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Geometry
Chapter 9
Circle Vocabulary
Arc Length
Angle & Segment Theorems with Circles
Proofs
Chapter 9: Circles
Date
Due
Section
Topics
Assignment
Written Exercises

Definitions




Tangents
Circumscribed vs. Inscribed Pg. 335(bottom)-337 #1-7,
8(not d), 10, 14-20 even
Common Tangent
Tangent Circles





Arcs (minor and major)
Central <’s
Arc Addition Postulate
Congruent Arcs
Length of an Arc

Arcs and Chord
Relationships
Pg. 347 # 1-9, 12, 18, 22


Pg. 354-355 #2-8, 19-21,
Pg. 359-361 #1-10, 12-24 even

Inscribed Angles
Angles formed by Chords,
Tangents and Secants
Lengths of Segments in a
Circle
Proofs


Study For Test
Extra Practice
9.1
9.2
9.3
9.4
9.5
9.6
9.7
More
Proofs
Review
Chapter 9

Worksheet (pg330 classroom ex.all)
Pg. 331 #4, 6, 7, 12-15, 17
Pg.337 #1-3 (mixed review-bottom
of page)
Pg. 341 (bottom)- 342 # 1-6, 10,
[note m<CAO = m<2] , #16
Pg. 364 (bottom)- 366 #2-8 even,
14-22 even
Worksheet
Pg.349 (Self Test 1) #1-6
Pg.367 (Self Test 2) #1-8
Pg.369-370 (Chpt Rev) #1-24
Pg.371 (Chpt Test) #1-18
1
Circle Introductory Vocabulary
Geometry
Name _______________________
Date ___________ Block _______
Use appropriate notation to name the following in the given diagram.
Write a short explanation or definition as needed.
circle:
center:
diameter:
radius:
chord:
arc:
semicircle:
major arc:
minor arc:
________________________________________________________________________
secant:
tangent:
________________________________________________________________________
inscribed polygon:
circumscribed polygon:
2
p.330 Class Exercises
1. Name three radii of
O.
2. Name a diameter.
3. Consider RS and RS . Which is a
chord and which is a secant?
4. Why is TK not a chord?
5. Name a tangent to
O.
6. What name is given to point L?
_______________________________________________________________
7. Name a line tangent to sphere Q.
8. Name a secant of the sphere and
a chord of the sphere.
9. Name 4 radii. (none are drawn
on the diagram)
________________________________________________________________
10. What is the diameter of a circle with radius 8? 5.2? 4 3 ? j?
11. What is the radius of a sphere with diameter 14? 13? 5.6? 6n?
__________________________________________________________________
The radius of circle O has a length of 20. Radii OA and OB are drawn in,
forming an angle with the given measure. Find the length of AB using
your knowledge of isosceles and special triangles.
a) m<AOB = 90
b) m<AOB = 60
c) m<AOB = 120
3
________________________________________________________________________
9-3: Arcs and Central <'s
& 11-6: Arc Length
An arc is measured in degrees - Its measure is equal in measure of the
central angle which intercepts it.
Arcs are ≅ iff. their central <'s are ≅, with angles 0<θ<360.
4
The central angles are equal in measure....
While the arcs are equal in measure, the arcs are different in length!
Arc length is related to circumference....
C = πd or  C = 2 π r
Arc length....
l =
l =
central
measure
360
central
measure
360
d
2 r
Think about it -- arc length is a fractional part of the circumference of the
circle & the circle is 360 degrees!!!!
________________________________________________________________________
Thm: the measure of a central angle = the measure of the intercepted arc
5
Central Angle & Arcs Notes
Geometry
Name _________________________
Date ________ Block _________
Find the measure of each arc in the diagram.
________________________________________________________________________
Use the diagram to answer the following:
________________________________________________________________________
6
Arc Length Practice WS
Determine the length of an arc with the given central angle measure,
m<M, in a circle with radius r. Give your answers in simplest form in terms
of  .
