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Page 1 of 36
Circles
Name: ___________________________________________________________
Homework: Quizzes: Exams:
Page 2 of 36
Do Nows:
1.
2.
3. The vertex angle of an isosceles triangle measures
15 degrees more than one of its base angles. How
many degrees are there in a base angle
of the triangle?
(1) 50 (2) 55 (3) 65 (4) 70
4.
5.
6.
Page 3 of 36
7.
10.
The three medians of a triangle intersect at a point.
Which measurements could represent the segments of
one of the medians? (1) 2 and 3 (3) 3 and 6
(2) 3 and 4.5 (4) 3 and 9
8.
11. A regular polygon has an exterior angle that measures
45°. How many sides does the polygon have?
(1) 10 (3) 6
(2) 8 (4) 4
9.
Two prisms with equal altitudes have equal
volumes. The base of one prism is a square
with a side length of 5 inches. The base of
the second prism is a rectangle with a side
length of 10 inches. Determine and state, in
inches, the measure of the width of the
rectangle.
12. Triangle RST is similar to _XYZ with RS _ 3 inches and
XY _ 2 inches. If the area of _RST is 27 square inches, determine and state the area of _XYZ,
in square inches.
Page 4 of 36
Intro to Circles - Definitions, Arcs and Central Angles
1. Name a circle using the center point. ⨀ 2. The angle measure of a circle is __________________ degrees 3. A chord is a segment whose endpoints are on the circle. ________ and ________ are chords.
4. A diameter is a chord that passes through the center of a circle. _________ is a diameter. 5. A radius a segment with one endpoint on the circle and one endpoint at the center. ________ and________ are radii. Theorem: All radii of a circle are congruent
6. A secant is line that passes through a circle at two points. ______ is a secant. 7. A tangent is a line that intersects a circle at exactly 1 point. _____ is a tangent. Sketch each of the following: Circle P with radius
Circle P with tangent Circle A with secant
Circle Z with diameter
Circle Z with chord
P
A
B
C
D
F
EG
H
tangentsecant
diameter / chord
chord
Page 5 of 36
9. An arc is a section of a circle, and is named with 2 or 3 letters and the arc symbol. or .
A major arc measures more than 180°. _________ is a major arc. (use three letters)
A minor arc measures less than 180°. ____________ is a minor arc. (use two letters)
A semicircle is an arc that measures 180°. ____________ is a semicircle. (use three letters) 10. A central angle is an angle with its vertex at the center of a circle and endpoints on the circle. ________ is a central angle. An arc is intercepted by a central angle, or a central angle intercepts an arc. __________ intercepts _____________ 11. Theorem: The measure of an arc is equal to the measure of its central angle.
If mAPB = 60, then m = ________.
Apply what you know:
A _____________________ intercepts a semicircle
P
A
B
m AB = m APB
P
C
D
m CD = 1/2 m CMD
M
Page 6 of 36
Keep these facts in mind: All radii are congruent (look for isosceles triangle) Arc measures sum to 360° Central angle and arc have equal measures Diameter intercepts a 180° arc
Examples: 1. Find the measure 2. is a diameter and m∠FMG = 46°.
of and Find m , m , and m .
3. Diameter SOT is drawn in circle O. 4. In ⊙C, mPCQ = 82 and PR QR .
If mROT= 110, find m RS and m STR . What is the measure of PR?
R
O
S
TU
C
82o
P
Q
R
Page 7 of 36
Practice with arcs and central angles
1. 2.
3. m =78° and m = 68°.
Find m and m .
4. is a diameter. Find the
measures of SR)
, TS)
, and RUTº .
5. Points D, J, and K are located on
circle O. The three points divide
the circle into arcs that are in a
3:4:5 ratio. What is the measure
of the three arcs?
6. In ⊙P, APB BPC, and m
ADC = 208. What is the measure
of AB ?
7. In ⊙P, m AB = 108. What is
the measure of A?
8. Diameter SOT is drawn in
circle O. If m = 118, find m RS
and mROT.
9/ In O, mAB = 60, AB BC,
and mCOD = 150. Find
mAOB, mAC, mBC, mADC,
mAOD.
