Download Geometry – Arcs, Central Angles, and Chords

Geometry – Arcs, Central Angles, and Chords
An arc is part of a circle. There are three types you need to understand:
X
A
P
B
Semicircle – exactly
half of a circle
180°
Geometry – Arcs, Central Angles, and Chords
An arc is part of a circle. There are three types you need to understand:
C
X
A
P
Semicircle – exactly
half of a circle
180°
D
B
P
Minor arc – less than a
semicircle ( < 180° )
Geometry – Arcs, Central Angles, and Chords
An arc is part of a circle. There are three types you need to understand:
A
P
D
B
Semicircle – exactly
half of a circle
180°
B
C
X
P
P
E
A
Minor arc – less than a
semicircle ( < 180° )
Major arc – bigger
than a semicircle
( > 180° )
Geometry – Arcs, Central Angles, and Chords
An arc is part of a circle. There are three types you need to understand:
A
P
D
B
Semicircle – exactly
half of a circle
180°
B
C
X
P
P
E
A
Minor arc – less than a
semicircle ( < 180° )
Major arc – bigger
than a semicircle
( > 180° )
The symbol for an arc (
letters naming the arc
) is placed above the
Geometry – Arcs, Central Angles, and Chords
An arc is part of a circle. There are three types you need to understand:
A
P
AXB
You need 3 letters to
name a semicircle
The symbol for an arc (
letters naming the arc
D
B
Semicircle – exactly
half of a circle
180°
B
C
X
P
P
E
A
Minor arc – less than a
semicircle ( < 180° )
Major arc – bigger
than a semicircle
( > 180° )
) is placed above the
Geometry – Arcs, Central Angles, and Chords
An arc is part of a circle. There are three types you need to understand:
A
P
AXB
You need 3 letters to
name a semicircle
The symbol for an arc (
letters naming the arc
D
B
Semicircle – exactly
half of a circle
180°
B
C
X
CD
P
P
E
A
Minor arc – less than a
semicircle ( < 180° )
- Use the ray endpoints
to name a minor arc
) is placed above the
Major arc – bigger
than a semicircle
( > 180° )
Geometry – Arcs, Central Angles, and Chords
An arc is part of a circle. There are three types you need to understand:
A
P
AXB
You need 3 letters to
name a semicircle
The symbol for an arc (
letters naming the arc
D
B
Semicircle – exactly
half of a circle
180°
B
C
X
BEA
CD
P
P
E
A
Minor arc – less than a
semicircle ( < 180° )
- Use the ray endpoints
to name a minor arc
) is placed above the
Major arc – bigger
than a semicircle
( > 180° )
- Use the ray
endpoints and a point
in between to name a
major arc
Geometry – Arcs, Central Angles, and Chords
A central angle is an angle whose vertex is at the center of a circle:
D
DPC
P
C
Geometry – Arcs, Central Angles, and Chords
A central angle is an angle whose vertex is at the center of a circle:
D
CD
DPC
P
C
If mDPC  40, arc CD  40
- This central angle creates an arc that is equal to the measure of the central angle.
Geometry – Arcs, Central Angles, and Chords
A central angle is an angle whose vertex is at the center of a circle:
D
CD
DPC
P
C
If mDPC  40, arc CD  40
The reverse is also true, if arc CD = 50°,
central angle DPC = 50°
-This central angle creates an arc that is equal to the measure of the central angle
Geometry – Arcs, Central Angles, and Chords
C
D
P
Y
Chord DC separates circle P into two arcs, minor arc DC, and major arc DYC.
Geometry – Arcs, Central Angles, and Chords
C
D
P
A
B
Theorem : if two chords of a circle have the same length, their intercepted arcs have
the same measure.
Geometry – Arcs, Central Angles, and Chords
C
D
P
A
B
Theorem : if two chords of a circle have the same length, their intercepted arcs have
the same measure.
If DC  AB, then mDC  mAB
Geometry – Arcs, Central Angles, and Chords
C
D
P
A
B
Theorem : if two chords of a circle have the same length, their intercepted arcs have
the same measure.
If DC  AB, then mDC  mAB
- The reverse is then also true, if intercepted arcs have the same measure,
their chord have the same length.
Geometry – Arcs, Central Angles, and Chords
C
D
P
A
B
Theorem : if two chords of a circle have the same length, their intercepted arcs have
the same measure.
If DC  AB, then mDC  mAB
- The reverse is then also true, if intercepted arcs have the same measure,
their chord have the same length.
EXAMPLE : CD = AB and the measure of arc AB = 86°.
What is the measure of arc CD ?
Geometry – Arcs, Central Angles, and Chords
C
D
P
A
B
Theorem : if two chords of a circle have the same length, their intercepted arcs have
the same measure.
If DC  AB, then mDC  mAB
- The reverse is then also true, if intercepted arcs have the same measure,
their chord have the same length.
EXAMPLE : CD = AB and the measure of arc AB = 86°.
What is the measure of arc CD ?
CD  86
Geometry – Arcs, Central Angles, and Chords
X
C
D
P
A
Y
B
Theorem : chords that are equidistant from the center have equal measure
If XP  YP , then CD  AB
Geometry – Arcs, Central Angles, and Chords
X
C
D
P
A
Y
B
Theorem : chords that are equidistant from the center have equal measure
If XP  YP , then CD  AB
EXAMPLE : XP = YP and the measure of AB = 30. What is the measure of CD ?
Geometry – Arcs, Central Angles, and Chords
X
C
D
P
A
Y
B
Theorem : chords that are equidistant from the center have equal measure
If XP  YP , then CD  AB
EXAMPLE : XP = YP and the measure of AB = 30. What is the measure of CD ?
CD  30
Geometry – Arcs, Central Angles, and Chords
P
A
X
B
Y
Theorem : If a diameter or radius is perpendicular to a chord, it
bisects that chord and its arc.
If PY  AB, then AX  XB and AY  BY
Geometry – Arcs, Central Angles, and Chords
P
A
X
B
Y
Theorem : If a diameter or radius is perpendicular to a chord, it
bisects that chord and its arc.
If PY  AB, then AX  XB and AY  BY
EXAMPLE : PY is perpendicular to and bisects AB, arc AB = 100°.
What is the measure of arc YB ?
Geometry – Arcs, Central Angles, and Chords
P
A
X
B
Y
Theorem : If a diameter or radius is perpendicular to a chord, it
bisects that chord and its arc.
If PY  AB, then AX  XB and AY  BY
EXAMPLE : PY is perpendicular to and bisects AB, arc AB = 100°.
What is the measure of arc YB ?
YB  50