Geometry

• Point D(4, -2) is 5 units below the line of reflection. D’ will

be 5 units above the line of reflection.
D(4, -2) ➜ D(4, 8).

Connect the vertices.

237

There are three common lines of reflection: the x-axis,

y-axis, and line y = x. Each has a rule that can be used to

plot points in an image.

LINE OF RULE EXAMPLE
REFLECTION

x-axis (x, y) ➜ (x, -y)
Multiply the
y-coordinate by -1.

y-axis (x, y) ➜ (-x, y)
Multiply the
x-coordinate by -1.

y=x (x, y) ➜ (y, x)
Reverse the order
the same as of the coordinates.

y = 1x + 0

238

EXAMPLE: Reflect AB across the x-axis.

Rule: (x, y) ➜ (x, -y)
A(1, 1) ➜ A’ (1, -1)
B(4, 3) ➜ B’ (4, -3)

1. Plot the image points.

2. Draw a line to connect

the points.

EXAMPLE: Reflect AB across the line y = x.
Rule: (x, y) ➜ (y, x)
A(1, 1) ➜ A’ (1, 1)
B(4, 3) ➜ B’ (3, 4)
1. Plot the image points.
2. Draw a line to connect the points.

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1. What is a geometric transformation?

2. What is a reflection?

3. Complete the sentence.

In a reflection, a point P and its image P’ are the same

distance to the .

For questions 4 and 5, draw the image of PQ, where
P(-1, -2) and Q(-2, 0) are reflected across the following lines.

4. x = 1

240

5. y = -1

answers 241

1. A geometric transformation changes the shape,
size, or position of a figure (preimage) to create
a new figure (the image).

2. A reflection is a type of rigid motion that flips
an image over a line.

3. Line of reflection
4.

5.

242

Chapter 21

TRANSLATIONS

A TRANSLATION is a type

of rigid motion that slides
a figure a certain distance
to the left or right, up or down,
or both horizontally and
vertically.

Each point in the figure
slides the same distance
in the same direction.

The figure’s shape, size,
and orientation remain
the same.

243

TRANSLATIONS moves 4 units
ON A COORDINATE along the x-axis
PLANE

A translation on the moves 2 units
coordinate plane moves along the y-axis

all the points in the image

the same distance and

in the same direction. In

ABC, each point moves

4 units right (x-axis)

and 2 units up (y-axis).

Translations can be
defined (described) using
a TRANSLATION VECTOR,
which states how many
units each point in the
graph moves in the
translation.

The translation vector
is (4, 2).

4 units in the 2 units in the
x-direction y-direction

244

If a translation vector moves a point a units
along the x-axis and b units along the y-axis,

then the translation vector is (a, b).

The translation rule is:

(x, y) ➜ (x + a, y + b),
where (a, b) is the translation vector.

For example: A translation vector of (-1, 3) has a translation

rule of (x, y) ➜ (x - 1, y + 3). This moves each point 1 unit to

the left and 3 units up.

With that translation vector, the point (5, -2) maps to:

(5, -2) ➜ (5 - 1, -2 + 3) which is (4, 1) 5-1=4
-2 + 3 = 1
The point (-4, 7) maps to:

(-4, 7) ➜ (-4 - 1, 7 + 3) or (-5, 10) -4 - 1 = -5
7 + 3 = 10

245

EXAMPLE: What are the translation vector and
translation rule that describe the translation of P ➜ P’ ?

P moves 3 units right and

2 units up to P’ :

The translation vector is (3, 2).

The translation rule is:

(x, y) ➜ (x + 3, y + 2).

EXAMPLE: What are

the translation vector and
translation rule that describe
the translation of quadrilateral

FGHI ➜ F’ G’ H’ I’ ?

Each point moves 4 units right
and 3 units up.

The translation vector is (4, 3).

The translation rule is
(x, y) ➜ (x + 4, y + 3).

246

EXAMPLE: Graph the

translation of LMN, given a
translation vector of (-2, 5).

Move each vertex 2 units to
the left and 5 units up.

Rule: (x, y) ➜ (x - 2, y + 5)

Plot the points.

(-1 – 2) (-3 + 5)

L(-1, -3) ➜ L’ (-3, 2)

(1 – 2) ( 1 + 5)

M(1, 1) ➜ M’ (-1, 6)

(3 – 2) ( 1 + 5)

N(3, 1) ➜ N’ (1, 6)

Connect the points.

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1. After translating any figure, what always remains the
same about the figure?

2. What is the translation vector that translates D to D’ ?

For questions 3-6, state whether the following graph shows
a translation or not.
3. 4.

248

5. 6.

