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Reading to Learn MathematicsReflections
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
© Glencoe/McGraw-Hill 483 Glencoe Geometry
Less
on
9-1
Pre-Activity Where are reflections found in nature?
Read the introduction to Lesson 9-1 at the top of page 463 in your textbook.
Suppose you draw a line segment connecting a point at the peak of a mountainto its image in the lake. Where will the midpoint of this segment fall?
Reading the Lesson1. Draw the reflected image for each reflection described below.
a. reflection of trapezoid ABCD in the line n b. reflection of �RST in point PLabel the image of ABCD as A�B�C�D�. Label the image of RST as R�S�T�.
c. reflection of pentagon ABCDE in the originLabel the image of ABCDE as A�B�C�D�E�.
2. Determine the image of the given point under the indicated reflection.a. (4, 6); reflection in the y-axisb. (�3, 5); reflection in the x-axisc. (�8, �2); reflection in the line y � xd. (9, �3); reflection in the origin
3. Determine the number of lines of symmetry for each figure described below. Thendetermine whether the figure has point symmetry and indicate this by writing yes or no.a. a square b. an isosceles triangle (not equilateral) c. a regular hexagon d. an isosceles trapezoid e. a rectangle (not a square) f. the letter E
Helping You Remember4. A good way to remember a new geometric term is to relate the word or its parts to
geometric terms you already know. Look up the origins of the two parts of the wordisometry in your dictionary. Explain the meaning of each part and give a term youalready know that shares the origin of that part.
A�B�
C� D�E�
AB
CDE
x
y
O
R�S�
T�
RS P
Tn
A�B�
C�
D�A
BCD
Reading to Learn MathematicsTranslations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
© Glencoe/McGraw-Hill 489 Glencoe Geometry
Less
on
9-2
Pre-Activity How are translations used in a marching band show?
Read the introduction to Lesson 9-2 at the top of page 470 in your textbook.
How do band directors get the marching band to maintain the shape of thefigure they originally formed?
Reading the Lesson1. Underline the correct word or phrase to form a true statement.
a. All reflections and translations are (opposites/isometries/equivalent).b. The preimage and image of a figure under a reflection in a line have
(the same orientation/opposite orientations).c. The preimage and image of a figure under a translation have
(the same orientation/opposite orientations).d. The result of successive reflections over two parallel lines is a
(reflection/rotation/translation).e. Collinearity (is/is not) preserved by translations.f. The translation (x, y) → (x � a, y � b) shifts every point a units
(horizontally/vertically) and y units (horizontally/vertically).
2. Find the image of each preimage under the indicated translation.a. (x, y); 5 units right and 3 units upb. (x, y); 2 units left and 4 units downc. (x, y); 1 unit left and 6 units upd. (x, y); 7 units righte. (4, �3); 3 units upf. (�5, 6); 3 units right and 2 units downg. (�7, 5); 7 units right and 5 units downh. (�9, �2); 12 units right and 6 units down
3. �RST has vertices R(�3, 3), S(0, �2), and T(2, 1). Graph �RSTand its image �R�S�T� under the translation (x, y) → (x � 3, y � 2).List the coordinates of the vertices of the image.
Helping You Remember4. A good way to remember a new mathematical term is to relate it to an everyday meaning
of the same word. How is the meaning of translation in geometry related to the idea oftranslation from one language to another?
R�
S�
T�
R
S
T
x
y
O
Reading to Learn MathematicsRotations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
© Glencoe/McGraw-Hill 495 Glencoe Geometry
Less
on
9-3
Pre-Activity How do some amusement rides illustrate rotations?
Read the introduction to Lesson 9-3 at the top of page 476 in your textbook.
What are two ways that each car rotates?
Reading the Lesson
1. List all of the following types of transformations that satisfy each description: reflection,translation, rotation.a. The transformation is an isometry.b. The transformation preserves the orientation of a figure.c. The transformation is the composite of successive reflections over two intersecting
lines.d. The transformation is the composite of successive reflections over two parallel
lines.e. A specific transformation is defined by a fixed point and a specified angle.f. A specific transformation is defined by a fixed point, a fixed line, or a fixed plane.
g. A specific transformation is defined by (x, y) → (x � a, x � b), for fixed values of a and b.
h. The transformation is also called a slide.i. The transformation is also called a flip.j. The transformation is also called a turn.
2. Determine the order and magnitude of the rotational symmetry for each figure.
a. b.
c. d.
Helping You Remember
3. What is an easy way to remember the order and magnitude of the rotational symmetryof a regular polygon?
© Glencoe/McGraw-Hill 492 Glencoe Geometry
Rotational Symmetry When the figure at the right is rotated about point P by 120° or 240°, the image looks like the preimage. The figure has rotational symmetry, which means it can be rotated less than 360° about a point and the preimage and image appear to be the same.
The figure has rotational symmetry of order 3 because there are 3 rotations less than 360° (0°, 120°, 240°) that produce an image that is the same as the original. The magnitude of the rotational symmetry for a figure is 360 degrees divided by the order. For the figure above,the rotational symmetry has magnitude 120 degrees.
Identify the order and magnitude of the rotational symmetry of the design at the right.
The design has rotational symmetry about the center point for rotations of 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°.
There are eight rotations less than 360 degrees, so the order of its rotational symmetry is 8. The quotient 360 � 8 is 45, so the magnitude of its rotational symmetry is 45 degrees.
Identify the order and magnitude of the rotational symmetry of each figure.
1. a square 2. a regular 40-gon
3. 4.
5. 6.
P
Study Guide and Intervention (continued)
Rotations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
ExercisesExercises
ExampleExample
Reading to Learn MathematicsTesselations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
© Glencoe/McGraw-Hill 501 Glencoe Geometry
Less
on
9-4
Pre-Activity How are tessellations used in art?
Read the introduction to Lesson 9-4 at the top of page 483 in your textbook.
• In the pattern shown in the picture in your textbook, how many smallequilateral triangles make up one regular hexagon?
• In this pattern, how many fish make up one equilateral triangle?
Reading the Lesson
1. Underline the correct word, phrase, or number to form a true statement.
a. A tessellation is a pattern that covers a plane with the same figure or set of figures sothat there are no (congruent angles/overlapping or empty spaces/right angles).
b. A tessellation that uses only one type of regular polygon is called a(uniform/regular/semi-regular) tessellation.
c. The sum of the measures of the angles at any vertex in any tessellation is(90/180/360).
d. A tessellation that contains the same arrangement of shapes and angles at everyvertex is called a (uniform/regular/semi-regular) tessellation.
e. In a regular tessellation made up of hexagons, there are (3/4/6) hexagons meeting ateach vertex, and the measure of each of the angles at any vertex is (60/90/120).
f. A uniform tessellation formed using two or more regular polygons is called a(rotational/regular/semi-regular) tessellation.
g. In a regular tessellation made up of triangles, there are (3/4/6) triangles meeting ateach vertex, and the measure of each of the angles at any vertex is (30/60/120).
h. If a regular tessellation is made up of quadrilaterals, all of the quadrilaterals must becongruent (rectangles/parallelograms/squares/trapezoids).
2. Write all of the following words that describe each tessellation: uniform, non-uniform,regular, semi-regular.
a. b.
Helping You Remember
3. Often the everyday meanings of a word can help you to remember its mathematicalmeaning. Look up uniform in your dictionary. How can its everyday meanings help youto remember the meaning of a uniform tessellation?
© Glencoe/McGraw-Hill 498 Glencoe Geometry
Tessellations with Specific Attributes A tessellation pattern can contain any typeof polygon. If the arrangement of shapes and angles at each vertex in the tessellation is thesame, the tessellation is uniform. A semi-regular tessellation is a uniform tessellationthat contains two or more regular polygons.
Determine whether a kite will tessellate the plane. If so, describe the tessellation as uniform, regular, semi-regular, or not uniform.A kite will tessellate the plane. At each vertex the sum of the measures is a � b � b � c, which is 360. The tessellation is uniform.
Determine whether a semi-regular tessellation can be created from each set offigures. If so, sketch the tessellation. Assume that each figure has a side length of1 unit.
1. rhombus, equilateral triangle, 2. square and equilateral triangleand octagon
Determine whether each polygon tessellates the plane. If so, describe thetessellation as uniform, not uniform, regular, or semi-regular.
3. rectangle 4. hexagon and square
c� a�
b�
b�
c�
c�
a�
a�b�
b�
b�
c�a�
b�
b�
c�a�
b�
b�
c�a�
b�
b�
b�
c� a�
b�
b�
Study Guide and Intervention (continued)
Tessellations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
ExercisesExercises
ExampleExample
Reading to Learn MathematicsDilations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
© Glencoe/McGraw-Hill 507 Glencoe Geometry
Less
on
9-5
Pre-Activity How do you use dilations when you use a computer?
Read the introduction to Lesson 9-5 at the top of page 490 in your textbook.
In addition to the example given in your textbook, give two everydayexamples of scaling an object, one that makes the object larger and anotherthat makes it smaller.
Reading the Lesson1. Each of the values of r given below represents the scale factor for a dilation. In each
case, determine whether the dilation is an enlargement, a reduction, or a congruencetransformation.a. r � 3 b. r � 0.5c. r � �0.75 d. r � �1
e. r � �23� f. r � ��
32�
g. r � �1.01 h. r � 0.999
2. Determine whether each sentence is always, sometimes, or never true. If the sentence isnot always true, explain why.a. A dilation requires a center point and a scale factor.
b. A dilation changes the size of a figure.
c. A dilation changes the shape of a figure.
d. The image of a figure under a dilation lies on the opposite side of the center from thepreimage.
e. A similarity transformation is a congruence transformation.
f. The center of a dilation is its own image.
g. A dilation is an isometry.
h. The scale factor for a dilation is a positive number.
i. Dilations produce similar figures.
Helping You Remember3. A good way to remember something is to explain it to someone else. Suppose that your
classmate Lydia is having trouble understanding the relationship between similaritytransformations and congruence transformations. How can you explain this to her?
Skills PracticeDilations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
© Glencoe/McGraw-Hill 505 Glencoe Geometry
Less
on
9-5
Draw the dilation image of each figure with center C and the given scale factor.
1. r � 2 2. r � �14�
Find the measure of the dilation image M���N��� or of the preimage M�N� using thegiven scale factor.
3. MN � 3, r � 3 4. M�N� � 7, r � 21
COORDINATE GEOMETRY Find the image of each polygon, given the vertices, aftera dilation centered at the origin with a scale factor of 2. Then graph a dilation centered at the origin with a scale factor of �
12�.
5. J(2, 4), K(4, 4), P(3, 2) 6. D(�2, 0), G(0, 2), F(2, �2)
Determine the scale factor for each dilation with center C. Determine whether thedilation is an enlargement, reduction, or congruence transformation. The dashedfigure is the dilation image.
7. 8.
CC
G�
F�
D�
G�
F �
D �x
y
O
J� K�
P�J � K �
P �x
y
O
CC
Reading to Learn MathematicsVectors
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
© Glencoe/McGraw-Hill 513 Glencoe Geometry
Less
on
9-6
Pre-Activity How do vectors help a pilot plan a flight?
Read the introduction to Lesson 9-6 at the top of page 498 in your textbook.
Why do pilots often head their planes in a slightly different direction fromtheir destination?
