Download UNIT 12: C - Humble ISD

Name:
Period
GL
UNIT 12: CIRCLES
I can define, identify and illustrate the following terms:
Interior of a circle
Chord
Exterior of a circle
Secant of a circle
Tangent to a circle
Point of tangency
Central angle of a circle
Arc
Minor Arc
Major Arc
Semicircle
Sector of a circle
Arc length
Inscribed angle
Intercepted arc
Interior Angle
Exterior Angle
Diameter
Dates, assignments, and quizzes subject to change without advance notice.
Monday
Tuesday
1
(only see 2nd, 4th and 6th)
Central angles, Arc measures,
Arc Length and Area Sectors
8
Angles in Circles
15
Circle Equation
Arcs, Chords, and Tangents
2
(only see 3nd, 5th and 7th)
Central angles, Arc measures,
Arc Length and Area Sectors
9
MOCK EOC
16
Arcs, Chords, and Tangents
Block Day
3
(only see 6th,
4th and 2nd)
4
(only see 5th,
3rd and 1st)
10-11
MOCK EOC
17-18
Review/Practice Day
Friday
5
Angles in Circles
12 extended advisory
Angles in Circles
19
TEST #12 – CIRCLES(1)
Monday – Thursday 4/1 – 4/4
11-2, 11-4, 11-5 : Angles in Circles
I can solve for the measure of a central angle or its inscribed arc
I can find the arc length
I can find the area of the sector
I can name an arc
I can classify an arc as minor, semicircle, or major
PRACTICE:
Friday, 4/5
11-2, 11-4, 11-5 : Angles in Circles
I can solve for the measure of central, inscribed, interior, and exterior angles.
I can solve for the measure of an arc given a central, inscribed, interior, or exterior angle.
I can solve for the arcs and angles of a circle using algebra.
PRACTICE: Inscribed, Interior, and Exterior Angles
Monday, 4/8
11-2, 11-4, 11-5 : Angles in Circles
I can solve for the measure of central, inscribed, interior, and exterior angles.
I can solve for the measure of an arc given a central, inscribed, interior, or exterior angle.
I can solve for the arcs and angles of a circle using algebra.
PRACTICE: Mixed Practice: Angles in Circles
Friday, 4/12
11-2, 11-4, 11-5 : Angles in Circles
I can solve for the measure of central, inscribed, interior, and exterior angles.
I can solve for the measure of an arc given a central, inscribed, interior, or exterior angle.
I can solve for the arcs and angles of a circle using algebra.
PRACTICE: Complex Pictures Worksheet
Monday, 4/15
Equations of circles, arcs, chords, and tangents
I can solve problems using the properties of chords and tangents in a circle.
I can write the equation of a circle given the radius and center (by graph or words)
I can determine the radius and center given the equation of a circle.
PRACTICE:
Tuesday, 4/16
Equations of circles, arcs, chords, and tangents
I can solve problems using the properties of chords and tangents in a circle.
I can write the equation of a circle given the radius and center (by graph or words)
I can determine the radius and center given the equation of a circle.
PRACTICE:
Wednesday or Thursday, 4/17 or 4/18
Review for Friday’s test.
Friday, 4/19
Test 12: Circles
Score:
Equations of Circles
One of the first ways you learned to graph a line was probably by using a y-intercept (the starting point)
and a slope. That’s why slope-intercept form of a line is so useful:
y = mx + b, where m is the slope and b is the is the y-intercept
To graph a circle, what you really need to know is where the center is and how long the radius is.
You can find both from the standard form equation of a circle:
The equation of a circle is ( x − h) 2 + ( y − k ) 2 = r 2
where (h, k) is the center of the circle and r is the radius.
Example 1: Find the center and radius from the equation ( x − 2)2 + ( y + 4)2 = 36
Compare this to the standard form equation: ( x − 2)2 + ( y + 4)2 = 36
( x − h) 2 + ( y − k ) 2 = r 2
Since h is 2, the x-coordinate of the
center is 2. (Notice that the coordinate
is positive when there is a subtraction
sign in the parentheses. The sign will
always be the opposite.)
Center =
Since the second
parentheses has y + 4, the
value of k, the y-coordinate
of the center will be –4.
(2, –4)
Now look at the other side of the equation to find the radius.