7. What is the translation vector in question 6?

8. What is the translation rule that describes the
translation in question 6?

9. Graph the translation of 10. Graph the translation of
EFG, given a translation the quadrilateral below,
given a translation
vector of (-3, 1). vector of (2, 0).

answers 249

1. The figure’s shape, size, and orientation
2. [-2, -4]
3. Yes, [-5, -3]
4. Yes, [-4, 3]
5. Yes, [3, -1]
6. Yes
7. [7, -2]
8. (x, y) ➜ (x + 7, y - 2)

9. 10.

250

Chapter 22
ROTATIONS

ROTATIONS are transformations

that turn a figure around a fixed
point. Rotations are rigid motions.
The shape, size, and measures of
angles of the figure stay the same,
but the orientation changes.

A rotation includes a:

CENTER OF ROTATION - the point around which a

figure is turned. The center of rotation can be located
outside the figure or anywhere inside or along the figure.

ANGLE OF ROTATION - the number of degrees each

point on the figure is turned. Rotation can be clockwise
or counterclockwise.

clockwise = turns right
counterclockwise = turns left

251

Any point and its image are the same distance from
the center of rotation.

EXAMPLE: Point T is rotated x ˚ counterclockwise

about point R.
The center of rotation is R.

The angle of rotation is x ˚.

T and T’ are the same distance

from the center of rotation, R.

This is written as: RT = RT’.

252

DRAWING ROTATIONS

You can use a protractor and a ruler to draw a rotation
about a point.

To rotate point K 70˚ counterclockwise

about point P:

Step 1: Draw a line from P to K.

Step 2: Use a protractor to draw

a 70˚ angle counterclockwise,

left, from PK.

Step 3: Measure the length of PK.
Draw a new point labeled K’ the same

distance from P on the new line.

253

Rotating a Square
To rotate a square 90˚ clockwise

about the center of rotation, P,
each point on the square must

rotate 90˚ clockwise.

Distance of A to P is the
same as distance of A' to P.

Since AP = A’ P , think of the line

AP rotating 90˚ clockwise.

Use a protractor to draw a

90˚ angle.

Plot a point at the location.

Repeat the same for
vertices B, C, and D.
Then connect the points.

254

EXAMPLE: Draw the image of

ABC rotated 110˚ counterclockwise

about point Q.
Rotate each vertex, one at a time,
using a protractor and ruler.

To rotate Point A:
1. Draw a line from point Q to point A.

2. Use a protractor to draw a 110˚ angle.

3. Measure the length of QA.
4. Draw a point A’ the same

distance on the new line.
Rotate points B and C in
the same way. Connect

points A’, B’, and C’.

255

FINDING THE ANGLE
OF ROTATION

You can find an angle of rotation with a protractor and
ruler. A figure is rotated counterclockwise about a point
located at (-1, 0). Point (2, 2) is rotated to (-3, 3).
To find the angle of rotation:

1. Draw a line from the center of rotation through each

point (2, 2) and (-3, 3).

256

2. Use a protractor to measure the angle.

The angle of rotation is 90˚.

ROTATIONS ON
THE COORDINATE PLANE

Three common rotation angles used The origin is the

on a coordinate plane are 90˚, 180˚, point (0, 0). It’s
and 270˚. There are rules that we where the x-axis
and y-axis meet.
can use for these rotations about

the origin.

257

ANGLE OF RULE EXAMPLE
ROTATION
counterclockwise
about the origin

90˚ (x, y) ➜ (-y, x)

Multiply the
y-coordinate
by -1, and
reverse the
order of the
coordinates.

180˚ (x, y) ➜ (-x, -y)

Multiply
the x- and
y-coordinates
by -1.

270˚ (x, y) ➜ (y, -x)
Multiply the
258 x-coordinate
by -1, and
reverse the
order of the
coordinates.

EXAMPLE: Rotate the

triangle 180˚ about the origin.

First rotate each point 180˚

about the origin:

Rule: (x, y) ➜ (-x, -y) Multiply the x and y
D(-6, 4) ➜ D’ (6, -4) coordinates by -1.
E(1, 2) ➜ E’ (-1, -2)
F(-3, -1) ➜ F’ (3, 1)

Next plot the new points.

Then connect all the
points.

When the rotation is 180˚, it doesn’t matter if

the direction is clockwise or counterclockwise,
because the image will end up in the same place.

259

FINDING THE CENTER
OF ROTATION

Steps for finding the center of rotation:

1. Draw a line to connect A and A’.

2. Construct a perpendicular bisector through AA’.

260

3. Repeat steps 1 and 2 on points B and B’.

The intersection of the two perpendicular bisectors is
the center of rotation.

If we draw the perpendicular bisector of CC’, it will also

pass through the point of rotation.

261

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1. True or False: In a rotation, the shape, size, and

orientation of a figure remain the same.

2. Rotate the triangle 90˚ counterclockwise about the

center of rotation, R.

3. Use a protractor to draw the rotation of GHI 60˚

counterclockwise about point R.