Reading the Lesson1. Supply the missing words or phrases to complete the following sentences.
a. A is a directed segment representing a quantity that has bothmagnitude and direction.
b. The length of a vector is called its .
c. Two vectors are parallel if and only if they have the same or direction.
d. A vector is in if it is drawn with initial point at the origin.
e. Two vectors are equal if and only if they have the same and the
same .
f. The sum of two vectors is called the .
g. A vector is written in if it is expressed as an ordered pair.
h. The process of multiplying a vector by a constant is called .
2. Write each vector described below in component form.
a. a vector in standard position with endpoint (a, b)
b. a vector with initial point (a, b) and endpoint (c, d)
c. a vector in standard position with endpoint (�3, 5)
d. a vector with initial point (2, �3) and endpoint (6, �8)
e. a� � b� if a� � ��3, 5� and b� � �6, �4�
f. 5u� if u� � �8, �6�
g. ��13�v� if v� � ��15, 24�
h. 0.5u� � 1.5v� if u� � �10, �10� and v� � ��8, 6�
Helping You Remember3. A good way to remember a new mathematical term is to relate it to a term you already
know. You learned about scale factors when you studied similarity and dilations. How is theidea of a scalar related to scale factors?
Skills PracticeVectors
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
© Glencoe/McGraw-Hill 511 Glencoe Geometry
Less
on
9-6
Write the component form of each vector.
1. 2.
Find the magnitude and direction of RS� for the given coordinates. Round to thenearest tenth.
3. R(2, �3), S(4, 9) 4. R(0, 2), S(3, 12)
5. R(5, 4), S(�3, 1) 6. R(1, 5), S(�4, �6)
Graph the image of each figure under a translation by the given vector(s).
7. �ABC with vertices A(�4, 3), B(�1, 4), 8. trapezoid with vertices T(�4, �2),C(�1, 1); t� � �4, �3� R(�1, �2), S(�2, �3), Y(�3, �3); a� � �3, 1�
and b� � �2, 4�
Find the magnitude and direction of each resultant for the given vectors.
9. y� � �7, 0�, z� � �0, 6� 10. b� � �3, 2�, c� � ��2, 3�
S�Y�
T� R�
SY
T Rx
y
O
C�
B�A�C
B
A
x
y
O
x
y
O
B(–2, 2)
D(3, –3)
x
y
OC(1, –1)
E(4, 3)
Reading to Learn MathematicsTransformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
9-79-7
© Glencoe/McGraw-Hill 519 Glencoe Geometry
Less
on
9-7
Pre-Activity How can matrices be used to make movies?
Read the introduction to Lesson 9-7 at the top of page 506 in your textbook.
• What kind of transformation should be used to move a polygon?
• What kind of transformation should be used to resize a polygon?
Reading the Lesson1. Write a vertex matrix for each figure.
a. �ABC b. parallelogram ABCD
2. Match each transformation from the first column with the corresponding matrix fromthe second or third column. In each case, the vertex matrix for the preimage of a figureis multiplied on the left by one of the matrices below to obtain the image of the figure.All rotations listed are counterclockwise through the origin. (Some matrices may be usedmore than once or not at all.)
a. reflection over the y-axis i. v.b. 90° rotationc. reflection over the line y � x ii. vi.d. 270° rotatione. reflection over the origin iii. vii.f. 180° rotationg. reflection over the x-axis iv. viii.h. 360° rotation
Helping You Remember3. How can you remember or quickly figure out the matrices for the transformations in
Exercise 2?
0 �11 0
0 �1�1 0
1 00 �1
0 11 0
�1 0 0 1
�1 0 0 �1
0 1�1 0
1 00 1
B C
A D x
y
O
B
C
A
x
y
O
Study Guide and InterventionTransformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
9-79-7
© Glencoe/McGraw-Hill 515 Glencoe Geometry
Less
on
9-7
Translations and Dilations A vector can be represented by the ordered pair �x, y� or
by the column matrix . When the ordered pairs for all the vertices of a polygon are
placed together, the resulting matrix is called the vertex matrix for the polygon.
For �ABC with A(�2, 2), B(2, 1), and C(�1, �1), the vertex
matrix for the triangle is .
For �ABC above, use a matrix to find the coordinates of thevertices of the image of �ABC under the translation (x, y) → (x � 3, y � 1).To translate the figure 3 units to the right, add 3 to each x-coordinate. To translate thefigure 1 unit down, add �1 to each y-coordinate.
Vertex Matrix Translation Vertex Matrixof �ABC Matrix of �A�B�C�
� �
The coordinates are A�(1, 1), B�(5, 0), and C�(2, �2).
For �ABC above, use a matrix to find the coordinates of the verticesof the image of �ABC for a dilation centered at the origin with scale factor 3.
Scale Vertex Matrix Vertex MatrixFactor of �ABC of �A�B�C�
3 �
The coordinates are A�(�6, 6), B�(6, 3), and C�(�3, �3).
Use a matrix to find the coordinates of the vertices of the image of each figureunder the given translations or dilations.
1. �ABC with A(3, 1), B(�2, 4), C(�2, �1); (x, y) → (x � 1, y � 2)
2. parallelogram RSTU with R(�4, �2), S(�3, 1), T(3, 4), U(2, 1); (x, y) → (x � 4, y � 3)
3. rectangle PQRS with P(4, 0), Q(3, �3), R(�3, �1), S(�2, 2); (x, y) → (x � 2, y � 1)
4. �ABC with A(�2, �1), B(�2, �3), C(2, �1); dilation centered at the origin with scalefactor 2
5. parallelogram RSTU with R(4, �2), S(�4, �1), T(�2, 3), U(6, 2); dilation centered at theorigin with scale factor 1.5
�6 6 �3 6 3 �3
�2 2 �1 2 1 �1
1 5 21 0 �2
3 3 3�1 �1 �1
�2 2 �1 2 1 �1
�2 2 �1 2 1 �1
x
y
B(2, 1)
C(–1, –1)
A(–2, 2)
x y
ExercisesExercises
Example 1Example 1
Example 2Example 2
© Glencoe/McGraw-Hill 516 Glencoe Geometry
Reflections and Rotations When you reflect an image, one way to find thecoordinates of the reflected vertices is to multiply the vertex matrix of the object by areflection matrix. To perform more than one reflection, multiply by one reflection matrixto find the first image. Then multiply by the second matrix to find the final image. Thematrices for reflections in the axes, the origin, and the line y � x are shown below.
For a reflection in the: x-axis y-axis origin line y � x
Multiply the vertex matrix by:
�ABC has vertices A(�2, 3), B(1, 4), and C(3, 0). Use a matrix to find the coordinates of the vertices of the image of �ABC after a reflection in the x-axis.To reflect in the x-axis, multiply the vertex matrix of �ABC by thereflection matrix for the x-axis.
Reflection Matrix Vertex Matrix Vertex Matrix for x-axis of �ABC of �A’B’C’
�
Use a matrix to find the coordinates of the vertices of the image of each figureunder the given reflection.
1. �ABC with A(�3, 2), B(�1, 3), C(1, 0); reflection in the x-axis
2. �XYZ with X(2, �1), Y(4, �3), Z(�2, 1); reflection in the y-axis
3. �ABC with A(3, 4), B(�1, 0), C(�2, 4); reflection in the origin
4. parallelogram RSTU with R(�3, 2), S(3, 2), T(5, �1), U(�1, �1); reflection in the line y � x
5. �ABC with A(2, 3), B(�1, 2), C(1, �1); reflection in the origin, then reflection in the liney � x
6. parallelogram RSTU with R(0, 2), S(4, 2), T(3, �2), U(�1, �2); reflection in the x-axis,then reflection in the y-axis
�2 1 3�3 �4 0
�2 1 3 3 4 0
1 00 �1
x
y
A�B�
AB
C
0 11 0
�1 0 0 �1
�1 0 0 1
1 00 �1
Study Guide and Intervention (continued)
Transformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
9-79-7
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill A4 Glencoe Geometry
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at
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.Whe
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ook
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igin
of
that
par
t.S
amp
le a
nsw
er:T
he
firs
t p
art
com
es f
rom
iso
s,w
hic
h m
ean
s eq
ual
,as
in is
osc
eles
.Th
e se
con
d p
art
com
es f
rom
met
ron
,wh
ich
mea
ns
mea
sure
,as
in g
eom
etry
.
A�
B�
C�
D�
E�
AB
CD
E
x
y
O
R�
S�
T�
RS
P
Tn
A�
B�
C�
D�
A
BC D
©G
lenc
oe/M
cGra
w-H
ill48
4G
lenc
oe G
eom
etry
Ref
lect
ion
s in
th
e C
oo
rdin
ate
Pla
ne
Stu
dy
the
dia
gram
at
the
righ
t.It
sh
ows
how
th
e tr
ian
gle
AB
Cis
map
ped
on
to t
rian
gle
XY
Zb
y th
etr
ansf
orm
atio
n (
x,y)
→(�
x�
6,y)
.Not
ice
that
�X
YZ
is t
he
refl
ecti
on i
mag
e w
ith
res
pec
t to
th
e ve
rtic
al
lin
e w
ith
eq
uat
ion
x�
3.
1.P
rove
th
at t
he
vert
ical
lin
e w
ith
equ
atio
n x
�3
is t
he
perp
endi
cula
r bi
sect
or o
f th
e se
gmen
t w
ith
en
dpoi
nts
(x
,y)
and
(�x
�6,
y).(
Hin
t:U
se t
he
mid
poin
t fo
rmu
la.)
Mid
po
int
���x
�(�
2x�
6)�
,�y
� 2y
��o
r (3
,y)
Th
e se
gm
ent
join
ing
(x,
y)
and
(�
x�
6,y
) is
h
ori
zon
tal a
nd
hen
ce is
per
pen
dic
ula
r to
th
e ve
rtic
al li
ne.
2.E
very
tra
nsf
orm
atio
n o
f th
e fo
rm (
x,y)
→(�
x�
2h
,y)
is
a re
flec
tion
wit
h r
espe
ct t
o th
e ve
rtic
al l
ine
wit
h e
quat
ion
x
�h
.Wh
at k
ind
of t
ran
sfor
mat
ion
is
(x,y
) →
(x,�
y�
2k)
?
refl
ecti
on
wit
h r
esp
ect
to t
he
ho
rizo
nta
l lin
e w
ith
eq
uat
ion
y�
k
Dra
w t
he
tran
sfor
mat
ion
im
age
for
each
fig
ure
an
d t
he
give
n
tran
sfor
mat
ion
.Is
it a
ref
lect
ion
tra
nsf
orm
atio
n?
If s
o,w
ith
re
spec
t to
wh
at l
ine?
3.(x
,y)
→(�
x�
4,y)
4.(x
,y)
→(x
,�y
�8)
yes;
x�
�2
yes;
y�
4
x
y
Ox
y
O
x �
3
A
B
C
X
Y
Z
(x, y
) → (�
x �
6, y
)
x
y
O
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
Answers (Lesson 9-1)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csTr
ansl
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
©G
lenc
oe/M
cGra
w-H
ill48
9G
lenc
oe G
eom
etry
Lesson 9-2
Pre-
Act
ivit
yH
ow a
re t
ran
slat
ion
s u
sed
in
a m
arch
ing
ban
d s
how
?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-2
at
the
top
of p
age
470
in y
our
text
book
.
How
do
ban
d di
rect
ors
get
the
mar
chin
g ba
nd
to m
ain
tain
th
e sh
ape
of t
he
figu
re t
hey
ori
gin
ally
for
med
?S
amp
le a
nsw
er:T
he
ban
d p
ract
ices
mar
chin
g in
un
iso
n,w
ith
eve
ryo
ne
mak
ing
iden
tica
l mov
es a
tth
e sa
me
tim
e.R
ead
ing
th
e Le
sso
n1.