( x − 2)2 + ( y + 4)2 = 36
( x − h) 2 + ( y − k ) 2 = r 2
Since r2 = 36, r = 36 = 6
To graph a circle, first plot the center:
Then count “r” units up, down,
left, and right to plot four points
around the circle.
Finally, use those four points
to sketch in the circle.
You can also use the standard
form of a circle to work backwards
and write the equation of a circle
with given information.
Example 2: Write the equation of a circle centered at (–4, 3) with a radius of 3 2 .
Using ( x − h) 2 + ( y − k ) 2 = r 2 , we’ll plug in h = –4 (the x-coordinate of the center),
k = 3 (the y-coordinate of the center),
(
( x − −4) 2 + ( y − 3) 2 = 3 2
)
2
and r = 3 2 (the length of the radius)
( x + 4) 2 + ( y − 3)2 = 9 ⋅ 4
( x + 4) 2 + ( y − 3) 2 = 18
Turn the page. Worksheet continues.
You can also write an equation of a circle by looking at it. All you need to do is locate the center and
count the length of the radius. (It’s usually easiest to find the center by drawing two diameters, one
vertical and one horizontal.)
Example 3: Find the center and radius, then write the equation of the circle.
Center appears to be at (–3, 2)
Radius is 4 units long
×
( x − h) 2 + ( y − k )2 = r 2
( x + 3) 2 + ( y − 2) 2 = 16
Your turn: Practice problems. Refer to examples 1, 2, and 3 for guidance.
Write the equation of each circle:
1. Centered at the origin with radius 10
7.
2. Center (–5, –5) and radius 2 5
3. Centered at the origin with radius
3
8.
4. Center (11, 4) and radius 2 7
5.
For each equation below, tell where the center
of the circle is and how long its radius is.
(Radius lengths should be exact and simplified.
9.
6.
2
( x − 4 ) + ( y + 1)
2
= 45
Center:
Radius:
2
10. x 2 + ( y − 3) = 256
Center:
Radius:
Name _______________________________________ Period ___
5/11-5/13/11 GH
Angles In Circles
Part I – Central Angles
The measure of a central angle is ____________ the measure or the intercepted arc created by the angle.
The total number of degrees in a circle is _____________.
F
The total number of degrees in a semicircle is _________.
Use the picture of C at right for the example questions.
Tell whether each arc is a major arc, minor arc, or a semicircle. Then, find the degree measure of each arc. BD is a
diameter and CD is an angle bisector.
Ex 1: m AB
Ex 5: mBD
Ex 2: m AF
Ex 6: mFD
Ex 3: mBE
Ex 7: mDAE
A
37°
B
85°
D
127° C
E
Ex 4: mFBE
B
C
Use the picture of A at right for questions 1-10. CF is a diameter
and AE is an angle bisector. Tell whether each arc is a major
arc, minor arc, or a semi-circle. Then, find the degree measure
of each arc.
1. mCB
2. mCG
3. mDF
4. mCF
5. mDFB
6. mCE
7. mCFE
8. mDC
9. mFG
10. mED
90°
A
5°
G
D
F
E
A
B
100°
K
40°
E
C
D
Use the picture at left to answer questions 11 -20. AC and BE
are diameters of K. Tell whether each arc is a major arc,
minor arc, or a semi-circle. Then, find the degree measure of
each arc.
11. mBC
16. m AKD
12. mBKC
17. m ABC
13. m AKE
18. mDKB
14. mED
19. mCD
15. mCD
20. m ACE
Arc Length and Sector Area NOTES
The area of sectors is actually finding a ____________ of the circle’s ___________ ____________.
part
You can set up a proportion to solve using
.
total
Ex 1.
Ex. 2.
15 cm
7 cm 75°
=
105°
=
The arc length is actually finding a _______________ of the circle’s ___________ ____________. You
part
can set up a proportion to solve using
.
total
Ex 3.
Ex. 4.
15 cm
7 cm 75°
=
105°
=
PARTS OF CIRCLES WORKSHEET
Find the area of each shaded sector and the length of AB . Round to the nearest hundredths place.
A
9.
B
16 in
A
10.
11.
9 ft
B
A
150º
2m
60º
250º
B
Area of sector:__________
Area of sector:__________
Area of sector:__________
Length of AB ___________
Length of AB :___________
Length of AB :__________
12.
13.
B
A
A
A
14.