P
A
B28
o (3x + 7)o
Find the value of x
O
M
142o
Find m MN and m MQN
N
Q
P
A
B
C
D
O
S
102o
T
R
U
PA
B
C
D
P
A
B
R
O
S
TU
C
O
A
B
D
Page 8 of 36
Inscribed Angles
Inscribed Angle – An angle whose vertex is on a circle and whose rays intersect the circle
Inscribed Angle Theorems:
The measure of a central angle equals ½ the measure of its intercepted arc
Inscribed angles which intercept congruent arcs are congruent
An inscribed right angle intercepts a semicircle
Examples:
In circle P, ACB is an inscribed angle and In ⊙P, APB is an central angle and
measures 28. What is the measure of AB ? ADB is an inscribed angle. If mAPB =
110, find mADB
C and D are inscribed angles in ⊙P. In ⊙O, is a diameter. If RS=4 and RT=10,
If mC = 21, what is mD? find the length of in simplest radical form.
Prove ∆ADQ ~ ∆BCQ
P
A
B
m AB = m APB
P
C
D
m CD = 1/2 m CMD
M
P
A
B28
o
CP
A
B
D
110o
P
A
B
C
D
Q
RO
S
T
Page 9 of 36
Practice with Inscribed and Central Angles
1. Find x 2. Find x, y, and z 3. Find x
4. Find x, y, and z 5. Find x 6. The endpoints of two
intersecting diameters form 2 triangles. Find w, x, y, and z
7. Given circle O,
If AOB = 40o, find m AB .
If m = 60o , find DEF
If OA = (3x + 6) and OB = (4x – 4), find OA.
m
m
Page 10 of 36
8. If CAB = 60o and
, find m , BDC, m , m , and m
9. In O, ROT = 90. Find m and mRST.
10. P Q. The radius of P equals (2x + 8) and the radius of Q equals (3x - 4).
Find the value of x 11. In O, m = m = 50, and mCOD = 150. Find: a) mAOB d) mAC b) AD e) mADC c) mAOD f) mBAD 12. In O, m = 110. Find mABC. 13. In O, mRST = 58. Find m .
m
m
C
O
A
B
D
O
A B
C
O
R
T
S
O
R
T
S
Page 11 of 36
Chord / Arc / Tangent Relationships
Theorem: The arcs between congruent chords are _____________________________________________________ Theorem: Congruent chords intercept __________________________________________________________________ example
In ⊙A, MN = JK. If m MN = equal to (3x + 8) Find x.
and m JK is equal to 83, what is the value of x?
Find x and y. In circle ⊙P, AB CD and AB //CD .
If m = 75, find m CD .
A
K
N
J
M
PA
B
C
D
P
A
BC
D
118o
78o
x
Page 12 of 36
Theorem: Given two chords, the longer chord is _______________________________________________________ Theorem: Two tangents from the same point to a circle are ______________________________ Theorem: A radius to a point of tangency forms a _______________________ with the tangent.
Ex) In circle P, chords AB, CD, and EF Ex) mQ = 70. and are tangents. and
Are drawn. If AB = 4, CD = 5, and . Find mRST and m .
EF = 6, which chord is furthest from
point P? Which is closest?
Ex) EFC, EHD, and CGD are tangents. Ex) In ⊙F radius FL is drawn to tangent KL .
EF = 6, HD =7, and CG = 4. If LF = 8 and KL = 12, what is the length Find the perimeter of ∆CDE of in simplest radical form?
L
K
F
M
P
A
B
Q
Page 13 of 36
.
Practice With Chord / Arc / Tangent Relationships
1. ABC is inscribed in
⊙O, and measures 35.
If AB BC , what is the
measure of AB ?
2. In ⊙F radius FL is
drawn to tangent KL . If
mK = 29, what is the
measure of F?
3. . Find m .
4. If CD = EF, find m .
5. In O, . If m = (5x-2) and m = (3x + 12), find x.
6. Arcs AB, BC, and AC are congruent to each other. Find mACB.
7. Tangent KL and radius FL are drawn in circle F. If FL = 10 and KL = 24, what is the length FK?
8. In P, chord RS = 7 and chord TU = 5. Which is closer to point P?
O
35o
x
A
B
C
L
K
F
M
O
A B
C
D
Page 14 of 36
Use this figure for problems 9 and 10:
9. ∆ABC is circumscribed about circle P. Which of the following must be true?