Un
derl
ine
the
corr
ect
wor
d or
ph
rase
to
form
a t
rue
stat
emen
t.a.
All
ref
lect
ion
s an
d tr
ansl
atio
ns
are
(opp
osit
es/is
omet
ries
/equ
ival
ent)
.b
.T
he
prei
mag
e an
d im
age
of a
fig
ure
un
der
a re
flec
tion
in
a l
ine
hav
e (t
he
sam
e or
ien
tati
on/o
ppos
ite
orie
nta
tion
s).
c.T
he
prei
mag
e an
d im
age
of a
fig
ure
un
der
a tr
ansl
atio
n h
ave
(th
e sa
me
orie
nta
tion
/opp
osit
e or
ien
tati
ons)
.d
.T
he
resu
lt o
f su
cces
sive
ref
lect
ion
s ov
er t
wo
para
llel
lin
es i
s a
(ref
lect
ion
/rot
atio
n/t
ran
slat
ion
).e.
Col
lin
eari
ty (
is/i
s n
ot)
pres
erve
d by
tra
nsl
atio
ns.
f.T
he
tran
slat
ion
(x,
y) →
(x�
a,y
�b)
sh
ifts
eve
ry p
oin
t a
un
its
(hor
izon
tall
y/ve
rtic
ally
) an
d y
un
its
(hor
izon
tall
y/ve
rtic
ally
).
2.F
ind
the
imag
e of
eac
h p
reim
age
un
der
the
indi
cate
d tr
ansl
atio
n.
a.(x
,y);
5 u
nit
s ri
ght
and
3 u
nit
s u
p(x
�5,
y�
3)b
.(x
,y);
2 u
nit
s le
ft a
nd
4 u
nit
s do
wn
(x�
2,y
�4)
c.(x
,y);
1 u
nit
lef
t an
d 6
un
its
up
(x�
1,y
�6)
d.
(x,y
);7
un
its
righ
t(x
�7,
y)e.
(4,�
3);3
un
its
up
(4,0
)f.
(�5,
6);3
un
its
righ
t an
d 2
un
its
dow
n(�
2,4)
g.(�
7,5)
;7 u
nit
s ri
ght
and
5 u
nit
s do
wn
(0,0
)h
.(�
9,�
2);1
2 u
nit
s ri
ght
and
6 u
nit
s do
wn
(3,�
8)
3.�
RS
Th
as v
erti
ces
R(�
3,3)
,S(0
,�2)
,an
d T
(2,1
).G
raph
�R
ST
and
its
imag
e �
R�S
�T�
unde
r th
e tr
ansl
atio
n (x
,y)
→(x
�3,
y�
2).
Lis
t th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f th
e im
age.
R�(
0,1)
,S�(
3,�
4),T
�(5,
�1)
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al t
erm
is
to r
elat
e it
to
an e
very
day
mea
ning
of t
he
sam
e w
ord.
How
is
the
mea
nin
g of
tra
nsl
atio
nin
geo
met
ry r
elat
ed t
o th
e id
ea o
ftr
ansl
atio
nfr
om o
ne
lan
guag
e to
an
oth
er?
Sam
ple
an
swer
:Wh
en y
ou
tra
nsl
ate
fro
m o
ne
lan
gu
age
to a
no
ther
,yo
u c
arry
ove
rth
e m
ean
ing
fro
m o
ne
lan
gu
age
to a
no
ther
.Wh
en y
ou
tra
nsl
ate
a g
eom
etri
c fig
ure
,yo
u c
arry
over
the
figu
re f
rom
on
e p
osi
tion
to
an
oth
er w
itho
ut
chan
gin
g it
s b
asic
R�
S�
T�
R
S
T
x
y
O
©G
lenc
oe/M
cGra
w-H
ill49
0G
lenc
oe G
eom
etry
Tran
slat
ion
s in
Th
e C
oo
rdin
ate
Pla
ne
You
can
use
alg
ebra
ic d
escr
ipti
ons
of r
efle
ctio
ns
to s
how
th
atth
e co
mp
osit
e of
tw
o re
flec
tion
s w
ith
res
pec
t to
par
alle
l li
nes
is
a tr
ansl
atio
n (
that
is,
a sl
ide)
.
1.S
upp
ose
aan
d b
are
two
diff
eren
t re
al n
um
bers
.Let
San
d T
be t
he
foll
owin
g re
flec
tion
s.
S:(
x,y)
→(�
x�
2a,
y)T
:(x,
y) →
(�x
�2
b,y)
Sis
ref
lect
ion
wit
h r
espe
ct t
o th
e li
ne
wit
h e
quat
ion
x�
a,an
d T
isre
flec
tion
wit
h r
espe
ct t
o th
e li
ne
wit
h e
quat
ion
x�
b.
a.F
ind
an a
lgeb
raic
des
crip
tion
(si
mil
ar t
o th
ose
abov
e fo
r S
and
T)
to d
escr
ibe
the
com
posi
te t
ran
sfor
mat
ion
“S
foll
owed
by
T.”
(x,y
) →
(x�
2 (b
�a)
,y)
b.F
ind
an a
lgeb
raic
des
crip
tion
for
th
e co
mpo
site
tra
nsf
orm
atio
n“T
foll
owed
by
S.”
(x,y
) →
(x�
2 (a
�b
),y
)
2.T
hin
k ab
out
the
resu
lts
you
obt
ain
ed i
n E
xerc
ise
1.W
hat
do
they
tell
you
abo
ut
how
th
e di
stan
ce b
etw
een
tw
o pa
rall
el l
ines
is
rela
ted
to t
he
dist
ance
bet
wee
n a
pre
imag
e an
d im
age
poin
t fo
r a
com
posi
te o
f re
flec
tion
s w
ith
res
pect
to
thes
e li
nes
?
Th
e d
ista
nce
bet
wee
n a
pre
imag
e an
d it
s im
age
po
int
istw
ice
the
dis
tan
ce b
etw
een
th
e p
aral
lel l
ines
.
3.Il
lust
rate
you
r an
swer
s to
Exe
rcis
es 1
an
d 2
wit
h s
ketc
hes
.Use
ase
para
te s
hee
t if
nec
essa
ry.
See
stu
den
ts’w
ork
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
Answers (Lesson 9-2)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csR
ota
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
©G
lenc
oe/M
cGra
w-H
ill49
5G
lenc
oe G
eom
etry
Lesson 9-3
Pre-
Act
ivit
yH
ow d
o so
me
amu
sem
ent
rid
es i
llu
stra
te r
otat
ion
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-3
at
the
top
of p
age
476
in y
our
text
book
.
Wh
at a
re t
wo
way
s th
at e
ach
car
rot
ates
?
Eac
h c
ar s
pin
s ar
ou
nd
its
ow
n c
ente
r an
d e
ach
car
ro
tate
sar
ou
nd
a p
oin
t in
th
e ce
nte
r o
f th
e ci
rcu
lar
trac
k.
Rea
din
g t
he
Less
on
1.L
ist
all
of t
he
foll
owin
g ty
pes
of t
ran
sfor
mat
ion
s th
at s
atis
fy e
ach
des
crip
tion
:ref
lect
ion
,tr
ansl
atio
n,r
otat
ion
.a.
Th
e tr
ansf
orm
atio
n i
s an
iso
met
ry.
refl
ecti
on
,tra
nsl
atio
n,r
ota
tio
nb
.T
he
tran
sfor
mat
ion
pre
serv
es t
he
orie
nta
tion
of
a fi
gure
.tr
ansl
atio
n,r
ota
tio
nc.
Th
e tr
ansf
orm
atio
n i
s th
e co
mpo
site
of
succ
essi
ve r
efle
ctio
ns
over
tw
o in
ters
ecti
ng
lin
es.
rota
tio
nd
.T
he
tran
sfor
mat
ion
is
the
com
posi
te o
f su
cces
sive
ref
lect
ion
s ov
er t
wo
para
llel
li
nes
.tr
ansl
atio
ne.
A s
peci
fic
tran
sfor
mat
ion
is
defi
ned
by
a fi
xed
poin
t an
d a
spec
ifie
d an
gle.
rota
tio
nf.
A s
peci
fic
tran
sfor
mat
ion
is
defi
ned
by
a fi
xed
poin
t,a
fixe
d li
ne,
or a
fix
ed p
lan
e.re
flec
tio
ng.
A s
peci
fic
tran
sfor
mat
ion
is
defi
ned
by
(x,y
) →
(x�
a,x
�b)
,for
fix
ed v
alu
es o
f a
and
b.tr
ansl
atio
nh
.T
he
tran
sfor
mat
ion
is
also
cal
led
a sl
ide.
tran
slat
ion
i.T
he
tran
sfor
mat
ion
is
also
cal
led
a fl
ip.
refl
ecti
on
j.T
he
tran
sfor
mat
ion
is
also
cal
led
a tu
rn.
rota
tio
n
2.D
eter
min
e th
e or
der
and
mag
nit
ude
of
the
rota
tion
al s
ymm
etry
for
eac
h f
igu
re.
a.b
.
ord
er 3
;m
agn
itu
de
120°
ord
er 2
;m
agn
itu
de
180°
c.d
.
ord
er 5
;m
agn
itu
de
72°
ord
er 8
;m
agn
itu
de
45°
Hel
pin
g Y
ou
Rem
emb
er
3.W
hat
is
an e
asy
way
to
rem
embe
r th
e or
der
and
mag
nit
ude
of
the
rota
tion
al s
ymm
etry
of a
reg
ula
r po
lygo
n?
Sam
ple
an
swer
:Th
e o
rder
is t
he
sam
e as
th
e n
um
ber
of
sid
es.T
o f
ind
th
em
agn
itu
de,
div
ide
360
by t
he
nu
mb
er o
f si
des
.
©G
lenc
oe/M
cGra
w-H
ill49
6G
lenc
oe G
eom
etry
Fin
din
g t
he
Cen
ter
of
Ro
tati
on
Su
ppos
e yo
u a
re t
old
that
�X
�Y�Z
�is
th
e ro
tati
on i
mag
e of
�X
YZ
,bu
t yo
u a
re n
ot t
old
wh
ere
the
cen
ter
of r
otat
ion
is
nor
th
e m
easu
re
of t
he
angl
e of
rot
atio
n.C
an y
ou f
ind
them
? Ye
s,yo
u c
an.C
onn
ect
two
pair
s of
cor
resp
ondi
ng
vert
ices
wit
h s
egm
ents
.In
th
e fi
gure
,th
e se
gmen
ts Y
Y�
and
ZZ
�ar
e u
sed.
Dra
w t
he
perp
endi
cula
r bi
sect
ors,
�an
d m
,of
thes
ese
gmen
ts.T
he
poin
t C
wh
ere
�an
d m
inte
rsec
t is
th
e ce
nte
r of
rot
atio
n.
1.H
ow c
an y
ou f
ind
the
mea
sure
of
the
angl
e of
rot
atio
n i
n t
he
figu
re a
bove
?D
raw
�X
CX
�(�
YC
Y�
or
�Z
CZ
�w
ou
ld
also
do
) an
d m
easu
re it
wit
h a
pro
trac
tor.
Loc
ate
the
cen
ter
of r
otat
ion
for
th
e ro
tati
on t
hat
map
s W
XY
Zon
to W
�X�Y
�Z�.
Th
en f
ind
th
e m
easu
re o
f th
e an
gle
of r
otat
ion
.