____________________________________________________________________
Determine the length of an arc with the given central angle measure,
m<M, in a circle with radius r. Give your answers rounded to the nearest
hundredth.
___________________________________________________________________
Determine the degree measure of an arc with the given length, L, in a
circle with radius r. Give your answers rounded to the nearest tenth.
Extra Practice: p.341 CE(1-13)
7
Thm: A line tangent to a circle  the line is perpendicular to the radius @
the pt of intersection
Converse: A line which is
perpendicular to the radius @
a point on the circle  the
line is tangent to the circle
* the circle & line must be
Coplanar!
Thm: parallel lines/chords in a circle  intercepted arcs are congruent
Thm: Tangent segments from an external point are congruent
Thm: If a line in the plane of a circle is perpendicular to the radius
(diameter) at its outer endpoint, then the line is tangent to the circle.
___________________________________________________________________
8
3. If AP = 12 and BP = 6, find AO.
______________________________________________________________________
Tangents with Circles
Internal Tangent Line
External Tangent Line
Tangent Circles
_______________________________________________________________________
9
Thm: In the same circle or congruent circles,
congruent arcs  congruent chords
Thm: If a diameter of a circle is perpendicular to a chord  it bisects both
the chord and its intercepted arc.
Converse: if diameter bisects a
chord  it is perpendicular to the
chord @ its midpoint
Thm: In the same circle or congruent circles,
If 2 chords are equidistant from the center  the chords are congruent.
Converse: if chords are congruent
 the chords are equidistant from
the center
________________________________________________________________________
10
Classwork:
1) p.349 ST1 (1-6)
2) p.346 Class Exercises (1-6)
_______________________________________________________________________
Thm: the measure of an inscribed angle = half the measure of the
intercepted arc
* 2 inscribed angles which
intercept the same arc
 angles are congruent
* An angle inscribed in a
semicircle  right angle
Proof of theorem on next page…
11
________________________________________________________________________
Corollary 1:
2 inscribed angles which intercept the same arc are congruent.
12
Corollary 2:
An angle inscribed in a semicircle is a right angle.
Corollary 3: quadrilateral inscribed in a circle  opposite angles are
supplementary
Thm: an angle formed by a tangent & a chord  its measure = half the
measure of the intercepted arc
________________________________________________________________________
Examples (p.353 #4 – 9) Find the value for x and y in each question.
13
Inscribed Angles Notes
Geometry
Name ___________________________
Date __________ Block ___________
______________________________________________________________________
14
Angles formed by a tangent and a chord – Notes
________________________________________________________________________
15
Thm: If 2 chords intersect inside a circle  the products of the segments
on each chord are equal
Proof of theorem:
16
Thm: an angle formed by 2 secants/tangents  its measure = half the
difference of the measures of the intercepted arcs
________________________________________________________________________
p.359 CE(1-10)
6
17
Other Angle Relationships
Geometry
Name _______________________
Date _____________ Block _____
___________________________________________________________________
18
19
Circle Formulas
Geometry
Name______________________
Date __________ Block _____
For each of the given diagrams, fill in the appropriate formulas for the
angle measures and lengths of the segments.
Q
m QOP 
P
O
m QRP 
R
Circle with Center O
D
C
m ACB 
A
B
O
m ABD 
Diameter AB and tangent BD