(A) mC = mB (C) AC = AB
(B) the perimeter of ∆ABC is equal (D) DB = DE
to one-half the circumference of circle P
10. ∆ABC is circumscribed about circle P. If DB = 8, AD = 14, and CF = 10, what is the perimeter of
∆ABC?
11. In circle O, tangents KL and KM are drawn from point K. OL and
OM are radii. If OL = 5 and KNO = 15, what is the perimeter of KMOL?
Use this figure for #11 and 12
12. In circle O, tangents and are drawn from point K. and
are radii. If m = 60, what is
mLKO?
13. , m = (5x + 2), and mFD = (3x + 12). Find x.
PA
B
C
D
E
F
L
K
O
M
N
Page 15 of 36
Mixed Review #1
1. m = 100, m = 110, and m = 96. Find m , mA, mB, mC , and mAEB.
2. In O, ABCD is a rectangle. If mCD = 120, find mBC, mAB and mAD.
3. Find the measure of angles A, B, C, and D.
4. In the diagram below, , , and are tangents to circle O at points F, E, and D, respectively, , , and .
What is the perimeter of ?
5. If measures 160o and measures 80o, find mJLH.
6. In the diagram below of circle O, is
parallel to .
Which statement must be true?
a. b.
c. d.
7. In the diagram below of , is tangent to circle O at point A, , and .
What is the length of ?
O
D C
A B
E
O
A B
C D
Page 16 of 36
8. In the diagram of circle O below, chords and
are parallel, and is a diameter of the circle.
If , what is ?
9. In the diagram below of circle O,
, and .
Which statement must be true?
a. b.
c. d.
10 Line segment AB is tangent to circle O at A. Which type of triangle is always formed when points A, B, and O are connected? a. right b. obtuse c. scalene d. isosceles
11. In the diagram below of circle O, chords
and intersect at E.
Which relationship must be true? a. b.
c. d.
12. In the diagram of circle O below, chord is
parallel to diameter and .
What is ?
13. How many common tangent lines can be drawn to the two externally tangent circles shown below?
Page 17 of 36
14. In the diagram below, circle O has a radius of 5,
and . Diameter is perpendicular to
chord at E.
What is the length of ?
15. In the diagram below, trapezoid ABCD, with
bases and , is inscribed in circle O, with
diameter . If , find .
16. GHIJKL is a regular pentagon. Find m
17. The measures of DF and CF are in a 4:3 ratio. If F = 110o, find the measure of .
18. Equilateral triangle BPL is inscribed in circle Q. Find BQL.
619. In the diagram below, circles X and Y have two
tangents drawn to them from external point T. The points of tangency are C, A, S, and E. The ratio of TA to AC is . If , find the length
of .
20. In the diagram below of circle O, chords , ,
, and are drawn such that
. Identify one pair of inscribed angles that are congruent to each other and give their measure.
m
m
Page 18 of 36
Angles formed by intersecting chords, secants, and tangents Vertex inside the circle angle measure = ½ the sum of the
intercepted arcs
m1= m2= ½ (m AC + m DB )
m3 = m4 = ½ (m BC + m AD )
Vertex on the circle angle measure = ½ of the intercepted arc
mABC = ½m BC
mDEF = ½
Vertex outside the circle angle measure = ½ the diference of the intercepted arcs
mP = ½(m ACB - m AB )
mP = ½(m CD - AB )
mP = ( AC - AB )
When the vertex is: ON the circle, angle = ½ the arc IN the circle, angle = ½ the sum of the arcs
OUTSIDE the circle, angle = ½ the difference of the arcs
O
A
B
C
D
12
4
3
O
AB
CD
E
F
O
B
A
C P
OB
A
D
P
C
OB
A
C
P
Page 19 of 36
Let’s Try It - Apply the Formulas Going Forward
1. m = 29 and m = 37, find mAMB. Write the equation: Substitute values: Evaluate:
2. m = 218. Find mWXY Find : Write the equation: Evaluate:
3. m = 82 and m = 22. Find mP Write the equation: Evaluate:
4. m = 280. Find mP. Find m : Write the equation: Evauate
P
A
B
C
D
M
C
Y
X
W
Z
OB
A
D
P
C
O
B
A
C P
Page 20 of 36
Let’s Try It - Apply the Formulas Going Backwards!
