Unit 4:
Arcs and Chords
Keystone Geometry
Measures of an Arc
Measures of an Arc:
1. The measure of a minor arc is the measure of its central angle.
2. The measure of a major arc is 360 - (measure of its minor arc).
3. The measure of any semicircle is 180.
Adjacent Arcs:
Arcs in a circle with exactly one point
in common.
List: Major Arcs
Minor Arcs
Semicircles
Adjacent Arcs
Example: Find the measure of each arc.
4x + 3x + (3x +10) + 2x + (2x-14) = 360°
14x – 4 = 360° so 14x = 364
x=
26°
4x = 4(26) = 104°
3x = 3(26) = 78°
3x +10 = 3(26) +10= 88°
2x = 2(26) = 52°
2x – 14 = 2(26) – 14 = 38°
Theorem
• In the same circle or in congruent circles, two
minor arcs are congruent if and only if their
central angles are congruent.
4
Arc Addition Postulate
• The measure of the arc formed by two
adjacent arcs is the sum of the measures of
these two arcs.
5
Example
• In circle J, find the measures
of the angle or arc named
with the given information:
HG = 70
KF = 80
ÐGJF = 80
• Find:
GF
HF
ÐHJG
HKG
6
In Circle C, find the measure of each
arc or angle named.
• Given: SP is a diameter of
the circle. Arc ST = 80 and
Arc QP=60.
• Find: SQ
SPQ
SPT
SP
ÐSCQ
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Theorem #1:
In a circle, if two chords are congruent then their
corresponding minor arcs are congruent.
A
B
E
C
D
Example:
8
Theorem #2:
In a circle, if a diameter (or radius) is perpendicular
to a chord, then it bisects the chord and its arc.
D
Example: If AB = 5 cm, find AE.
A
B
E
C
9
Theorem #3:
In a circle, two chords are congruent if and only if they
are equidistant from the center.
D
F
C
Example: If AB = 5 cm, find CD.
Since AB = CD, CD = 5 cm.
O
A
B
E
10
Try Some:
• Draw a circle with a chord that is 15 inches long and 8
inches from the center of the circle.
• Draw a radius so that it forms a right triangle.
• How could you find the length of the radius?
Solution:
∆ODB is a right triangle and OD bisects AB
A
15cm
B
D
8cm
O
x
11
Try Some Sketches:
• Draw a circle with a diameter that is 20 cm long.
• Draw another chord (parallel to the diameter) that is 14cm long.
• Find the distance from the small chord to the center of Circle O.
Solution:
∆EOB is a right triangle. OB (radius) = 10 cm
14 cm
A
10 cm
C
7.1 cm
E
x
20cm
B
10 cm
O
D
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