2.3.
Th
e se
gm
ents
stu
den
ts c
ho
ose
to
bis
ect
may
var
y.S
ee s
tud
ents
’wo
rk.
Z
Z�
W
W�
X�
X
Y
Y�
cen
ter
of
rota
tio
n
ang
le o
fro
tati
on
:ab
ou
t 11
0
Z�
Z
W
W�
X�
X
Y�
Y
cen
ter
of
rota
tio
n
ang
le o
fro
tati
on
:ab
ou
t 10
0
Z�
Z
C
X�
X
Y�
Y
m�
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
Answers (Lesson 9-3)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Ro
tati
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
©G
lenc
oe/M
cGra
w-H
ill49
1G
lenc
oe G
eom
etry
Lesson 9-3
Dra
w R
ota
tio
ns
A t
ran
sfor
mat
ion
cal
led
a ro
tati
ontu
rns
a fi
gure
th
rou
gh a
spe
cifi
edan
gle
abou
t a
fixe
d po
int
call
ed t
he
cen
ter
of r
otat
ion
.To
fin
d th
e im
age
of a
rot
atio
n,o
ne
way
is
to u
se a
pro
trac
tor.
An
oth
er w
ay i
s to
ref
lect
a f
igu
re t
wic
e,in
tw
o in
ters
ecti
ng
lin
es.
�A
BC
has
ver
tice
s A
(2,1
),B
(3,4
),an
d
C(5
,1).
Dra
w t
he
imag
e of
�A
BC
un
der
a r
otat
ion
of
90°
cou
nte
rclo
ckw
ise
abou
t th
e or
igin
.•
Fir
st d
raw
�A
BC
.Th
en d
raw
a s
egm
ent
from
O,t
he
orig
in,
to p
oin
t A
.•
Use
a p
rotr
acto
r to
mea
sure
90°
cou
nte
rclo
ckw
ise
wit
h O �
A�as
on
e si
de.
•D
raw
OR
� �� .
•U
se a
com
pass
to
copy
O �A�
onto
OR
��� .
Nam
e th
e se
gmen
t O�
A���.
•R
epea
t w
ith
seg
men
ts f
rom
th
e or
igin
to
poin
ts B
and
C.
Fin
d t
he
imag
e of
�A
BC
un
der
ref
lect
ion
in
lin
es m
and
n.
Fir
st r
efle
ct �
AB
Cin
lin
e m
.Lab
el t
he
imag
e �
A�B
�C�.
Ref
lect
�A�
B�C
�in
lin
e n.
Lab
el t
he
imag
e �
A�B
�C�.
�A
�B�C
�is
a r
otat
ion
of
�A
BC
.Th
e ce
nte
r of
rot
atio
n i
s th
ein
ters
ecti
on o
f li
nes
man
d n.
Th
e an
gle
of r
otat
ion
is
twic
e th
em
easu
re o
f th
e ac
ute
an
gle
form
ed b
y m
and
n.
Dra
w t
he
rota
tion
im
age
of e
ach
fig
ure
90°
in t
he
give
n d
irec
tion
ab
out
the
cen
ter
poi
nt
and
lab
el t
he
coor
din
ates
.
1.P�
Q�w
ith
en
dpoi
nts
P(�
1,�
2)
2.�
PQ
Rw
ith
ver
tice
s P
(�2,
�3)
,Q(2
,�1)
,an
d Q
(1,3
) co
un
terc
lock
wis
e an
d R
(3,2
) cl
ockw
ise
abou
t th
e po
int
T(1
,1)
abou
t th
e or
igin
Fin
d t
he
rota
tion
im
age
of e
ach
fig
ure
by
refl
ecti
ng
it i
n l
ine
man
d t
hen
in
lin
e n
.
3.4.
mn
B�
C�
A�
AB
C
B�
C�
A�
mn
Q�
P�
Q
PQ
�
P�
x
yP
�
Q�
R�
P
Q
RT
x
y
P�
Q�
P
Q
m
n
A� B
�C
�
A�
B�
C�
B
C
A
x
y
O
A�
B�
C�
A
B
C
R
Exer
cises
Exer
cises
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
©G
lenc
oe/M
cGra
w-H
ill49
2G
lenc
oe G
eom
etry
Ro
tati
on
al S
ymm
etry
Wh
en t
he
figu
re a
t th
e ri
ght
is r
otat
ed
abou
t po
int
Pby
120
°or
240
°,th
e im
age
look
s li
ke t
he
prei
mag
e.T
he
figu
re h
as r
otat
ion
al s
ymm
etry
,wh
ich
mea
ns
it c
an b
e ro
tate
d le
ss
than
360
°ab
out
a po
int
and
the
prei
mag
e an
d im
age
appe
ar t
o be
th
e sa
me.
Th
e fi
gure
has
rot
atio
nal
sym
met
ry o
f or
der
3 be
cau
se t
her
e ar
e 3
rota
tion
s le
ss t
han
360
°(0
°,12
0°,2
40°)
th
at p
rodu
ce a
n i
mag
e th
at
is t
he
sam
e as
th
e or
igin
al.T
he
mag
nit
ud
eof
th
e ro
tati
onal
sym
met
ry
for
a fi
gure
is
360
degr
ees
divi
ded
by t
he
orde
r.F
or t
he
figu
re a
bove
,th
e ro
tati
onal
sym
met
ry h
as m
agn
itu
de 1
20 d
egre
es.
Iden
tify
th
e or
der
an
d m
agn
itu
de
of t
he
rota
tion
al
sym
met
ry o
f th
e d
esig
n a
t th
e ri
ght.
Th
e de
sign
has
rot
atio
nal
sym
met
ry a
bou
t th
e ce
nte
r po
int
for
rota
tion
s of
0°,
45°,
90°,
135°
,180
°,22
5°,2
70°,
and
315°
.
Th
ere
are
eigh
t ro
tati
ons
less
th
an 3
60 d
egre
es,s
o th
e or
der
of i
ts
rota
tion
al s
ymm
etry
is
8.T
he
quot
ien
t 36
0 �
8 is
45,
so t
he
mag
nit
ude
of
its
rot
atio
nal
sym
met
ry i
s 45
deg
rees
.
Iden
tify
th
e or
der
an
d m
agn
itu
de
of t
he
rota
tion
al s
ymm
etry
of
each
fig
ure
.
1.a
squ
are
2.a
regu
lar
40-g
on
ord
er:
4;m
agn
itu
de:
90°
ord
er:
40;
mag
nit
ud
e:9°
3.4.
ord
er:
2;m
agn
itu
de:
180°
ord
er:
5;m
agn
itu
de:
72°
5.6.
ord
er:
16;
mag
nit
ud
e:22
.5°
ord
er:
3;m
agn
itu
de:
120°
P
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Ro
tati
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 9-3)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csTe
ssel
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
©G
lenc
oe/M
cGra
w-H
ill50
1G
lenc
oe G
eom
etry
Lesson 9-4
Pre-
Act
ivit
yH
ow a
re t
esse
llat
ion
s u
sed
in
art
?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-4
at
the
top
of p
age
483
in y
our
text
book
.
•In
th
e pa
tter
n s
how
n i
n t
he
pict
ure
in
you
r te
xtbo
ok,h
ow m
any
smal
leq
uil
ater
al t
rian
gles
mak
e u
p on
e re
gula
r h
exag
on?
6•
In t
his
pat
tern
,how
man
y fi
sh m
ake
up
one
equ
ilat
eral
tri
angl
e?1�
1 2�
Rea
din
g t
he
Less
on
1.U
nde
rlin
e th
e co
rrec
t w
ord,
phra
se,o
r n
um
ber
to f
orm
a t
rue
stat
emen
t.
a.A
tes
sell
atio
n i
s a
patt
ern
th
at c
over
s a
plan
e w
ith
th
e sa
me
figu
re o
r se
t of
fig
ure
s so
that
th
ere
are
no
(con
gru
ent
angl
es/o
verl
appi
ng
or e
mpt
y sp
aces
/rig
ht
angl
es).
b.
A t
esse
llat
ion
th
at u
ses
only
on
e ty
pe o
f re
gula
r po
lygo
n i
s ca
lled
a(u
nif
orm
/reg
ula
r/se
mi-
regu
lar)
tes
sell
atio
n.
c.T
he
sum
of
the
mea
sure
s of
th
e an
gles
at
any
vert
ex i
n a
ny
tess
ella
tion
is
(90/
180/
360)
.
d.
A t
esse
llat
ion
th
at c
onta
ins
the
sam
e ar
ran
gem
ent
of s
hap
es a
nd
angl
es a
t ev
ery
vert
ex i
s ca
lled
a (
un
ifor
m/r
egu
lar/
sem
i-re
gula
r) t
esse
llat
ion
.
e.In
a r
egu
lar
tess
ella
tion
mad
e u
p of
hex
agon
s,th
ere
are
(3/4
/6)
hex
agon
s m
eeti
ng
atea
ch v
erte
x,an
d th
e m
easu
re o
f ea
ch o
f th
e an
gles
at
any
vert
ex i
s (6
0/90
/120
).
f.A
un
ifor
m t
esse
llat
ion
for
med
usi
ng
two
or m
ore
regu
lar
poly
gon
s is
cal
led
a(r
otat
ion
al/r
egu
lar/
sem
i-re
gula
r) t
esse
llat
ion
.
g.In
a r
egu
lar
tess
ella
tion
mad
e u
p of
tri
angl
es,t
her
e ar
e (3
/4/6
) tr
ian
gles
mee
tin
g at
each
ver
tex,
and
the
mea
sure
of
each
of
the
angl
es a
t an
y ve
rtex
is
(30/
60/1
20).
h.
If a
reg
ula
r te
ssel
lati
on i
s m
ade
up
of q
uad
rila
tera
ls,a
ll o
f th
e qu
adri
late
rals
mu
st b
eco
ngr
uen
t (r
ecta
ngl
es/p
aral
lelo
gram
s/sq
uar
es/t
rape
zoid
s).
2.W
rite
all
of
the
foll
owin
g w
ords
th
at d
escr
ibe
each
tes
sell
atio
n:u
nif
orm
,non
-un
ifor
m,
regu
lar,
sem
i-re
gula
r.
a.n
on
-un
iform
b.
un
iform
,sem
i-re
gu
lar
Hel
pin
g Y
ou
Rem
emb
er
3.O
ften
th
e ev
eryd
ay m
ean
ings
of
a w
ord
can
hel
p yo
u t
o re
mem
ber
its
mat
hem
atic
alm
ean
ing.
Loo
k u
p u
nif
orm
in y
our
dict
ion
ary.
How
can
its
eve
ryda
y m
ean
ings
hel
p yo
uto
rem
embe
r th
e m
ean
ing
of a
un
ifor
mte
ssel
lati
on?
Sam
ple
an
swer
:Th
ree
sim
ilar
mea
nin
gs
of
un
iform
are
unv
aryi
ng
,co
nsi
sten
t,an
d id
enti
cal.
In a
un
iform
tes
sella
tio
n,t
he
arra
ng
emen
t o
f sh
apes
an
d a
ng
les
at e
ach
vert
ex is
th
e sa
me.
Eac
h o
f th
e th
ree
ever
yday
mea
nin
gs
des
crib
es t
his
situ
atio
n.