R
m QUR 
T
U
O
Q
Segment Relationship:
S
H
m JHL 
J
O
m HJL 
L
Segment Relationship:
K
HJ and LJ are tangents
20
U
m VUW 
X
Segment Relationship:
Y
O
V
W
Secants UXV and UYW
M
R
m PMN 
N
O
Segment Relationship:
P
Secant MRN and tangent MP
Segment Relationship:
B
O
Arc Relationship:
E
C
D
BD  OC
m FG
F
O
length of FG 
G
21
Review for quiz WS1
Geometry -- chapter 9
Name _____________________________
Date ______________ Block _________
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22
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23
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24
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25
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26
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27
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28
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29
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30
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31
Review for Quiz WS2
Geometry – chapter 9
Name ____________________________
Date ______________ Block ______
In Circle E, m BD =200, m DF =1800, and m
AF
=450.
GA is tangent to
circle E at A. Find the following:
1. m<CAG = _______
2. m<GCA = _______
3. m<CGA = _______
_____________________________________________
In the figure, XY is tangent to circle Z at X.
4. If m XW = 950, find m<YXW. _______
5. If m<YXW = 1000, find m XW . _______
6. If m XW = x + 15, find m<YXW in terms of x. _______
________________________________________________________________________
In Circle P, m<LPJ = 300 and m<KMJ = 450. Find the following:
7. m<LMP = _______
8. m<JPK = _______
9. m<MJK = _______
10. m<LPM = _______
11. m MLJ = _______
12. m<MPK = _______
13. m<JPK = _______
14. m LJ = _______
15. m KM = _______
16. m<PLM = _______
32
In Circle J,
17.
SL 
JP  KL at S.
_______
For question 18 & 19,
give answers in simplest radical form!
18. IF JL = 4, and JS = 1,
What is KS? _______
What is KL? _______
19. IF JK = 26 and JS = 11,
What is KS? _______
What is KL? _______
______________________________________________________________________
Determine the length of an arc with the given central angle measure,
m<P, in a circle with radius r. Give answers in terms of  and then
rounded to the nearest tenth.
20. m<P = 400; r = 6
21. m>P = 200; r = 8
22. m>P = 1180; r = 30
23. m>P = 1300; r = 61
______________________________________________________________________
Determine the degree measure of an arc with the given length, L, in a
circle with radius r. Give answers rounded to the nearest degree.
24. L = 27; r = 5
25. L = 100; r = 79
26. L = 35; r = 11
27. L = 2.3; r = 85
33
Thm: If 2 chords intersect inside a circle  the products of the segments
on each chord are equal
____________________________________________________________________
34
Thm: If 2 secants are drawn to a circle from an external point  the
products of the external secant and the whole secant are equal
Proof of theorem:
______________________________________________________________________
35
Thm: If a secant segment and a tangent segment are drawn to a circle
from an external point the product of the secant segment and its
external segment is equal to the square of the tangent segment
Proof of theorem:
_____________________________________________________________________
36
Segments in Circles WS
Geometry
Name _____________________
Date __________ Block _____
B
C
1. Chords AC and BD intersect at point E.
a) If AE = 5, AC = 13, and DE = 10, find BE.
A
b) If AE = 3, CE = 4x+1, DE = 9, and BE = 2x-1, find x.
D
c) If AC bisects BD , AE = 8, and EC = 32, find BD.
2. In Circle O, diameter HJ is perpendicular to
chord FG at K. If HO = 13 and FG = 10, how far
is the chord from the center of the circle?
H
G
O
K
J
F
P
3. In Circle O, tangent PT and secant PBA
