Transcript Chords, Arcs & Central Angles

Geometry
Arcs and Chords
Goals
 Identify arcs & chords in circles
 Compute arc measures and angle
measures
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Central Angle
A
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An angle whose
vertex is the
center of a
circle.
Minor Arc
C
Part of a circle.
The measure of
the central
T angle is less
than 180.
A
CT
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Semicircle
C
A
D
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Half of a circle. The
endpoints of the arc
are the endpoints of
a diameter. The
central angle
T
measures 180.
CTD
Major Arc
C
A
D
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T
Part of a circle.
The measure of
the central
angle is greater
than 180.
CTD
Major Arc
CTD
C
BUT NOT
A
D
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T
CDT
Measuring Arcs
 An arc has the same measure as the
central angle.
 We say, “a central angle subtends an arc
of equal measure”.
m  A C B  42 
A
42
42
C
B
m A B  42 
Central Angle Demo
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Measuring Major Arcs
 The measure of an major arc is given by
360  measure of minor arc.
m  A C B  42 
A
42
42
D
C
m A B  42 
B
m ADB  360   42   318 
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Arc Addition Postulate
Postulate Demonstration
R
T
C
A
m RAT  m RA  m AT
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What have you learned so far?
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Page 607
Do problems 3 – 8.
Answers…
3) m R S  6 0 
4) m R P S  300 
5) m P Q R  180 
6) mQ S  1 0 0 
7) mQ SP  220 
8)m  Q T R  40 
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Q
T
40
R
60
S
P
120
Subtending Chords
A
O
C
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Chord AB
subtends AB.
B
Chord BC
subtends BC.
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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
If A B  C D , th en A B  C D .
B
A
C
D
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Example
Solve for x.
120
(5x + 10)
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
Solve for x.
52
2x
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2x = 52
x = 26
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
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.
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Practice Problems
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