B
22 in
3 ft
134º
8m
223º
44º
Area of sector:__________
Area of sector:__________
B
Area of sector:__________
Length of AB ___________
Length of AB :___________
Length of AB :__________
2004 Exit
15) A frozen dinner is divided into 3 sections on a
circular plate with a 12-inch diameter.
What is the approximate length of the
arc of the section containing peas?
F 3 in.
G 21 in.
H 16 in.
J 5 in.
2004 Summer Exit
16) The shaded area in the circle below represents
the section of a park used by the chamber of
commerce for a fund-raising event.
What is the approximate area of the section of the park
used for the fund-raiser?
F
G
H
J
339 square feet
1,357 square feet
4,071 square feet
12,214 square feet
17) Given the following circle, which equation is not set up correctly to find the shaded area?
A
95
x
=
360 π 202
B
x
95
=
40
π ( ) 2 360
2
C
360 x = 3800π
D
95
x
=
360 π 402
30°
30°
• Vertex is at the
center of the circle
• Angle Measure = Arc
Measure
80°
80°
40
°
160°
• Vertex is on the circle
• Angle Measure = Arc
Measure ÷ 2
• Arc Measure = Angle
Measure × 2
• Vertex inside circle (not at
center)
• Angle = ½ (
+
)
50°
75°
100°
50°
• Vertex outside circle
• Angle = ½ (
-
(360 – x)°
35°
x
)
UNIT 15 - TOPIC 2: ANGLES IN CIRCLES
EXAMPLE 1 Name ALL of the inscribed angles and their
corresponding intercepted arcs below.
Inscribed angles/Intercepted Arc:
____________________________
O
____________________________
G
D
____________________________
THEOREM: If an angle is inscribed in a circle, then the measure of the
angle is half the measure of the intercepted arc.
Inscribed ∠ = ½ arc
A
B
EXAMPLE 2 Given that m BC = 100°, find the value of ‘x’ in
circle O.
O
x°
x = _______________
EXAMPLE 3 In circle Q, m ST = 68°. Find the m∠1 and m∠2.
C
R
1
S
m∠1 = _______________
Q
m∠2 = _______________
2
T
EXAMPLE 4 Find the value of ‘x’.
x = _______________
E
x°
F
EXAMPLE 5 Find the value of the inscribed angle.
P
G
P
H
X
Angle = __________
O
Z
EXAMPLE 6 Find the value of ‘x’.
32°°
x°°
x = _________
Y
EXAMPLE 7 Find the value of ‘x’.
x = ________
x°°
110°°
EXAMPLE 8 Find the value of ‘x’.
30°°
x°°
x = _______________
EXAMPLE 9 Find the value of ‘x’.
x°°
89°°
x = _______________
THEOREM: If two secants intersect in the interior of a circle, then the measure of the angle
formed is half the sum of the measures of the arcs intercepted by the angle and its
vertical angle.
Angle inside = ½(arc + arc)
EXAMPLE 10 Find the value of ‘x’.
x = _______________
70
°
x°°
30
°
110°°
EXAMPLE 11 Find the value of ‘x’.
x
°
x = ______________
70
THEOREM: If two secants, a secant and a tangent, or two tangents intersect in the exterior of
a circle, then the measure of the angle formed is half the POSITIVE difference of the
measures of the intercepted arcs.
Angle outside = ½(arc - arc)
EXAMPLE 12 Find the value of ‘x’.
x = _______________
150°° x°°
EXAMPLE 13 Find the value of ‘x’.
x = _______________
150°°
x°°
70°°
INSCRIBED/ Interior/ Exterior ANGLES Assignment
Find the indicated value(s).
32°°
1. x = __________
x°°
2. x = __________
x°°
110°°
3. x = __________
(x + 30)°°
120
°
14x°°
4. x =________
4x°°
5. x = __________
x°
6. x = __________
x°
267°
85°
7. x = __________
x°
230°
41°
8. x = __________
x°
81°
9. x = __________
x°
140°
40°
35°
10. x = __________
x°
87°
11. x = __________
148°
128°
x°
12. x = __________
x°
106°
54°
13. x = __________
84°
52°
x°
14. x = __________
25°
95°
x°
Notes Angles in Circles Day 3:
Use the following patterns to make a conjecture about circles.
In each of the circles below, four angles are formed by the intersection of 2
secant lines. The measures of two intercepted arcs and one angle are shown for
the first three circles.