intersect at point P, outside the circle.
a) If PT = x, PB = 3, and AB = 13, find x.
B
T
b) If PT = 3, PB = x, and AB = 8, find x.
A
P
4. Secants PBA and PCD intersect at point P,
outside the circle.
a) If PB = 8, PA = 18, PC = 9, and PD = x, find x.
B
C
b) If PB = 4, AB = 17, PC = x, and CD = 5, find x.
c) If PB = 6, AB = 9, PC = 8, and PD = x, find x.
D
A
37
Circle Proofs WS 1
Geometry
Name ___________________
Date _______ Block _______
1. Given: Circle O with diameter AOD ,
tangent CA , mAB  2mAE
Prove: ADOA = OCBD
2. Given: Circle O, tangents PR and PV
Prove:
RPO  VPO
3. Given: CT  CH
Prove: HA  TD
4. Given: chords AB and CD of circle O intersect at E, an interior point of
circle O;
chords AD and CB are drawn.
Prove: (AE)(EB) = (CE)(ED)
38
5)
6)
7)
8)
________________________________________________________________________
9)
39
Circle Proof WS 2
Geometry
Name __________________
Date ___________ Block _______
______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
40
______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________
41
_______________________________________________________________________
_______________________________________________________________________
______________________________________________
________________________________________
42
Review Chapter 9 WS 1
Geometry
1.
2.
Name _____________________________
Date ___________ BlockA_________
3.
8
|----------x------------|
y
3
2
x
4
E
x
B
C
T
D
R S
4
10
________________________________________________________________________
80 o
o
280
4. A
5.
6.
7. 310 o
8.
x
B
x
O
C
O
x
120 o
90
o
100 o
B
x
A
A
B
x
B
A
B
70
A
D
________________________________________________________________________
C
H
A
3y
9.o
10. o
11.
12. 100 o
13. B
B
100
70
C
E
O
x
G
40 o
F
x
x
B
o
y
y
O
60
O
x
A
o
A
C
O
x
C
O
B
D
________________________________________________________________________
80 o
x
100 o
80 o
14.
15.
16.
17.
18.
70
A
B
A
o
A
B
x
O
x C
C
B
30 o
O
D
O
C
B
A
O
D
D
B
A
70 o O
x
C
C
D
D
x
________________________________________________________________________
120 o
19.
20.
21.
22.
23.
4y
A
260 o
A
x
D
x
y
E
70 o
x
H
T
B
T
G
T
________________________________________________________________________
C
C
A
o
24. 100 o
25. A 50 C 26.
27. 40 o
28. 2y
D
A
40 o
C
100 o
x
x
O
T
100
A
B
A
0
O
B
D
1400
B
O
x
x
B
x
T
y
B
O
O
D
C
A
O
O
y
80 o
x
O
O
O
x
A
2y
A
D
D
43
29.
A
5y
30.
60 o
31.
32.
260 o
33.
3y
3y
E
O
O
O
y
y
100 o
B
O
O
2y
y
y
D
x
x
x
x
x
C
________________________________________________________________________
34.
given: AB is tangent; AC is secant; BC, DE, FC
B
A
4
1
are chords; mEB  50 ; mBC  4x  50 ;
E
mCD  x ; mDF  x  25 ; mFE  x  15
Find:
mBC 
m 1
O
F
3
2
mCD 
m 2
mDF 
m 3
mFE 
m 4
________________________________________________________________________
C
C
D
35.
36.
37.
38.
D
x
8
B
E
12
C
4
A
6
E
A
16
O
C
B
2
A
B
5
B
A 4
O
O
D
Find AB
C
AB = 8
Radius = ?
D
PD = 12
________________________________________________________________________
39.
40.
41.
O
O
3
x
50 o
O
x
60 o
x
radius = 4
________________________________________________________________________
A
D
42.
43.
44.
In circle O, radii OA, OB ,
G
and chord AB are drawn.
If OA = 2x+8, OB = x+24, and
AB = 3x-8, find OA, AB, and
m<AOB.
O 60 o
C
O
x B
x F
E
m DFE = 170
o
44
P
x
Circles Chapter Review WS 2
Geometry
Name _______________________
Date __________ Block ______
NOTE: Diagrams may not be to scale!!!!!!
1. Find the arc length for Circle P with radius 6 if m<P = 40 (nearest tenth).
2. Find the measure of the central angle that intercepts an arc of length
27 on a circle with a radius of 5 (nearest degree).
__________________________________________
3. If JP  KL at S, JK = 26, and JS = 11, find:
J
KS = _____
KL = _____
_____________________________________
4. If m<LPJ = 30 and m<KMJ = 45, find:
JK = ______
S
K
J
L
P
L
MK = ______
P
K
LM = ______
