G.C.2 Wkst 2

G.C.2 STUDENT NOTES WS #2
1
Arc – An Arc is a piece or portion of the circumference of a circle. Arcs are classified into two
types, major and minor arcs. Major arcs have a value greater than 180° and minor arcs have
a value less than 180°. These arcs also have naming conventions; minor arcs are named
D
E
using its two endpoints, such as DE or FD while major arcs include a third point between its
two endpoints to help distinguish the direction and size of the arc, such as DEF or EDF .
F
Semi-Circle – You might have noticed that major and minor arcs were either bigger or
smaller than 180°. So how do we refer to the arc when it is exactly 180°? It is called a
semi-circle. A semi-circle is an arc that is exactly half of the circle and has an arc measure
of 180°. There is no distinction as to whether semi-circles should be written as a major
H
I
M
or minor arc - actually either is acceptable, such as GH , GMH , GIH . The reason you
might use three letters is to help distinguish which half of the circle you are referring to.
Major Arc, ABC
Minor Arc, AC
B
Semi-Circle, AED
B
D
G
Arc Addition
95°
E
H
C
G
65°
C
F
A
A
A
Greater than 180°
Less than 180°
Equal to 180°
mFG + mGH = mFH
65° + 95° = 160°
Arc Length (Distance) & Arc Angle (Angle Measure)
Arcs have two ways to measure them - their length and their angle measure. Arc length refers to the physical
distance along the arc from one end to the other. Arc length is a portion or percentage of the circumference
distance. The arc angle or arc measure refers to the portion or percentage of the circles angle sum. When we
refer to the measurement of an arc (its angle measure) we place a lowercase italic m in front of the arc name,
mGH . The m denotes the measure of the arc -- just as we did when we referenced the measurement of an
angle, m∠ABC.
Determine the arc measure.
mCE = ??
F
34°
mEF = ??
mECK = ??
mCE is equal to 87° because we add the two arcs together.
E
mEF is equal to 93° because we subtract 87° from 180°
D
A
53°
K
mDFC = ??
mECK is equal to 207° because we add 87° to 120°.
C
H
mDFC is equal to 307° because we subtract 53° from 360°.
G.C.2 STUDENT NOTES WS #2
2
Central Angles and Arc Measurement
A central angle of a circle is an angle that has its vertex at the center of the circle and has radii as its sides. We
looked at central angles when we discussed the regular polygons. We found that the central angles of the
regular polygons were 360°/n, where n was the number of sides of the polygon.
x
x
x
x
Central Angle = 120°
Central Angle = 90°
Central Angle = 72°
Central Angle = 60°
The central angle is always equal to the intercepted arc measure. This is a very important relationship for
solving many problems dealing with circles.
90°
B
B
90°
76°
A
76°
180°
180°
90°
Central angle is equal to the
intercepted arc measure.
∠ABC is a central
angle.
90°
C
A
C
90°
90° 90°
90°
The square has 4 ≅ arcs
and 4 ≅ central angles.
A diameter is a central angle of
180°, thus the arc is also 180°.
Determine the missing information.
Given circle B with EC as a diameter.
mAC = ??
a) mAC is equal to 39° because arc angle is equal to central angle.
E
mAE = ??
b) mAE is equal to 141° because we subtract 39° from 180°
B
39°
mEK = ??
m∠KBD = ??
A
c) mEK is equal to 71° because we subtract 70° and 39° from 180°.
K
C
D
d) m∠KBD is equal to 39° because arcs are equal.
70°
Given a regular hexagon. Determine the missing information.
D
a) m∠ATB = ??
a) m∠ATB = 60° because 360/6 = 60°
E
b) mAC = ??
b) mAC is equal to 120° because central ∠ is 2(60°)
C
T
c) m∠AEB = ??
c) m∠AEB = 30° because 180° - 60° - 90° = 30°.
F
d) If AB = 5 cm,
what is TB = ??
d) TB = 5 cm because ∆ABT is an equilateral ∆.
B
A
e) If AB = 5 cm,
what is EA = ??
e) EA = 5 3 cm because it is the long side of a 30-60-90 ∆
G.C.2 WORKSHEET #2
Name: ________________________________ Period ______
1
1. Determine the arc measure.
a)
b)
D
mDF = _________
mAC = _________
100°
41°
146°
E
E
mECA = _________
B
55°
mAF = _________
120°
mAD = _________
D
F
G
C
A
mDAF = _________
F
c)
31°
B
A
mCFD = _________
mAE = _________
mDAG = _________
C
108°
d)
E
36°
mLJ = _________
D
L
K
mAB = _________
mKJ = _________
C
mGJK = _________
I
F
mCDB = _________
mBD = _________
87°
B
A
H
40°
G
mKLI = _________
J
85°
2. Determine the measure of the missing arcs on the circle.
a) Circle F
b) Circle H
m AC = 117°
JLK = 302°
mBE = 111°
LG = 168°
mBA = 91°
3. From the given diagram, determine whether the arcs are Major, Minor or Semi-Circle.
To describe the arc without giving it way through notation we will refer to clockwise and
counterclockwise (counter cw).