Which expression can be used to represent m <ABC in degrees?
A 1 [(5x + 2) − (4 x + 4)]
B 1 [(5x + 2) + (4 x + 4)]
C 2[(5x + 2) − (4 x + 4)]
D
2
2[(5 x + 2) + (4 x + 4)]
2
_______________________________________
_______________________________________
In each of the circles below, 1 angle is formed by the intersection of 2 secant lines. The measures of two
intercepted arcs and the angle are shown for the first three circles.
B
96°
62°
52°
(2x + 7)°
14°
71°
26°
A
206°
150°
(3x + 5)°
175°
Which expression can be used to represent m<ABC in degrees?
A. ½[(3x + 5) + (2x + 7)]
B. 2[(3x + 5) - (2x + 7)]
C. ½[(3x + 5) - (2x + 7)]
D. 2[(3x + 5) + (2x + 7)]
_______________________________________
_______________________________________
Assignment Day 3 Mixed Practice
1. z = ___________
2. b =______ c =_______
3. m ZXY = __________
Z
Y
b
60°
113°
c
X
z°
mYZ=85°
4. p =______ r =_______
= _______
5. mCD
6. x = ________
J
42 °
(4 x − 2)°
25°
120 °
p
r
D
B
C
mBD=90
K
(7x − 18)°
L
(6x + 6)°
C
7. p = ________
8. x = _________
9. p =_________
x
161°
p
A
147°
67°
67°
25
Ci rcl e A
p
11. mSPT = ________
10. c = ______
12. x = ________
Q
3c
111 °
S
10z°
M
6z°
P
6x − 6
30°
P
Q
N
T
R
c
61 °
http://teachers.henrico.k12.va.us/math/igo/08Circles/8_5.html
13. y = ________
14. x = ________
m ∠ A = 20°
15. x = ______, y = ______
A
(15y − 4)°
T
R
V
(3x+2)°
2
(3 y − 18)°
(14 + 4x )°
U
W
(6x − 14)°
U
S
(8x +2)°
16. In E , the measures of AEB , BEC ,
and CED are in the ratio 3 : 4 : 5.
, and mCD
.
Find m AB , mBC
T
(12 y − 5)°
= _______ mDEF
=_________
17. mDF
64°
C
B
D
F
D
E
A
E
18. x = ______
19. x = ______, y = ______
(2x+5)°
52°
(22x -20)°
y°
(5x - 6)°
16x°
20. x = ______
21. x = ______
A
(2x - 10)°
A
x°
C
(18x - 15 )°
E
D
B
(8x + 1)°
4x°
D
B
160°
Complex Circles Notes
Complex pictures are formed by using multiple types of angles in one picture.
Start with a central angle.
Add an inscribed angle
Add an exterior angle
AB = ______
Find m Find mADB = _____
Find m AD = ____
Find mACD = _____
= ____
Find mFD
Find mCAD = _____
= ____
Find mFB
B
A
93°
C
Find mGEB = ____
B
A
93°
C
Example 1
= _______
1. mEG
D
J
4°
H
2. mEHG = ______
M
= _______
3. mEK
= _______
4. mML
L
F
125°
G
E
15°
K
C
Example 2
J
5. mGFL = ______
= _______
6. mLG
H
M
15°
= _______
7. mEG
L
8. mEHG = ______
74°
K
= _______
9. mHL
F
125°
= _______
10. mML
G
= _______
11. mEM
E
12. mEJK = ______
Complex Pictures Worksheet
= _______
2. mCD
1. z = ________
3. x =_____ y = _____
y
60°
x
25°
130°
D
B
z°
110°
C
4. x =_____ z=_____
5. a =_____ b=_____
6. t = ______
mBD=90
w =_____ y=_____
48 °
a
95 °
106°
z
y
60°
165 °
t
b
x
w
O
7.
B
G
B
8.