©G
lenc
oe/M
cGra
w-H
ill50
2G
lenc
oe G
eom
etry
Po
lyg
on
al N
um
ber
sC
erta
in n
um
bers
rel
ated
to
regu
lar
poly
gon
s ar
e ca
lled
pol
ygon
al n
um
ber
s.T
he
char
tsh
ows
seve
ral
tria
ngu
lar,
squ
are,
and
pen
tago
nal
nu
mbe
rs.T
he
ran
kof
a p
olyg
on n
um
ber
is t
he
nu
mbe
r of
dot
s on
eac
h “
side
”of
th
e ou
ter
poly
gon
.For
exa
mpl
e,th
e pe
nta
gon
aln
um
ber
22 h
as a
ran
k of
4.
Pol
ygon
al n
um
bers
can
be
desc
ribe
d w
ith
for
mu
las.
For
exa
mpl
e,a
tria
ngu
lar
nu
mbe
r T
of r
ank
r ca
n b
e de
scri
bed
by T
��r(
r2�
1)�
.
An
swer
eac
h q
ues
tion
.
1.D
raw
a d
iagr
am t
o fi
nd
the
2.D
raw
a d
iagr
am t
o fi
nd
the
tria
ngu
lar
nu
mbe
r of
ran
k 5.
15pe
nta
gon
al n
um
ber
of r
ank
5.35
3.W
rite
a f
orm
ula
for
a s
quar
e n
um
ber
4.W
rite
a f
orm
ula
for
a p
enta
gon
alS
of r
ank
r.n
um
ber
Pof
ran
k r.
S�
r2P
��r(
3r2�
1)�
5.W
hat
is
the
ran
k of
th
e pe
nta
gon
al6.
Lis
t th
e h
exag
onal
nu
mbe
rs f
or r
anks
nu
mbe
r 70
?7
1 to
5.(
Hin
t:D
raw
a d
iagr
am.)
1,6,
15,2
8,45
Ran
k 1
Tria
ngle
13
610
14
916
15
1222
Squ
are
Pen
tago
n
Ran
k 2
Ran
k 3
Ran
k 4
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
Answers (Lesson 9-4)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Tess
ella
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
©G
lenc
oe/M
cGra
w-H
ill49
7G
lenc
oe G
eom
etry
Lesson 9-4
Reg
ula
r Te
ssel
lati
on
sA
pat
tern
th
at c
over
s a
plan
e w
ith
rep
eati
ng
copi
es o
f on
e or
mor
e fi
gure
s so
th
at t
her
e ar
e n
o ov
erla
ppin
g or
em
pty
spac
es i
s a
tess
ella
tion
.A r
egu
lar
tess
ella
tion
uses
onl
y on
e ty
pe o
f re
gula
r po
lygo
n.In
a t
esse
llat
ion,
the
sum
of
the
mea
sure
sof
the
ang
les
of t
he p
olyg
ons
surr
ound
ing
a ve
rtex
is 3
60.I
f a
regu
lar
poly
gon
has
an
in
teri
oran
gle
that
is
a fa
ctor
of
360,
then
th
e po
lygo
n w
ill
tess
ella
te.
regu
lar
tess
ella
tion
tess
ella
tion
Cop
ies
of a
reg
ular
hex
agon
C
opie
s of
a r
egul
ar p
enta
gon
can
form
a t
esse
llatio
n.ca
nnot
for
m a
tes
sella
tion.
Det
erm
ine
wh
eth
er a
reg
ula
r 16
-gon
tes
sell
ates
th
e p
lan
e.E
xpla
in.
If m
�1
is t
he
mea
sure
of
one
inte
rior
an
gle
of a
reg
ula
r po
lygo
n,t
hen
a f
orm
ula
for
m�
1
is m
�1
��18
0(n n
�2)
�.U
se t
he
form
ula
wit
h n
�16
.
m�
1��18
0(n n
�2)
�
��18
0(1 16 6
�2)
�
�15
7.5
Th
e va
lue
157.
5 is
not
a f
acto
r of
360
,so
the
16-g
on w
ill
not
tes
sell
ate.
Det
erm
ine
wh
eth
er e
ach
pol
ygon
tes
sell
ates
th
e p
lan
e.If
so,
dra
w a
sam
ple
fig
ure
.
1.sc
alen
e ri
ght
tria
ngl
eye
s2.
isos
cele
s tr
apez
oid
yes
Det
erm
ine
wh
eth
er e
ach
reg
ula
r p
olyg
on t
esse
llat
es t
he
pla
ne.
Exp
lain
.
3.sq
uar
e4.
20-g
on
Yes;
the
mea
sure
of
each
inte
rio
r N
o;
the
mea
sure
of
each
inte
rio
r an
gle
is 9
0,an
d 9
0 is
a f
acto
r an
gle
is 1
62,a
nd
162
is n
ot
a fa
cto
r o
f 36
0.o
f 36
0.
5.se
ptag
on6.
15-g
on
No
;th
e m
easu
re o
f ea
ch in
teri
or
No
;th
e m
easu
re o
f ea
ch in
teri
or
ang
le is
128
.6,a
nd
128
.6 is
no
t a
ang
le is
156
,an
d 1
56 is
no
t a
fact
or
fact
or
of
360.
of
360.
7.oc
tago
n8.
pen
tago
n
No
;th
e m
easu
re o
f ea
ch in
teri
or
No
;th
e m
easu
re o
f ea
ch in
teri
or
ang
le is
135
,an
d 1
35 is
no
t a
ang
le is
108
,an
d 1
08 is
no
t a
fact
or
fact
or
of
360.
of
360.
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill49
8G
lenc
oe G
eom
etry
Tess
ella
tio
ns
wit
h S
pec
ific
Att
rib
ute
sA
tes
sell
atio
n p
atte
rn c
an c
onta
in a
ny
type
of p
olyg
on.I
f th
e ar
ran
gem
ent
of s
hap
es a
nd
angl
es a
t ea
ch v
erte
x in
th
e te
ssel
lati
on i
s th
esa
me,
the
tess
ella
tion
is
un
ifor
m.A
sem
i-re
gula
r te
ssel
lati
onis
a u
nif
orm
tes
sell
atio
nth
at c
onta
ins
two
or m
ore
regu
lar
poly
gon
s.
Det
erm
ine
wh
eth
er a
kit
e w
ill
tess
ella
te t
he
pla
ne.
If s
o,d
escr
ibe
the
tess
ella
tion
as
un
ifor
m,r
egu
lar,
sem
i-re
gula
r,or
not
un
ifor
m.
A k
ite
wil
l te
ssel
late
th
e pl
ane.
At
each
ver
tex
the
sum
of
th
e m
easu
res
is a
�b
�b
�c,
wh
ich
is
360.
Th
e te
ssel
lati
on i
s u
nif
orm
.
Det
erm
ine
wh
eth
er a
sem
i-re
gula
r te
ssel
lati
on c
an b
e cr
eate
d f
rom
eac
h s
et o
ffi
gure
s.If
so,
sket
ch t
he
tess
ella
tion
.Ass
um
e th
at e
ach
fig
ure
has
a s
ide
len
gth
of
1 u
nit
.
1.rh
ombu
s,eq
uil
ater
al t
rian
gle,
2.sq
uar
e an
d eq
uil
ater
al t
rian
gle
and
octa
gon
no
yes
Det
erm
ine
wh
eth
er e
ach
pol
ygon
tes
sell
ates
th
e p
lan
e.If
so,
des
crib
e th
ete
ssel
lati
on a
s u
nif
orm
,not
un
ifor
m,r
egu
lar,
or s
emi-
regu
lar.
3.re
ctan
gle
4.h
exag
on a
nd
squ
are
yes,
un
iform
no
c�a�
b� b�
c�c�
a�
a�b�
b� b�
c�a�
b� b�
c�a�
b� b�
c�a�
b� b�
b�
c�a�
b� b�
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Tess
ella
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 9-4)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Dila
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
©G
lenc
oe/M
cGra
w-H
ill50
5G
lenc
oe G
eom
etry
Lesson 9-5
Dra
w t
he
dil
atio
n i
mag
e of
eac
h f
igu
re w
ith
cen
ter
Can
d t
he
give
n s
cale
fac
tor.
1.r
�2
2.r
��1 4�
Fin
d t
he
mea
sure
of
the
dil
atio
n i
mag
e M�
��N���o
r of
th
e p
reim
age
M�N�
usi
ng
the
give
n s
cale
fac
tor.
3.M
N�
3,r
�3
4.M
�N�
�7,
r�
21
M�N
��
9M
N�
�1 3�
CO
OR
DIN
ATE
GEO
MET
RYF
ind
th
e im
age
of e
ach
pol
ygon
,giv
en t
he
vert
ices
,aft
era
dil
atio
n c
ente
red
at
the
orig
in w
ith
a s
cale
fac
tor
of 2
.Th
en g
rap
h a
dil
atio
n
cen
tere
d a
t th
e or
igin
wit
h a
sca
le f
acto
r of
�1 2� .
5.J
(2,4
),K
(4,4
),P
(3,2
)6.
D(�
2,0)
,G(0
,2),
F(2
,�2)
Det
erm
ine
the
scal
e fa
ctor
for
eac
h d
ilat
ion
wit
h c
ente
r C
.Det
erm
ine
wh
eth
er t
he
dil
atio
n i
s an
en
larg
emen
t,re
du
ctio
n,o
r co
ng
ruen
ce t
ran
sfor
ma
tion
.Th
e d
ash
edfi
gure
is
the
dil
atio
n i
mag
e.
7.8.
4;en
larg
emen
t1;
con
gru
ence
tra
nsf
orm
atio
n
CC
G�
F�
D�
G�
F�
D�
x
y
O
J�K
�
P�
J�
K�
P�
x
y
O
CC
©G
lenc
oe/M
cGra
w-H
ill50
6G
lenc
oe G
eom
etry
Dra
w t
he
dil
atio
n i
mag
e of
eac
h f
igu
re w
ith
cen
ter
Can
d t
he
give
n s
cale
fac
tor.
1.r
��3 2�
2.r
��2 3�
Fin
d t
he
mea
sure
of
the
dil
atio
n i
mag
e A�
��T���o
r of
th
e p
reim
age
A�T�
usi
ng
the
give
nsc
ale
fact
or.
3.A
T�
15,r
��3 5�
4.A
T�
30,r
��
�1 6�5.
A�T
��
12,r
��4 3�
A�T
��
9A
�T�
�5
AT
�9
CO
OR
DIN
ATE
GEO
MET
RYF
ind
th
e im
age
of e
ach
pol
ygon
,giv
en t
he
vert
ices
,aft
era
dil
atio
n c
ente
red
at
the
orig
in w
ith
a s
cale
fac
tor
of 2
.Th
en g
rap
h a
dil
atio
n
cen
tere
d a
t th
e or
igin
wit
h a
sca
le f
acto
r of
�1 2� .
6.A
(1,1
),C
(2,3
),D
(4,2
),E
(3,1
)7.
Q(�
1,�
1),R
(0,2
),S
(2,1
)
Det
erm
ine
the
scal
e fa
ctor
for
eac
h d
ilat
ion
wit
h c
ente
r C
.Det
erm
ine
wh
eth
er t
he
dil
atio
n i
s an
en
larg
emen
t,re
du
ctio
n,o
r co
ng
ruen
ce t
ran
sfor
ma
tion
.Th
e d
otte
dfi
gure
is
the
dil
atio
n i
mag
e.
8.�1 3� ;
red
uct
ion
9.2;
enla
rgem
ent
10.P
HO
TOG
RA
PHY
Est
ebe
enla
rged
a 4
-in
ch b
y 6-
inch
ph
otog
raph
by
a fa
ctor
of
�5 2� .W
hat
are
the
new
dim
ensi
ons
of t
he
phot
ogra
ph?
