Lesson 10.2
Arcs and Chords
Arcs of Circles
• Central Angle-angle whose vertex is the center of the circle.
P
A
BC
central angle
Minor Arc
• formed from a central angle less than 180°
P
A
BC
minor arc
Major Arc
• formed from a central angle that measures between 180 ° - 360 °
P
A
BC
major arc
Semicircle
• formed from an arc of 180 °• Half circle!• Endpoints of an arc are endpoints
of the diameter
Naming Arcs
• How do we name minor arcs, major arcs, and semicircles??
Minor Arc
• Named by the endpoints of the arc.
P
A
BC
Minor Arc: AB or BA
Major Arc
• Named by the endpoints of the arc and one point in between the arc
P
A
BC
Major Arc: ACB or BCA
Could we name this major arc BAC?
Semicircle
• Named by the endpoints of the diameter and one point in between the arc
CA
B
mABC = 180°
Example
Measuring Arcs• A Circle measures 360 °
Measure of a Minor Arc
• Measure of its central angle
P
A
BC
95°m AB=95 °
Measure of a Major Arc
• difference between 360° and measure of minor arc
P
A
BC
95°mACB=360°– 95° = 265°
Arc Addition Postulate
• Measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
55
45A C
B
D
What is the measure of BD?
m BD=100 °
Example for #1-10
Congruent Arcs
• Two arcs of the same circle or congruent circles are congruent arcs if they have the same measure.
60
60
C
A
D
B
AB is congruent to DC since their arc measures are the same.
Theorem 10.4
• Two minor arcs are congruent iff their corresponding chords are congruent.
P
A
B
C
Chords are congruent
Example 1Solve for x
E
F
HG
2x X+40
Theorem 10.5
• If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
E
G
D
F
If DE = EF, then DG = GF
Example. Find DC.
40
A ED
B
C
m DC = 40º
Theorem 10.6
• If one chord is a perpendicular bisector to another chord, then the first chord is a diameter.
C
D
A
B
Since AB is perpendicular to CD, CD is the diameter.
Example. Solve for x.
x
7 x = 7
Theorem 10.7
• Two chords are congruent iff they are equidistant from the center.
Congruent Chords
Example. Solve for x.
15
x x = 15