C
A
90°
O
35°
A
F
30°
D
E
126°
O
C
80 °
= ________
mBC
F
E
mAGE = ______
D
mBOA = _______
m
AF = ______
= _____
mEB
m
AB = ______
m
AD = _____
= ______
mECB
mC = ______
= ______
mED
= ______
mDF
= ______
mCD
mA = ______
= ______
mBC
= ____
9. mDC
= ____
mFG
mFDG = ____
mDGC = ____
mCEG = ____
= ____
mDF
Circle E
10. mXYZ = ____
mVXY = ____
mXVA = ____
m
XB = ____
mZ = ____
= ____
mYB
112 °
D
Y
Circle W
V
A
83°
E
F
W
21°
C
G
76 °
B
Z
H
X
ARC, CHORDS, and TANGENTS Notes
GP
THEOREM: In a circle (or congruent circles), 2 minor arcs are if and only if their
corresponding chords are congruent.
A
B
EXAMPLE 1 Use the figure to answer the questions below.
a) Which two chords are congruent? _____________________________
20
b) Which two arcs are congruent? _____________________________ °
c)
What are the measures of their arcs?________________________
220
°
C
D
EXAMPLE 2 If PS = 12 and TR = 15, then find QR.
T 20°S 60
50°
R
QR = _______________
70°
P
Q
EXAMPLE 3 Find HI.
78°
G
H
9
HI = _______________
91°
10
J
11
I
113°
THEOREM: In a circle, if a diameter (or radius) is perpendicular to a
chord, then it bisects the chord and its arc.
EXAMPLE 4 AD ⊥ BC, AE = 12, and the radius is 13. Find the following.
a)
ED = __________
b)
AC = __________
c)
AB = __________
d)
EB = __________
e)
EC = __________
f)
BC = __________
EXAMPLE 5 if the measure of CFB = 220°, find the following.
a)
m CB = __________
b)
m∠CAB = __________
c)
m∠BAD = __________
d)
m CD = __________
A
B
E
D
C
F
A
B
E
C
D
EXAMPLE 6 In circle A, SQ = 12 and AT = 8. Find AR.
A
AR = __________
T
S
Q
R
EXAMPLE 7 Using the diagram below, find the indicated values.
AD = __________
CD = __________
C
m AB = __________
6
60°°
A
B
D
THEOREM: In a circle (or congruent circles), two chords are congruent if and only if they are
equidistant from the center.
y
EXAMPLE 8 Find the values of ‘x’ and ‘y’.
x = __________
y = __________
8
x
G
EXAMPLE 9 In circle O, FL = 3, GO = 5, and OP = 4. Find HJ.
HJ = __________
L
F
O
J
P
H
THEOREM: If two segments are tangent to a circle from the same external point, then the segments are
congruent.
A
5y - 28
EXAMPLE 10 DE and DF are tangent to circle P.
C
P
y = _________
AC = ________
CB = ________
B
ARC, CHORDS, and TANGENTS Assignment
Find the indicated values.
Y
1. m VX = _______________
110°°
22
m VY = _______________
15
U
X
10
Z 8
W
12
V
105°°
2. AD = _______________
C
21
B
120°°
24
6
30°°
D
A
110
°
3. m PR = _______________
P
Q
m RQ = _______________
R
4. PR
5. = _______________
R
Q
8
S
17
P
A
6. AC = 8; AQ = __________
B 3 Q
C
K
7. m GMJ = 200°
H
G
m GK = _______________
J
Q
M
8. CD = _______________
A
6
8
B
Y
O
D
8
X
C
F
9. HG = 12, OM = 6
EF = 12,
K O
E
OK = __________
M
H
G
C
CD is a diameter of circle O. Find the following.
10. __________
O
EB = ?
12
A
E
D
30°°
B
11. __________
OB = ?
12. __________
m DB = ?
13. __________
m AB = ?
14. __________
m AC = ?
15. __________
DE = ?
Find the indicated value in each of the following.
15. x = _________
20
15
C
x
16. x = _________
A
8
P
15
C
B
x
A
17. OC = _______
If AO = 6 and BC = 8, then OC = ?
C
O
B
A
18. d = ________
If OC = 15 and BC = 12, then the diameter = ?
C
O
B
A
19. x = _________
2x + 1
CB = __________
C
P
B
B
20. __________
3
x
C
A
D
P
4
E
1.8
20. P = ________
3.4
3.7
3.6
AB and CD are tangents of circle O. Find each of the following.
21.__________
If OB = 6 and AB = 8, then EA = ?
22. __________
If DE = 16 and EA = 9, then BA = ?
C
B
A
E
O
23. __________
m∠ABO = ?
24. __________
m∠ODC = ?
25. __________
If m∠DOB = 120°, then m∠EAB = ?
D