)°
A
B
C
D
Example 8:
Using Chords of Circles
Find the measure of: BDCandDCBC ,,
Using Chords of Circles
Example 9:
B
D
C
J
Locate the center of the following circle.