5. mAMB = 32 and m = 40. Find m .
6. mP = 45 and m = 75. Find m .
7. mP = 18 and m = 24. Find m .
P
A
B
C
D
M
OB
A
C
P
OB
A
D
P
C
Page 21 of 36
Practice
1. Chords AGB and CGD intersect at G in circle P.
m AC = 41 and m BD = 133. Find mAGC,
mBGD, mAGD, and mBGC.
2. In ⊙A, MAKN is a diameter, m GN = 105 and
mGKM = 64. Find m MF and m NF .
3. In circle P, chord ST intersects tangent TR at
point T. m ST = 152. What is the measure of
RTS?
4. In circle P, tangent TR intersects chord TV at
T. If m TV = 116, what is the measure of RTV?
5. In ⊙P, secants RST and RUV intersect at R.
If the measure of TV = 140 and the measure of
arc SU is 42, what is the measure of R?
6. AM andAN are tangent to ⊙P at M and N.
If the measure of A is 32, what are the
measures of arcs MLN and MN
PA
BC
D
G
A
M
N
F
G
K
P
T
R
S
152o
ox
PV
T
R
S
116o
ox
P
R
S
T
U
V
P
AM
N
L
Page 22 of 36
7. In ⊙A, m LF = 48 and m OP = 52. What
is the measure of 1?
8. In ⊙A, m LO = 59 and m FP = 151.
What is the measure of 2?
9. In ⊙O, chords UES and TER intersect at
point E. If m UT =38
and m RS = 74, what is the measure of
TES?
10. Secant RST intersects ⊙C at S and T, and
secant RUV intersects ⊙C at U and V. If
m TV = 117 and m SU = 21, what is the
measure of angle R?
`
11. Secant RST intersects ⊙C at S and T, and
secant RUV intersects ⊙C
at U and V. If mR = 39 and m SU = 29,
find m TV .
12. AE is tangent to ⊙C at E, and secant
ABD intersects ⊙C at B and D. If m EB =
98 and mA = 49, what is the measure
of ED ?
A
2
O
P
1
F
L
A
2
O
P
1
F
L
OT
R
U
S
E
C
R
S
T
U
V
C
R
S
T
U
V
C
A
B
E
D
Page 23 of 36
13. Tangent AE and secant ABD intersect
⊙C. If m EB = 94 and m BD = 76, find
mA.
14. Tangents FP and FL intersect circle O at
P and L. If mF = 62, what is the
measure of PL?
15. Given ⊙P with diameter BPD
intersectingAEC at E, m AB = 112 and
m BC = 52, what is the measure of
AED?
16. In ⊙C, tangent XW intersects chord XY
at X. If m WY = 142, what is the measure
of WXY?
17. PNM is tangent to circle C at N. If
mONP = 85, what is the measure of
NLO ?
18. PNM is tangent to circle C and intersects
chord ON at N. If m ON :m OLN = 2:3,
what is the measure of PNO?
C
A
B
E
D
O
IP
FL
PA
B
C
D
E
C
Y
X
W
Z
C
M
N
O
P
L
C
M
N
O
P
L
Page 24 of 36
Segments Formed by Chords, Tangents and Secants
Intersecting chords
products of the parts are equal
ab = cd
Intersecting Secants
outside whole = outside whole
PA PC = PB PD
Intersecting Tangent and Secant
outside whole = outside whole
PA2 = PB PC
Intersecting Chords – product of the parts are equal
BE = 6, AE = 12 and DE = 8. b) DC = 12, BE = 4, and AE = 9.
Find EC. If DE > EC, find DE and EC.
a
b
c
d
OB
A
D
P
C
OB
A
C
P
O E
A
DC
B
O E
A
DC
B
Page 25 of 36
Intersecting Secants and Tangents– outside x whole = outside x whole
If BC = 9, AD = 12, and AE = 30, find AB. If AB = 9, BC = 3, AD = 6 , find DE.