m<LMP = ______
m<PLM = ______
______________________________________
5. If mBD  20, mDF  180, mAF  45 , find:
M
A
m<CAG = ______
B
C
G
m<GCA = ______
F
E
D
m<CGA = ______
________________________________________________________________________
Find the value of x. Show algebra for questions 6 - 25.
11
6.
7.
8.
16
12
x
6
x
4
x
16
4
7
45
9.
10.
6
3
100o
11.
250 o
x
x
x
50o
12.
13.
14.
x
145 o
x
63 o
15
10
x
15.
16. (nearest tenth)
17. (nearest tenth)
6
x+3
2x
x+3
130 o
x
x
4
70 o
18.
4x
19.
20.
S
R
87 o
94 o
x
54
T
m TSR  4 x  15
m RTS  5 x  15
3x+9
46
21.
22.
23.
1
87 o
26 o
3x-9
92o
5x+9
2
x
m 1  2.5 x
m 2  1.5 x  14
24.
P
4x
3x
Q
R
5x
Find m R .
________________________________________________________________________
26. In Circle O, FA is tangent, FEDB is a secant, ADC and AB are
chords, m CE = 40, m AB = 130, and m<CAB = 60.
C
a) m BC = _____
b) m<EBA = _____
c) m<ADE = _____
B
D
E
F
O
d) m<F = _____
e) m<FAC = _____
A
47
27. In the accompanying diagram of Circle O with inscribed isosceles
triangle ABC, AB  AC , m BC = 60, FC is a tangent and secant FBA
intersects diameter CD at E.
A
D
a) m<ABC = _____
E
b) m AD = _____
O
c) m<DEB = _____
d) m<AFC = _____
B
C
e) m<BCF = _____
F
_______________________________________________________________________
28. In the accompanying diagram of Circle O, secant ABP , secant CDP ,
and chord AC is drawn; chords AD and BC intersect at E, tangent GCF
intersects circle O at C, and m AB : m BD : m DC : m CA = 8:2:5:3.
A
G
a) m CA = ______
b) m<ACB = ______
c) m<P = ______
d) m<AEB = ______
O
B
C
E
e) m<DCF = ______
D
P
F
________________________________________________________________________
29. In the accompanying diagram of Circle O, AOED is a diameter, PD is
a tangent, PBA is a secant, chords BD and BEC are drawn, m<DAB = 43,
and m<DEC = 72.
a) m<BDP = ______
A
B
b) m AB = ______
c) m AC = ______
P
O
d) m<P = ______
E
e) m<CBD = _____
D
C
48
Chapter 9--Theorems/Corollaries/Postulates
Formulas to know:
C d
A   r2
x
length of arc: l 
 d  ; x = measure of central
360
________________________________________________________________________
Hints: draw radii to endpts. of a chord [look for special right s]
find isosceles s formed w/ radii and a chord
find right s formed w/ tangent [radii or diameter a side of the ]
_______________________________________________________________________
Basics
1. line tangent to
 line is  to radius @ pt. of tangency
2. [coplanar line & ] line  to radius @ pt. on
 line tangent to
3. tangents to
from exterior pt. are 
4. [in 1
or 
s]  arcs   chords
5. diameter  to chord  bisects the chord & its intercepted arc
[then can use bisect to midpoint to congruent segs]
[then to congruent arcs]
6. diameter bisects a chord   to the chord at midpoint of chord
7. [in 1 or 
s] 2 chords equidist. from center   chords
8. 2 inscribed angles which intercept the same arc are 
9. an angle inscribed in a semicircle is a right angle
10. quad inscribed in
 opp. angles are supplementary
11. parallel lines which intersect circle intercept  arcs
_______________________________________________________________________
Angles
1. central
= measure of intercepted arc
2. inscribed
= 12 measure of intercepted arc
3.
formed by tangent & chord has measure = 12 the intercepted arc
4.
[notice this does not work for a secant and chord!]
formed by 2 chords which  inside
has measure =
1
2
sum of the 2
intercepted arcs
formed by 2 secants
 measure = 12
formed by 2 tangents
difference of the
formed by 1 secant & 1 tangent
intercepted arcs
_______________________________________________________________________
Segments

1. 2 chords  inside
products of segments formed on each chord are =

2. 2 secant segs to
products of external seg & whole secant for each secant seg are =
3. secant & tangent seg to 
5.
product of ext. seg & whole secant =  tan seg 
2
49