10 in
.by
15 in
.
CC
S�
Q�
R�
S�
Q�
R�
x
y
O
C�
D�
E�
A�
C�
D�
E�
A�
x
y
O
C
C
Pra
ctic
e (
Ave
rag
e)
Dila
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
Answers (Lesson 9-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csD
ilati
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
©G
lenc
oe/M
cGra
w-H
ill50
7G
lenc
oe G
eom
etry
Lesson 9-5
Pre-
Act
ivit
yH
ow d
o yo
u u
se d
ilat
ion
s w
hen
you
use
a c
omp
ute
r?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-5
at
the
top
of p
age
490
in y
our
text
book
.
In a
ddit
ion
to
the
exam
ple
give
n i
n y
our
text
book
,giv
e tw
o ev
eryd
ayex
ampl
es o
f sc
alin
g an
obj
ect,
one
that
mak
es t
he
obje
ct l
arge
r an
d an
oth
erth
at m
akes
it
smal
ler.
Sam
ple
an
swer
:la
rger
:en
larg
ing
ap
ho
tog
rap
h;
smal
ler:
mak
ing
a s
cale
mo
del
of
a bu
ildin
gR
ead
ing
th
e Le
sso
n1.
Eac
h o
f th
e va
lues
of
rgi
ven
bel
ow r
epre
sen
ts t
he
scal
e fa
ctor
for
a d
ilat
ion
.In
eac
hca
se,d
eter
min
e w
het
her
th
e di
lati
on i
s an
en
larg
emen
t,a
red
uct
ion
,or
a co
ngr
uen
cetr
ansf
orm
atio
n.
a.r
�3
enla
rgem
ent
b.r
�0.
5re
du
ctio
nc.
r�
�0.
75re
du
ctio
nd
.r�
�1
con
gru
ence
tra
nsf
orm
atio
ne.
r�
�2 3�re
du
ctio
nf.
r�
��3 2�
enla
rgem
ent
g.r
��
1.01
enla
rgem
ent
h.r
�0.
999
red
uct
ion
2.D
eter
min
e w
het
her
eac
h s
ente
nce
is
alw
ays,
som
etim
es,o
r n
ever
tru
e.If
th
e se
nte
nce
is
not
alw
ays
tru
e,ex
plai
n w
hy.
Fo
r ex
pla
nat
ion
s,sa
mp
le a
nsw
ers
are
giv
en.
a.A
dil
atio
n r
equ
ires
a c
ente
r po
int
and
a sc
ale
fact
or.
alw
ays
b.
A d
ilat
ion
ch
ange
s th
e si
ze o
f a
figu
re.
So
met
imes
;if
th
e d
ilati
on
is a
con
gru
ence
tra
nsf
orm
atio
n,t
he
size
of
the
fig
ure
is u
nch
ang
ed.
c.A
dil
atio
n c
han
ges
the
shap
e of
a f
igu
re.
Nev
er;
all d
ilati
on
s p
rod
uce
sim
ilar
fig
ure
s,an
d s
imila
r fi
gu
res
hav
e th
e sa
me
shap
e.d
.T
he
imag
e of
a f
igu
re u
nde
r a
dila
tion
lie
s on
th
e op
posi
te s
ide
of t
he
cen
ter
from
th
epr
eim
age.
So
met
imes
;th
is is
on
ly t
rue
wh
en t
he
scal
e fa
cto
r is
neg
ativ
e.e.
A s
imil
arit
y tr
ansf
orm
atio
n i
s a
con
gru
ence
tra
nsf
orm
atio
n.
So
met
imes
;th
is is
tru
e o
nly
wh
en t
he
scal
e fa
cto
r is
1 o
r �
1.f.
Th
e ce
nte
r of
a d
ilat
ion
is
its
own
im
age.
alw
ays
g.A
dil
atio
n i
s an
iso
met
ry.
So
met
imes
;th
is is
tru
e o
nly
wh
en t
he
dila
tio
n is
a co
ng
ruen
ce t
ran
sfo
rmat
ion
.h
.T
he
scal
e fa
ctor
for
a d
ilat
ion
is
a po
siti
ve n
um
ber.
So
met
imes
;th
e sc
ale
fact
or
can
be
any
po
siti
ve o
r n
egat
ive
nu
mb
er.
i.D
ilat
ion
s pr
odu
ce s
imil
ar f
igu
res.
alw
ays
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r so
met
hin
g is
to
expl
ain
it
to s
omeo
ne
else
.Su
ppos
e th
at y
our
clas
smat
e L
ydia
is
hav
ing
trou
ble
un
ders
tan
din
g th
e re
lati
onsh
ip b
etw
een
sim
ilar
ity
tran
sfor
mat
ion
san
d co
ngr
uen
ce t
ran
sfor
mat
ion
s.H
ow c
an y
ou e
xpla
in t
his
to
her
?S
amp
le a
nsw
er:
A c
on
gru
ence
tra
nsf
orm
atio
n p
rese
rves
th
e si
ze a
nd
shap
e o
f a
fig
ure
,th
at is
,th
e im
age
is c
on
gru
ent
to t
he
pre
imag
e.A
sim
ilari
ty t
ran
sfo
rmat
ion
pre
serv
es t
he
shap
e o
f a
fig
ure
,th
at is
,th
eim
age
is s
imila
r to
th
e p
reim
age.
A s
imila
rity
tra
nsf
orm
atio
n is
aco
ng
ruen
ce t
ran
sfo
rmat
ion
if t
he
scal
e fa
cto
r is
1 o
r �
1.
©G
lenc
oe/M
cGra
w-H
ill50
8G
lenc
oe G
eom
etry
Sim
ilar
Cir
cles
You
may
be
surp
rise
d to
lea
rn t
hat
tw
o n
onco
ngr
uen
t ci
rcle
s th
at l
ie i
nth
e sa
me
plan
e an
d h
ave
no
com
mon
in
teri
or p
oin
ts c
an b
e m
appe
don
e on
to t
he
oth
er b
y m
ore
than
on
e di
lati
on.
1.H
ere
is d
iagr
am t
hat
su
gges
ts o
ne
way
to
m
ap a
sm
alle
r ci
rcle
on
to a
lar
ger
one
usi
ng
a di
lati
on.T
he
circ
les
are
give
n.T
he
lin
essu
gges
t h
ow t
o fi
nd
the
cen
ter
for
the
dila
tion
.Des
crib
e h
ow t
he
cen
ter
is f
oun
d.U
se s
egm
ents
in
th
e di
agra
m t
o n
ame
the
scal
e fa
ctor
.
Dra
w t
he
two
co
mm
on
ext
ern
alta
ng
ents
.Th
e p
oin
t w
her
e th
ey in
ters
ect
is t
he
cen
ter
of
a d
ilati
on
th
at m
aps
�A
on
to �
A�;
scal
e fa
cto
r �
�A A�M M�
�.
2.H
ere
is a
not
her
pai
r of
non
con
gru
ent
circ
les
wit
h n
o co
mm
onin
teri
or p
oin
t.F
rom
Exe
rcis
e 1,
you
kn
ow y
ou c
an l
ocat
e a
poin
t of
f to
th
e le
ft o
f th
e sm
alle
r ci
rcle
th
at i
s th
e ce
nte
r fo
r a
dila
tion
map
pin
g �
Con
to �
C�.
Fin
d an
oth
er c
ente
r fo
r an
oth
er d
ilat
ion
that
map
s �
Con
to �
C�.
Mar
k an
d la
bel
segm
ents
to
nam
e th
esc
ale
fact
or.
Fin
d t
he
inte
rsec
tio
n o
f th
e tw
o c
om
mo
n in
tern
al t
ang
ents
;sc
ale
fact
or:
��C C�X X
��
.
X
X�
C�
CO
A �
M�
M
AO
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
Answers (Lesson 9-5)
© Glencoe/McGraw-Hill A18 Glencoe Geometry
Skil
ls P
ract
ice
Vec
tors
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
©G
lenc
oe/M
cGra
w-H
ill51
1G
lenc
oe G
eom
etry
Lesson 9-6
Wri
te t
he
com
pon
ent
form
of
each
vec
tor.
1.2.
�3,4
��5
,�5�
Fin
d t
he
mag
nit
ud
e an
d d
irec
tion
of
RS�
for
the
give
n c
oord
inat
es.R
oun
d t
o th
en
eare
st t
enth
.
3.R
(2,�
3),S
(4,9
)4.
R(0
,2),
S(3
,12)
2�37�
�12
.2,8
0.5°
�10
9�
�10
.4,7
3.3°
5.R
(5,4
),S
(�3,
1)6.
R(1
,5),
S(�
4,�
6)
�73�
�8.
5,20
0.6°
�14
6�
�12
.1,2
45.6
°
Gra
ph
th
e im
age
of e
ach
fig
ure
un
der
a t
ran
slat
ion
by
the
give
n v
ecto
r(s)
.
7.�
AB
Cw
ith
ver
tice
s A
(�4,
3),B
(�1,
4),
8.tr
apez
oid
wit
h v
erti
ces
T(�
4,�
2),
C(�
1,1)
;t��
�4,�
3�R
(�1,
�2)
,S(�
2,�
3),Y
(�3,
�3)
;a��
�3,1
�an
d b�
��2
,4�
Fin
d t
he
mag
nit
ud
e an
d d
irec
tion
of
each
res
ult
ant
for
the
give
n v
ecto
rs.
9.y �
��7
,0�,
z��
�0,6
�10
.b��
�3,2
�,c�
���
2,3�
�85�
�9.
2,40
.6°
�26�
�5.
1,78
.7°
S�
Y�
T�
R�
SY
TR
x
y
O
C�
B�
A�
CB
A
x
y
O
x
y
O
B( –
2, 2
)
D( 3
, –3)
x
y
OC
( 1, –
1)
E( 4
, 3)
©G
lenc
oe/M
cGra
w-H
ill51
2G
lenc
oe G
eom
etry
Wri
te t
he
com
pon
ent
form
of
each
vec
tor.
1.2.
�6,4
��5
,�8�
Fin
d t
he
mag
nit
ud
e an
d d
irec
tion
of
FG�
for
the
give
n c
oord
inat
es.R
oun
d t
o th
en
eare
st t
enth
.
3.F
(�8,
�5)
,G(�
2,7)
4.F
(�4,
1),G
(5,�
6)
6�5�
�13
.4,6
3.4°
�13
0�
�11
.4,3
22.1
°
Gra
ph
th
e im
age
of e
ach
fig
ure
un
der
a t
ran
slat
ion
by
the
give
n v
ecto
r(s)
.
5.�
QR
Tw
ith
ver
tice
s Q
(�1,
1),R
(1,4
),6.
trap
ezoi
d w
ith
ver
tice
s J
(�4,
�1)
,T
(5,1
);s �
���
2,�
5�K
(0,�
1),L
(�1,
�3)
,M(�
2,�
3);c�
��5
,4�
and
d��
��2,
1�
Fin
d t
he
mag
nit
ud
e an
d d
irec
tion
of
each
res
ult
ant
for
the
give
n v
ecto
rs.
7.a�
���
6,4�
,b��
�4,6
�8.
e��
��4,
�5�
,f��
��1,
3�
2�26�
�10
.2,1
01.3
°�
29��
5.4,
201.