Secant KNG intersects ⊙P at N and G. Tangent KI intersects ⊙P at I. If KN = 5 and NG = 7, find IK.
Express your answer in simplest radical form.
P
B AC
D
E
P
B AC
D
E
P
K
I
N
G
Page 26 of 36
Practice with Segments Formed by Chords, Secants and Tangents
1. In circle P, chords KL and MN
intersect at E. If KE = 8, EL = 11, and
EM = 5, what is the length of EN ?
2. Chords KL and MN interset at E in
⊙P, KE = 24, EL = 32, and NE:EM = 4:1.
Find NE in simplest radical form.
3. Chords ST and QR intersect at X in
⊙A. If XT = 14, XS = 16 and QX = 12,
find XR to the nearest tenth.
4. In circle A, XR = 15, QX = 8, and TX =
10. What is the length of XS?
5. Secant and tangent intersect
at P. BP = 4 and PC = 12. Find AP in
simplest radical from.
6. Secants BSU and BHT intersect at B. If
BS = 8, SU = 2, and BH = 5, what is the
length of HT ?
P EM
N
K
L
P EM
N
K
L
A
X
R
TS
Q
A
X
R
TS
Q
OB
A
C
P
O
B
H
T
S U
Page 27 of 36
7. BS = 6, BH = 3 and BT = 15. Find SU.
8. AR = 7 and RS = 9, what is the length of
AQ ?
9. Tangent AQ intersects ⊙O at Q. If AR
= 3 and AQ = 6, findRS ?
10. AB = 20, KI = 5 and IJ = 20. Find AI?
11. Tangents and intersect at P. AP
= 4x + 12 and PB = 8x – 4. Find PA and
PB
12. If a = b, c = 14 and d = 21, find
a and b in simplest radical form.
O
B
H
T
S U
O
Q
A R
S
O
Q
A R
S
OA
B
J
K
I
O
B
A
C P
a
b
c
d
Page 28 of 36
Mixed Review #2
1. In the diagram below of circle O, is tangent to
circle O at A, and is a secant with points B and C on the circle.
If and , what is the length of ?
2. In the diagram below of circle O, diameter
is perpendicular to chord at point E, , and .
What is the length of ?
3. In the diagram below, is inscribed in circle P. The distances from the center of circle P to each side of the triangle are shown.
Which statement about the sides of the triangle is true? a. b. c. d.
4. In the diagram below, tangent and secant
are drawn to circle O from external point P.
If and , what is the length of ?
5. In the diagram below of circle O, chords and
intersect at E, , and .
What is the degree measure of ?
6. In the diagram below of circle O, chords and
intersect at point B, such that and
.
What is ?
Page 29 of 36
7. In the diagram below of circle O, secant
intersects circle O at D, secant intersects circle O at E, , , and . What
is the length of ?
8. Tangents and are drawn to circle O from
an external point, P, and radii and are drawn. If , what is the measure of
?
9. In the diagram below of circle O, chords and intersect at E. If , , and ,
what is the length of ?
10. In the diagram below of circle C, ,
and . What is ?
11. In the diagram below, is a tangent to circle O
at point S, is a secant, , , and
. What is the length of ?
Page 30 of 36
12. In the diagram below of circle O, chord bisects chord at E. If AE = 8 and BE = 9, find the length CE in simplest radical form.
15. In the diagram below, tangent and secant are drawn to circle O. The ratio
m : m :m is 3:4:5 . Find mLMK.
13. and are all tangent to circle O . JA = 9, AL = 10, and CK = 14. Find the perimeter of JKL.
14. Given diameter and mBOC = 57, find the m BAC.
16. is a diameter and m = 40, what is mYAC? (The figure is not drawn to scale.)
17. m = 96 and m = 67. Find m A.
18. Find the value of x for m = 46 and m = 25.
L
J
K
O
C
A B
A
C
O
B
B
A
Y
O
C
D
A
E
C
B
C
A
D
B
O
x
Page 31 of 36
19. Find the value of x. Round to the nearest hundredth if needed.
20. Find the value of x. Round to the nearest
hundredth if needed.