a) F to G, clockwise
Major Minor Semi
b) A to F, clockwise
Clockwise
Major Minor Semi
J
G
H
C
c) J to C, clockwise
Major Minor Semi
d) K to D, clockwise
Major Minor Semi
K
A
E
e) D to I, counter cw Major Minor Semi
B
f) C to A, counter cw Major Minor Semi
D
j) F to J, clockwise
Major Minor Semi
k) G to I, counter cw Major Minor Semi
F
I
G.C.2 WORKSHEET #2
2
4. The teacher asks a student to write the name for the arc from A to B on the board.
Jackie comes up writes AB or BA . Jeff raises his hand and says that he has a different
answer. What might his answer be if it is different than Jackie’s?
D
B
C
A
5. Given Circle B with diameters HC , EG and DA .
a) m∠DBH = ___________
G
H
b) mDCE = ___________
D
c) mHG = ___________
d) mHCF = ___________
e) m∠HBA = ___________
f) m∠DBA = ___________
F
B
22°
61°
38°
A
C
E
6. Given circle M, HG ≅ HTG . How could this be?
7. Determine the missing information.
a) Given concentric circles with mGF = 76°,
b) Given concentric circles with mBC = 31°,
m∠HIE = 147° and CA & FH are diameters.
m∠FKJ = 68° and EB is a diameter.
mCB = __________
mED = __________
C
B
G
H
m∠GKH= __________
mHE = __________
I
F
A
E
G
A
m∠CIB = _________
B
C
H
K
68°
D
F
mBDC = _________
31°
mABD = _________
I
J
m∠AKB = _________
D
E
8. Given a regular octagon. Determine the missing information
a) m∠APB = __________
b) m∠HPF = __________
c) mAE = __________
d) mGEA = __________
A
B
H
C
P
e) m∠GPF = __________
f) m∠PAH = __________
G
g) m∠PGE = __________
D
h) If HD = 12 cm, what is GE = _________ (E)
F
9. Points A, B, C, D, and E are placed on circle R in this order such that there are five congruent arcs.
What is the mBCE ?
E
G.C.z WORKSHEET #2
Nome:
1. Determine the arc measure.
a)
*6i =
b)
lblo
mAC
mECA=
*rt=
wCFD
------------\-
\\
f
*ilG=
F
{t{'
mAD
=
*-o
(5
=
wDAF
c)
31'
=
d)
mAE
=
*7i
lLtqc
mL.I
=
rr("
*fu=
=
*68= 1u',
mBD
3(
=
mGJK =
=
wKLI
Af,o'
=
2' Determine the measure of the missing arcs
on the circte.
a) Circle
F
mAC =
ll7"
mBE =
lll"
b)Circle
H
JLK = 302o
tb"
LG = 168"
mfu=9P
E
3' From the given diagram, determine whether
the arcs are Major, Minor or semi-circle
To describe the arc without giving it way
through notation we wilt refer to clockwise
anr
counterclockwise (counter cw).
a)
F
to G, ctockwis"
ctockwise
@
c)J to C, clockwise Major
e) D to t, counter
Minor
@
Semi
b) A to F, clockwise
Major Minor
Semi
,.r,
d) K to D, clockwise
Major Minor
Semi
cw Major@ii6DSemi
j) F to J, clockwise
@*inor
semi
f)
C
to A, counter cw Major Minor Semi
k) G to l, counter cw
Major Minor
Semi
J6
Q
G.C.2 WORKSHEET #2
2
4. The teacher asks a student to
Jackie comes up
write the name for the arc from A to
B on
the board.
7i o, fu . Wt raises his hand and says that he has a different
writes
answer. What might his answer be if it is different than Jackie's?
5. Given Circle B with diameters
al nIDBH
c\ mHG
IlC
,
j&o
=
?_
b) wDCE
,_
gL"
=
=
e) mZHBA
EGand DA.
d) wHCF
l,Yt "
=
=
f) nZDBA=
5. Given circle M, HG =
HTG. How could this be?
7. Determine the missing information.
a) Given concentric circles with mdF =76o,
mlHlE = !47" and CA & FH are diameters.
*eE=
b) Given concentric circles
nIFKJ
= 68o
*fD=
[o\"
mHE =
mIGKH=
*fie= A56
*1ED=
mZ-ClB =
nIAKB
and
wilh mBC
= 31o,
EBis a diameter.
btr'
31'
Lvl'
=
8. Given a regular octagon. Determine the missing information
a')
n/APB
i-O
=
t{5
bl tIHPF
c) mAE =
e) nIGPF =
E)
Lo*
=
d) mGEA=
Ltr
f) IZPAH
nIPGE=
b1,t
h) lfHD=12 cm, what is GE =
9. Points A, B, C, D, and
What is the wBCE
-o
=
?
E
are placed on circle
R
(E)
in this order such that there are five congruent arcs.