8°
AV
IATI
ON
For
Exe
rcis
es 9
an
d 1
0,u
se t
he
foll
owin
g in
form
atio
n.
A je
t be
gins
a f
light
alo
ng a
pat
h du
e no
rth
at 3
00 m
iles
per
hour
.A w
ind
is b
low
ing
due
wes
tat
30
mile
s pe
r ho
ur.
9.F
ind
the
resu
ltan
t ve
loci
ty o
f th
e pl
ane.
abo
ut
301.
5 m
ph
10.F
ind
the
resu
ltan
t di
rect
ion
of
the
plan
e.ab
ou
t 5.
7°w
est
of
du
e n
ort
h
L�M
�
J�K
�
LM
JK
x
y
O
T�
R�
Q�
T
R
Qx
y
O
x
y
O
K( –
2, 4
)
L(3,
–4)
x
y
O
A( –
2, –
2)
B( 4
, 2)
Pra
ctic
e (
Ave
rag
e)
Vec
tors
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
Answers (Lesson 9-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csV
ecto
rs
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
©G
lenc
oe/M
cGra
w-H
ill51
3G
lenc
oe G
eom
etry
Lesson 9-6
Pre-
Act
ivit
yH
ow d
o ve
ctor
s h
elp
a p
ilot
pla
n a
fli
ght?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-6
at
the
top
of p
age
498
in y
our
text
book
.
Wh
y do
pil
ots
ofte
n h
ead
thei
r pl
anes
in
a s
ligh
tly
diff
eren
t di
rect
ion
fro
mth
eir
dest
inat
ion
?S
amp
le a
nsw
er:
to c
om
pen
sate
fo
r th
e ef
fect
of
the
win
d s
o t
hat
th
e re
sult
will
be
that
th
e p
lan
e w
ill a
ctu
ally
fly
in t
he
corr
ect
dir
ecti
on
Rea
din
g t
he
Less
on
1.S
upp
ly t
he
mis
sin
g w
ords
or
phra
ses
to c
ompl
ete
the
foll
owin
g se
nte
nce
s.
a.A
is
a d
irec
ted
segm
ent
repr
esen
tin
g a
quan
tity
th
at h
as b
oth
mag
nit
ude
an
d di
rect
ion
.
b.
Th
e le
ngt
h o
f a
vect
or i
s ca
lled
its
.
c.T
wo
vect
ors
are
para
llel
if
and
only
if
they
hav
e th
e sa
me
or
dire
ctio
n.
d.
A v
ecto
r is
in
if
it
is d
raw
n w
ith
in
itia
l po
int
at t
he
orig
in.
e.T
wo
vect
ors
are
equ
al i
f an
d on
ly i
f th
ey h
ave
the
sam
e an
d th
e
sam
e.
f.T
he
sum
of
two
vect
ors
is c
alle
d th
e .
g.A
vec
tor
is w
ritt
en i
n
if i
t is
exp
ress
ed a
s an
ord
ered
pai
r.
h.
Th
e pr
oces
s of
mu
ltip
lyin
g a
vect
or b
y a
con
stan
t is
cal
led
.
2.W
rite
eac
h v
ecto
r de
scri
bed
belo
w i
n c
ompo
nen
t fo
rm.
a.a
vect
or i
n s
tan
dard
pos
itio
n w
ith
en
dpoi
nt
(a,b
)�a
,b�
b.
a ve
ctor
wit
h i
nit
ial
poin
t (a
,b)
and
endp
oin
t (c
,d)
�c�
a,d
�b
�c.
a ve
ctor
in
sta
nda
rd p
osit
ion
wit
h e
ndp
oin
t (�
3,5)
��3,
5�d
.a
vect
or w
ith
in
itia
l po
int
(2,�
3) a
nd
endp
oin
t (6
,�8)
�4,�
5�e.
a��
b�if
a��
��3,
5�an
d b�
��6
,�4�
�3,1
�f.
5u�if
u��
�8,�
6��4
0,�
30�
g.�
�1 3� v�if
v��
��15
,24�
�5,�
8�h
.0.
5u��
1.5v�
if u�
��1
0,�
10�
and
v��
��8,
6���
7,4�
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al t
erm
is
to r
elat
e it
to
a te
rm y
ou a
lrea
dykn
ow.Y
ou le
arne
d ab
out
scal
e fa
ctor
sw
hen
you
stud
ied
sim
ilari
ty a
nd d
ilati
ons.
How
is t
heid
ea o
f a
scal
arre
late
d to
sca
le f
acto
rs?
Sam
ple
an
swer
:A s
cala
ris
th
e te
rm u
sed
for
a co
nst
ant
(a s
pec
ific
real
nu
mb
er)
wh
en w
ork
ing
with
vec
tors
.A v
ecto
rh
as b
oth
mag
nit
ud
e an
d d
irec
tio
n,w
hile
a s
cala
r is
just
a m
agn
itu
de.
Mu
ltip
lyin
g a
vec
tor
by a
po
siti
ve s
cala
r ch
ang
es t
he
mag
nit
ud
e o
f th
eve
cto
r,bu
t n
ot
the
dir
ecti
on
,so
it r
epre
sen
ts a
ch
ang
e in
sca
le.
scal
ar m
ult
iplic
atio
nco
mp
on
ent
formre
sult
ant
dir
ecti
on
mag
nit
ud
est
and
ard
po
siti
on
op
po
site
mag
nit
ud
e
vect
or
©G
lenc
oe/M
cGra
w-H
ill51
4G
lenc
oe G
eom
etry
Rea
din
g M
ath
emat
ics
Man
y qu
anti
ties
in
nat
ure
can
be
thou
ght
of a
s ve
ctor
s.T
he
scie
nce
of
phys
ics
invo
lves
man
y ve
ctor
qu
anti
ties
.In
rea
din
g ab
out
appl
icat
ion
sof
mat
hem
atic
s,as
k yo
urs
elf
wh
eth
er t
he
quan
titi
es i
nvo
lve
only
mag
nit
ude
or
both
mag
nit
ude
an
d di
rect
ion
.Th
e fi
rst
kin
d of
qu
anti
tyis
cal
led
scal
ar.T
he
seco
nd
kin
d is
a v
ecto
r.
Cla
ssif
y ea
ch o
f th
e fo
llow
ing.
Wri
te s
cala
ror
vec
tor.
1.th
e m
ass
of a
boo
ksc
alar
2.a
car
trav
elin
g n
orth
at
55 m
phve
cto
r
3.a
ball
oon
ris
ing
24 f
eet
per
min
ute
vect
or
4.th
e si
ze o
f a
shoe
scal
ar
5.a
room
tem
pera
ture
of
22 d
egre
es C
elsi
us
scal
ar
6.a
wes
t w
ind
of 1
5 m
phve
cto
r
7.th
e ba
ttin
g av
erag
e of
a b
aseb
all
play
ersc
alar
8.a
car
trav
elin
g 60
mph
vect
or
9.a
rock
fal
lin
g at
10
mph
vect
or
10.y
our
age
scal
ar
11.t
he
forc
e of
Ear
th’s
gra
vity
act
ing
on a
mov
ing
sate
llit
eve
cto
r
12.t
he
area
of
a re
cord
rot
atin
g on
a t
urn
tabl
esc
alar
13.t
he
len
gth
of
a ve
ctor
in
th
e co
ordi
nat
e pl
ane
scal
ar
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
Answers (Lesson 9-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Tran
sfo
rmat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-7
9-7
©G
lenc
oe/M
cGra
w-H
ill51
5G
lenc
oe G
eom
etry
Lesson 9-7
Tran
slat
ion
s an
d D
ilati
on
sA
vec
tor
can
be
repr
esen
ted
by t
he
orde
red
pair
�x,
y�or
by t
he
colu
mn
mat
rix
.Wh
en t
he
orde
red
pair
s fo
r al
l th
e ve
rtic
es o
f a
poly
gon
are
plac
ed t
oget
her
,th
e re
sult
ing
mat
rix
is c
alle
d th
e ve
rtex
mat
rix
for
the
poly
gon
.
For
�A
BC
wit
h A
(�2,
2),B
(2,1
),an
d C
(�1,
�1)
,th
e ve
rtex
mat
rix
for
the
tria
ngl
e is
.
For
�A
BC
abov
e,u
se a
mat
rix
to f
ind
th
e co
ord
inat
es o
f th
eve
rtic
es o
f th
e im
age
of �
AB
Cu
nd
er t
he
tran
slat
ion
(x,
y) →
(x�
3,y
�1)
.T
o tr
ansl
ate
the
figu
re 3
un
its
to t
he
righ
t,ad
d 3
to e
ach
x-c
oord
inat
e.T
o tr
ansl
ate
the
figu
re 1
un
it d
own
,add
�1
to e
ach
y-c
oord
inat
e.V
erte
x M
atri
xT
ran
slat
ion
V
erte
x M
atri
xof
�A
BC
Mat
rix
of �
A�B
�C�
��
Th
e co
ordi
nat
es a
re A
�(1,
1),B
�(5,
0),a
nd
C�(
2,�
2).
For
�A
BC
abov
e,u
se a
mat
rix
to f
ind
th
e co
ord
inat
es o
f th
e ve
rtic
esof
th
e im
age
of �
AB
Cfo
r a
dil
atio
n c
ente
red
at
the
orig
in w
ith
sca
le f
acto
r 3.
Sca
leV
erte
x M
atri
xV
erte
x M
atri
xFa
ctor
of �
AB
Cof
�A
�B�C
�
3
�
Th
e co
ordi
nat
es a
re A
�(�
6,6)
,B�(
6,3)
,an
d C
�(�
3,�
3).
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
eac
h f
igu
reu
nd
er t
he
give
n t
ran
slat
ion
s or
dil
atio
ns.
1.�
AB
Cw
ith
A(3
,1),
B(�
2,4)
,C(�
2,�
1);(
x,y)
→(x
�1,
y�
2)A
�(2,
3),B
�(�
3,6)
,C�(
�3,
1)2.
para
llel
ogra
m R
ST
Uw
ith
R(�
4,�
2),S
(�3,
1),T
(3,4
),U
(2,1
);(x
,y)
→(x
�4,
y�
3)
R�(
�8,
�5)
,S�(
�7,
�2)
,T�(
�1,
1),U
�(�
2,�
2)3.
rect
angl
e P
QR
Sw
ith
P(4
,0),
Q(3
,�3)
,R(�
3,�
1),S
(�2,
2);(
x,y)
→(x
�2,
y�
1)P
�(2,
1),Q
�(1,
�2)
,R�(
�5,
0),S
�(�
4,3)
4.�
AB
Cw
ith
A(�
2,�
1),B
(�2,
�3)
,C(2
,�1)
;dil
atio
n c
ente
red
at t
he
orig
in w
ith
sca
lefa
ctor
2A
�(�
4,�
2),B
�(�
4,�
6),C
�(4,
�2)
5.pa
rall
elog
ram
RS
TU
wit
h R
(4,�
2),S
(�4,
�1)
,T(�
2,3)
,U(6
,2);
dila
tion
cen
tere
d at
th
eor
igin
wit
h s
cale
fac
tor
1.5
R�(
6,�
3),S
�(�
6,�
1.5)
,T�(
�3,
4.5)
,U�(
9,3)
�6
6�
3
63
�3
�2
2�
1
21
�1
15
2 1
0�
2
33
3 �
1�
1�
1�
22
�1
2
1�
1
�2
2�
1
21
�1
x
y
B( 2
, 1)
C( –
1, –
1)
A( –
2, 2
)
x
y
Exer
cises
Exer
cises
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
©G
lenc
oe/M
cGra
w-H
ill51
6G
lenc
oe G
eom
etry
Ref
lect
ion
s an
d R
ota
tio
ns
Wh
en y
ou r
efle
ct a
n i
mag
e,on
e w
ay t
o fi
nd
the
coor
din
ates
of
the
refl
ecte
d ve
rtic
es i
s to
mu
ltip
ly t
he
vert
ex m
atri
x of
th
e ob
ject
by
are
flec
tion
mat
rix.