22. AB = 20, BC = 6, and CD = 8. Find the value of x.
23. In circle P, chord measures 5 inches, chord
measures 6 inches, chord measures 7
inches, and chord measures 8 inches.Which chord is closest to the center of the circle?
21. PS = x, PQ = 3, and PR = x + 18. Find PS. 24. Circles A and B do not intersect. Sketch the circles and all common tangents.
a
D
C
BA
P
x°
x
23
7
C
A
B
x
D
Page 32 of 36
Radian Measure and Arc Length Degrees are not the only units for measuring an angle! Radian – A unit of angle measure. In a circle, a central angle measuring one
radian intercepts an arc with length equal to the radius of the circle.
Degree – Radian Conversion Formula
(degrees)
p
180° = (radians)
(radians)
180°
p = (degrees)
Convert the following to radians: Convert the following to degrees:
(round to the nearest hundredth) (round to the nearest tenth)
42 = ______________ radians 0.5 rad = ______________ degrees
90 = ______________ radians 1.21 rad = ______________ degrees
152 = ______________ radians π/3 rad = ______________ degrees
(in terms of π) 3π/2 rad = ______________ degrees
180 = ______________ radians 0.24 rad = ______________ degrees
60 = ______________ radians 0.39 rad = ______________ degrees
PP A
B
S
R
1 radian
arc length S = radius R
Page 33 of 36
Arc length
a) What is the meaning of circumference? _______________________________________
b) What is the formula for circumference of a circle? _____________________________
c) How many radians is it around a circle? ________________________________
d) An arc with an angle measure of 2π radians has a length of _________________
e) An arc with an angle measure of 1 radian has a length of ________________
Arc Length = ______________ (angle in radians)
Or
Arc length = _______________ (angle in degrees)
Examples:
1. The radius of a circle is 4 inches. Find the length of the arc intercepted by a central angle measuring 1.2
radians.
2. The radius if a circle is 3cm. What is the length of the arc intercepted by a central angle measuring 42?
Round your answer to the nearest tenth of a centimeter.
3. What is the measure of the central angle that intercepts an arc with length equal to 12 if the circle has a
radius of 20? Express your answer in radians.
Page 34 of 36
4. Frannie is planting a flower bed in the area around a corner of her backyard fence. The two sections of
fence come together at an angle of
4p
3 radians, and she wants the flow bed to be a have a radius of 6 feet.
Frannie wants to put a brick border around the flower bed. What will be the length of the border, to the
nearest foot?
5 FKG is an inscribed angle measuring
p
6 radians. What is the length of FG if the radius of the circle is 10
meters? Express your answer in terms of π.
.6
4π/3
Page 35 of 36
Practice with Arc Length and Radian Measure
1. What is the degree measure of an angle
measuring
5p
6radians?
2. What is the degree measure of an angle
measuring
p
9 radians?
3. What is the measure in radians of an angle
whose measure is 45?
4. What is the measure in radians of an angle
whose measure is 270?
5. Arc RW is centered at point O. If mROW
= 115 and RO = 9, what is the length of
RW the nearest hundredth.
6. What is the length of arc S to the nearest
tenth?
7. What is the length of an arc of a circle
whose diameter equals 15 and whose
central angle measures 0.6 radians?
8. Inscribed angle I measures
2p
15 radians
and radius BO = 5. What is the length of
arc BG ?
R
O
W
4π rad9
S
12
O
B
I
G
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9. PAQ is a central angle in ⊙A and RBS
is a central angle in ⊙B. If AP/BR = 4, the
ratio
length PQ
length RS equals
The image of ⊙X after D4 is ⊙Y. What is
the ratio of an arc length on ⊙X to an arc
length on ⊙Y if the arcs are intercepted
by congruent central angles?
(1)
1
4 (2)
1
16
(3) 4 (4) 16
10. Two arc lengths, S1 and S2, are defined by
the accompanying figure. Which arc is
longer? Justify your answer.
13. ∆ABC is inscribed in ⊙P. AB = BC, mB =54
and PB = 4. What is the length of arc BC ? Round
to the nearest tenth.
0.28 radians
7
0.6 radians
3 S2S1
