Geometry
Arcs and Chords
April 19, 2023
Goals
Identify arcs & chords in circles Compute arc measures and angle
measures
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Central Angle
An angle whose vertex is the center of a circle.A
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Minor Arc
Part of a circle. The measure of the central angle is less than 180.
A
C
T
CT
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Semicircle
Half of a circle. The endpoints of the arc are the endpoints of a diameter. The central angle measures 180.
A
C
T
CTDD
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Major Arc
Part of a circle. The measure of the central angle is greater than 180.
A
C
T
CTDD
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Major Arc
CTDBUT NOT
A
C
T
D
CDT
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Measuring Arcs An arc has the same measure as the
central angle. We say, “a central angle subtends an arc
of equal measure”.
4242
A
BC
42
42
m ACB
mAB
Central Angle Demo
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Measuring Major Arcs The measure of an major arc is given by
360 measure of minor arc.
42
42
m ACB
mAB42
42A
BCD
360 42 318 mADB
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Arc Addition Postulate
R
ACT
mRAT mRA mAT
Postulate Demonstration
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What have you learned so far? Page 607 Do problems 3 – 8. Answers… 3) 4) 5) 6) 7) 8)
P
Q
R
S
T
120
60
40
60mRS 300mRPS 180mPQR 100mQS 220mQSP
40m QTR
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Subtending Chords
A
BO
C Chord BC subtends BC.
Chord AB subtends AB.
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Theorem 12.4
Two minor arcs are congruent if and only if corresponding chords are congruent.
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Theorem 12.4
AB
CD
If AB CD, then AB CD.
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Example
120 (5x + 10)
Solve for x.
5x + 10 = 120
5x = 110
x = 22
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Theorem 12.5
If a diameter is perpendicular to a chord, then it bisects the chord and the subtended arc.
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Example
52
2x
Solve for x.
2x = 52
x = 26
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Theorem 12.6
If a chord is the perpendicular bisector of another chord, then it is a diameter.
Diameter
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Theorem 12.7
Two chords are congruent if and only if they are equidistant from the center of a circle.
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The red wires are the same length because they are the same distance from the center of the grate.
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Example
16
4x – 2
Solve for x.
4x – 2 = 16
4x = 18
x = 18/4
x = 4.5
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Summary Chords in circles subtend major and
minor arcs. Arcs have the same measure as their
central angles. Congruent chords subtend congruent
arcs and are equidistant from the center.
If a diameter is perpendicular to a chord, then it bisects it.
April 19, 2023
Practice Problems