To
perf
orm
mor
e th
an o
ne
refl
ecti
on,m
ult
iply
by
one
refl
ecti
on m
atri
xto
fin
d th
e fi
rst
imag
e.T
hen
mu
ltip
ly b
y th
e se
con
d m
atri
x to
fin
d th
e fi
nal
im
age.
Th
em
atri
ces
for
refl
ecti
ons
in t
he
axes
,th
e or
igin
,an
d th
e li
ne
y�
xar
e sh
own
bel
ow.
Fo
r a
refl
ecti
on
in t
he:
x-ax
isy-
axis
ori
gin
line
y�
x
Mu
ltip
ly t
he
vert
ex
mat
rix
by:
�A
BC
has
ver
tice
s A
(�2,
3),B
(1,4
),an
d
C(3
,0).
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
th
e im
age
of �
AB
Caf
ter
a re
flec
tion
in
th
e x-
axis
.T
o re
flec
t in
th
e x-
axis
,mu
ltip
ly t
he
vert
ex m
atri
x of
�A
BC
by t
he
refl
ecti
on m
atri
x fo
r th
e x-
axis
.
Ref
lect
ion
Mat
rix
Ver
tex
Mat
rix
Ver
tex
Mat
rix
for
x-ax
isof
�A
BC
of �
A’B
’C’
�
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
eac
h f
igu
reu
nd
er t
he
give
n r
efle
ctio
n.
1.�
AB
Cw
ith
A(�
3,2)
,B(�
1,3)
,C(1
,0);
refl
ecti
on i
n t
he
x-ax
is
A�(
�3,
�2)
,B�(
�1,
�3)
,C�(
1,0)
2.�
XY
Zw
ith
X(2
,�1)
,Y(4
,�3)
,Z(�
2,1)
;ref
lect
ion
in
th
e y-
axis
X�(
�2,
�1)
,Y�(
�4,
�3)
,Z�(
2,1)
3.�
AB
Cw
ith
A(3
,4),
B(�
1,0)
,C(�
2,4)
;ref
lect
ion
in
th
e or
igin
A�(
�3,
�4)
,B�(
1,0)
,C�(
2,�
4)
4.pa
rall
elog
ram
RS
TU
wit
h R
(�3,
2),S
(3,2
),T
(5,�
1),U
(�1,
�1)
;ref
lect
ion
in t
he li
ne y
�x
R�(
2,�
3),S
�(2,
3),T
�(�
1,5)
,U�(
�1,
�1)
5.�
AB
Cw
ith
A(2
,3),
B(�
1,2)
,C(1
,�1)
;ref
lect
ion
in
th
e or
igin
,th
en r
efle
ctio
n i
n t
he
lin
ey
�x
A�(
�3,
�2)
,B�(
�2,
1),C
�(1,
�1)
6.pa
rall
elog
ram
RS
TU
wit
h R
(0,2
),S
(4,2
),T
(3,�
2),U
(�1,
�2)
;ref
lect
ion
in
th
e x-
axis
,th
en r
efle
ctio
n i
n t
he
y-ax
is
R�(
0,�
2),S
�(�
4,�
2),T
�(�
3,2)
,U�(
1,2)
�2
13
�3
�4
0�
21
3
34
01
0 0
�1
x
y
A�
B�
AB
C
01
10
�1
0
0�
1�
10
0
11
00
�1
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Tran
sfo
rmat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-7
9-7
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 9-7)
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csTr
ansf
orm
atio
ns
wit
h M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-7
9-7
©G
lenc
oe/M
cGra
w-H
ill51
9G
lenc
oe G
eom
etry
Lesson 9-7
Pre-
Act
ivit
yH
ow c
an m
atri
ces
be
use
d t
o m
ake
mov
ies?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-7
at
the
top
of p
age
506
in y
our
text
book
.
•W
hat
kin
d of
tra
nsf
orm
atio
n s
hou
ld b
e u
sed
to m
ove
a po
lygo
n?
refl
ecti
on
,tra
nsl
atio
n,o
r ro
tati
on
•W
hat
kind
of
tran
sfor
mat
ion
shou
ld b
e us
ed t
o re
size
a p
olyg
on?
dila
tio
n
Rea
din
g t
he
Less
on
1.W
rite
a v
erte
x m
atri
x fo
r ea
ch f
igu
re.
a.�
AB
Cb
.par
alle
logr
am A
BC
D
2.M
atch
eac
h t
ran
sfor
mat
ion
fro
m t
he
firs
t co
lum
n w
ith
th
e co
rres
pon
din
g m
atri
x fr
omth
e se
con
d or
th
ird
colu
mn
.In
eac
h c
ase,
the
vert
ex m
atri
x fo
r th
e pr
eim
age
of a
fig
ure
is m
ult
ipli
ed o
n t
he
left
by
one
of t
he
mat
rice
s be
low
to
obta
in t
he
imag
e of
th
e fi
gure
.A
ll r
otat
ion
s li
sted
are
cou
nte
rclo
ckw
ise
thro
ugh
th
e or
igin
.(S
ome
mat
rice
s m
ay b
e u
sed
mor
e th
an o
nce
or
not
at
all.)
a.re
flec
tion
ove
r th
e y-
axis
vii.
v.b
.90
°ro
tati
onvi
iic.
refl
ecti
on o
ver
the
lin
e y
�x
iiiii
.vi
.d
.27
0°ro
tati
onv
e.re
flec
tion
ove
r th
e or
igin
iiii
i.vi
i.f.
180°
rota
tion
iig.
refl
ecti
on o
ver
the
x-ax
isvi
iiv
.vi
ii.
h.
360°
rota
tion
i
Hel
pin
g Y
ou
Rem
emb
er3.
How
can
you
rem
embe
r or
qu
ickl
y fi
gure
ou
t th
e m
atri
ces
for
the
tran
sfor
mat
ion
s in
Exe
rcis
e 2?
Vis
ual
ize
or
sket
ch h
ow t
he
“un
it p
oin
ts”
(1,0
) o
n t
he
x-ax
is a
nd
(0,1
) o
n t
he
y-ax
is a
re m
oved
by
the
tran
sfo
rmat
ion
.Wri
te t
he
ord
ered
pai
rfo
r th
e im
age
po
ints
in a
2 �
2 m
atri
x,w
ith t
he
coo
rdin
ates
of
the
imag
e o
fth
e x-
axis
un
it p
oin
t in
th
e fir
st c
olu
mn
an
d t
he
coo
rdin
ates
of
the
imag
e o
fth
e y-
axis
un
it p
oin
t in
th
e se
con
d c
olu
mn
.
0�
1 1
0
0�
1 �
10
10
0�
10
1 1
0
�1
0
01
�1
0
0�
1
0
1 �
10
10
01
�4
�1
41
0
33
03
�3
�1
1
4�
4
BC
AD
x
y
O
B
C
A
x
y
O
©G
lenc
oe/M
cGra
w-H
ill52
0G
lenc
oe G
eom
etry
Vec
tor
Add
itio
nV
ecto
rs a
re p
hys
ical
qu
anti
ties
wit
h m
agn
itu
de a
nd
dire
ctio
n.F
orce
an
d ve
loci
ty a
re t
wo
exam
ples
.We
wil
l in
vest
igat
e ad
din
g ve
ctor
qu
anti
ties
.Th
e su
m o
f tw
o ve
ctor
s is
cal
led
are
sult
ant
vect
oror
just
th
e re
sult
ant.
Tw
o se
par
ate
forc
es,o
ne
mea
suri
ng
20 u
nit
s an
d t
he
oth
er m
easu
rin
g 40
un
its,
act
on a
n
obje
ct.I
f th
e an
gle
bet
wee
n t
he
forc
es i
s 50
°,fi
nd
th
em
agn
itu
de
and
dir
ecti
on o
f th
e re
sult
ant
forc
e.
Fir
st,t
he
vect
ors
mu
st b
e re
arra
nge
d by
pla
cin
g th
e ta
il o
f th
e 20
-un
it v
ecto
r at
th
e h
ead
of t
he
40-u
nit
vec
tor.
Sin
ce t
hes
e ve
ctor
s ar
e n
ot p
erpe
ndi
cula
r,th
e h
oriz
onta
l an
d ve
rtic
alco
mpo
nen
ts o
f on
e of
th
e ve
ctor
s m
ust
be
fou
nd.
Usi
ng
trig
onom
etry
,th
e h
oriz
onta
l co
mpo
nen
t m
ust
be
(20
cos
50°)
u
nit
s an
d th
e ve
rtic
al c
ompo
nen
t m
ust
be
(20
sin
50°
) u
nit
s.R
epla
cin
g th
e 20
-un
it v
ecto
r w
ith
th
ese
com
pon
ents
,we
can
n
ow f
orm
tw
o ve
ctor
s pe
rpen
dicu
lar
and
use
th
e P
yth
agor
ean
T
heo
rem
to
fin
d th
e re
sult
ant.
r2�
(40
�20
cos
50°
)2�
(20
sin
50°
)2
r2�
(52.
9)2
�(1
5.3)
2
r2�
3032
.5r2
�55
.1
tan
O�� 40
2 �02s 0in
c5 os0°50
°�
�0.
2898
m�
O�
16
Th
eref
ore,
the
resu
ltan
t fo
rce
is 5
5.1
un
its
dire
cted
16°
from
th
e 40
-un
it f
orce
.
Sol
ve.R
oun
d a
ll a
ngl
e m
easu
res
to t
he
nea
rest
deg
ree.
Rou
nd
all
oth
er m
easu
res
to t
he
nea
rest
ten
th.
1.A
pla
ne
flie
s du
e w
est
at 2
50 k
ilom
eter
s pe
r h
our
wh
ile
the
win
d bl
ows
sou
th a
t 70
kilo
met
ers
per
hou
r.F
ind
the
plan
e’s
resu
ltan
t ve
loci
ty.
259.
6 km
/h,1
6�so
uth
of
wes
t
2.A
pla
ne
flie
s ea
st f
or 2
00 k
m,t
hen
60°
sou
th o
f ea
st f
or 8
0 km
.Fin
d th
e pl
ane’
s di
stan
cean
d di
rect
ion
fro
m i
ts s
tart
ing
poin
t.
249.
8 km
,16�
sou
th o
f ea
st
3.O
ne
forc
e of
100
un
its
acts
on
an
obj
ect.
An
oth
er f
orce
of
80 u
nit
s ac
ts o
n t
he
obje
ct a
t a
40°
angl
e fr
om t
he
firs
t fo
rce.
Fin
d th
e re
sult
ant
forc
e on
th
e ob
ject
.
169.
3 u
nit
s,18
�fr
om
th
e 10
0-u
nit
fo
rce
O
20
40 �
20
cos
50�
20 s
in 5
0�r
20
20 c
os 5
0�
20 s
in 5
0�
50�
O
20 u
nits
40 u
nits
50�
O
20 u
nits
40 u
nits
50�
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-7
9-7
Exam
ple
Exam
ple
Answers (Lesson 9-7)