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G E O M E T R Y
10.2 & 10.3 Arcs and Chords
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
minorarcmajor
arcP
B
A
C
Using Arcs of Circles
AB
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
minorarcmajor
arcP
B
A
C
Naming Arcs
Arcs are named by their endpoints.
EH F
G
E
60°
60°
180°
GF
Naming Arcs
Major arcs and semicircles are named by their endpoints and by a point on the arc.
The measure of a minor arc is defined to be the measure of its or equal to the central angle.
EGF
EH F
G
E
60°
60°
180°
Naming Arcs
For instance, m = mGHF = 60°.
m 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°.
EH F
G
E
60°
60°
180°
GF
GF
Naming Arcs
The measure of a
major arc is defined as
the difference between
360° and the measure
of its associated minor
arc. For example, m
= 360° - 60° = 300°.
The measure of the
whole circle is 360°.
GEF
EH F
G
E
60°
60°
180°
GF
Ex. 1: Finding Measures of Arcs
Find the measure of each arc of R.
a.
b.
c.
MNMPN
PMN PR
M
N80°
Ex. 1: Finding Measures of Arcs
Find the measure of each arc of R.
a.
b.
c.
Solution:
is a minor arc, so m = mMRN = 80°
MNMPN
PMN PR
M
N80°
MN
MN
Ex. 1: Finding Measures of Arcs
Find the measure of each arc of R.
a.
b.
c.
Solution:
is a major arc, so m = 360° – 80° = 280°
MNMPN
PMN PR
M
N80°
MPN
MPN
Ex. 1: Finding Measures of Arcs
Find the measure of each arc of R.
a.
b.
c.
Solution:
is a semicircle, so m = 180°
MNMPN
PMN PR
M
N80°
PMN
PMN
Arc Addition Postulate Two arcs of the same
circle are adjacent if they intersect at exactly one point. You can add the measures of adjacent areas.
Arc Addition Postulate. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
ABCAB
B
A
C
m = m + m BC
Ex. 2: Finding Measures of Arcs
Find the measure of each arc.
a.
b.
c.
m = m + m =
40° + 80° = 120°
GE
GEFR
EF
G
H
GFGEGHHE
40°
80°
110°
Ex. 2: Finding Measures of Arcs
Find the measure of each arc.
a.
b.
c.
m = m + m =
120° + 110° = 230°
GE
GEFR
EF
G
H
GF
EF
40°
80°
110° GEFGE
Ex. 2: Finding Measures of Arcs
Find the measure of each arc.
a.
b.
c.
m = 360° - m =
360° - 230° = 130°
GE
GEFR
EF
G
H
GF
40°
80°
110° GF
GEF
Ex. 3: Identifying Congruent Arcs
Find the measures of the blue arcs. Are the arcs congruent?
C
D
A
BAB• and are in
the same circle and
m = m = 45°.
So,
DC
ABDCDC
AB
45°
45°
Q
S
P
R
Ex. 3: Identifying Congruent Arcs
Find the measures of the blue arcs. Are the arcs congruent?
RS
PQ• and are in
congruent circles and
m = m = 80°.
So,
PQRSRS
PQ
80°
80°
X
W
Y
Z
• m = m = 65°, but
and are not arcs of the
same circle or of
congruent circles, so
and are NOT
congruent.
Ex. 3: Identifying Congruent Arcs Find the measures of the
blue arcs. Are the arcs congruent?
XY
65° ZW
XYZW
XY
ZW
Ch. 10.3: What’s a chord again?
A chord is a segment inside a circle with endpoints on the circle.
Any chord will divide a circle into 2 arcs, a major and minor arc.
Using Chords of Circles
A point Y is called the midpoint if
.
Any line, segment, or ray that contains Y bisects . XYZ
XYYZ
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
C
B
A
What are the measures of arcs AB, AE, and BD?
360 – 75 – 75 = 210
210÷2 = 105
105º
105º
Find the measures of arcs AB and DB…Hint:
they should be the same!
360 – 128= 232 232÷2 = 116
Theorem 10.5
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
E
D
G
F
DG
GF
, DE EF
Theorem 10.6
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
JK is a diameter of
the circle.
J
L
K
M
Ex. 4: Using Theorem 10.4
You can use Theorem 10.4 to find m . AD
• Because AD DC,
and . So,
m = m
ADDC
ADDC
2x = x + 40 Substitute
x = 40 Subtract x from each
side.
BA
C
2x°
(x + 40)°
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.
Ex. 5: Finding the Center of a Circle
Step 2: Draw the perpendicular bisector of each chord. These are the diameters.
Ex. 5: Finding the Center of a Circle
Step 3: The perpendicular bisectors intersect at the circle’s center.
center
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. F
G
E
B
A
C
D
Ex. 7: Using Theorem 10.7
AB = 8; DE = 8, and CD = 5. Find CF.
58
8F
G
C
E
D
A
B
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, DG = =4.
2
8
58
8F
G
C
E
D
A
B
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
58
8F
G
C
E
D
A
B