4. Draw the center of rotation that rotates ABC to A’ B’ C’.

262

Use the graphed line below to answer questions 5 and 6:

5. RS is rotated 180˚ counterclockwise about the origin.

What are the coordinates of R’ and S’ ?

6. Draw the rotation of RS 270˚ counterclockwise about

the origin.

answers 263

1. False. The size and shape remain the same, but the
orientation is rotated.

2. 3.

4. 5. R’ (-1, 3), S’ (-2, -2)

6.

264

Chapter 23

COMPOSITIONS

COMPOSITIONS OF
TRANSFORMATIONS

COMPOSITIONS OF TRANSFORMATIONS combine two

or more transformations to form a new transformation.
In a composition, you perform each transformation on the
image from the previous transformation.
Example of a composition of

transformation: The green

fish is reflected to the pink
fish, and then translated to
the yellow fish.

265

EXAMPLE: Draw the

graph of JK rotated 90˚

counterclockwise about the

origin, and then reflected

across line y = 1.

Step 1: Rotate JK 90˚

counterclockwise about the origin.

Use the rule (x, y) ➜ (-y, x)
to find the endpoints.

• J(3, 6) ➜ J’ (-6, 3)
• K(7, 2) ➜ K’ (-2, 7)

Connect the endpoints. Label the points with double
prime ('' ) when an image is
Step 2: Reflect J'K' across line y = 1. reflected a second time.

J’ is 2 units above y = 1, so
place J’’ 2 units below y = 1.

K’ is 6 units above y = 1, so
place K’’ 6 units below y = 1.

Connect the endpoints.

266

COMPOSITIONS OF
TRANSLATIONS

A COMPOSITION OF TRANSLATIONS combines two or

more translations.

Example of a composition

of translations: The yellow

dog is translated to the pink
dog, and then translated to
the blue dog.

In this image:

A is translated to B.
B is translated to C.

The composition of two translations
is another translation.

267

EXAMPLE: Describe the

transformation from DE to

D’’E’’.

DE is translated along
vector (-4, -1) to D'E'.

D'E' is translated along

vector (4, 3) to D’’E’’.

This is a composition of two translations, so the result is
a translation.

To find the translation vector, we can either:

• Count the units from D to D’’ (or E to E’’ ):

D moves 0 units left/right and 2 units up

to map to D’’ (0, 2).

OR

• Add the coordinates of translation vectors (-4, -1) and (4, 3):

(-4 + 4, -1 + 3) = (0, 2).

The transformation from DE to D’’E’’ is a translation along

vector (0, 2).

268

GLIDE REFLECTIONS

A GLIDE REFLECTION is a translation followed by a

reflection. The reflection line is parallel to the direction of
the translation.

Example of glide reflection:

The gray cat translates to
the purple cat, then reflects
to the green cat.

EXAMPLE: Draw the glide

reflection where ABC is
translated along vector (-4, 0) and
then reflected across the x-axis.

Translate along vector (-4, 0):

A(1, -2) ➜ A’ (-3, -2)

B(2,-1) ➜ B’ (-2, -1)

C(4, -3) ➜ C’ (0, -3)

Plot the points and
connect the vertices.

269

Reflect A’ B’ C’ across
the x-axis:

A’ (-3, -2) ➜ A’’ (-3, 2)

B’ (-2, -1) ➜ B’’ (-2, 1)

C’ (0, -3) ➜ C’’ (0, 3)

Plot the points and
connect the vertices.

KEY PROPERTIES OF RIGID MOTIONS

Rigid Size stays Angle measure Orientation
Motion the same? stays the same? stays the same?

Reflection Yes Yes No

Translation Yes Yes Yes

Rotation Yes Yes No
Yes Yes No
Glide
reflection

270

COMPOSITIONS OF
REFLECTIONS

Compositions of reflections have different rules, depending
on whether the lines of reflection are parallel or intersect.

Parallel Intersect
A composition of two A composition of two
reflections across two reflections across two
parallel lines forms a intersecting lines forms a
translation. rotation about the point of
intersection.

271

EXAMPLE: Reflect LMN

across the y-axis and then
across x = 5. What is the
single transformation that

maps LMN to L’’ M’’ N’’ ?

Reflecting across the y-axis

gives L’ M’ N’ .

Reflecting L’ M’ N’ across
x = 5 gives L’’ M’’ N’’ .

LMN moves 10 units to the

right to L’’ M’’ N’’ . Therefore,

a translation along vector

(10, 0) maps LMN to L’’ M’’ N’’ .

272

EXAMPLE: Reflect

quadrilateral PQRS across

line l and then m. What’s

the single transformation
that maps PQRS to

P’’ Q’’ R’’ S’’ ?

Reflecting PQRS across

line l gives P’ Q’ R’ S’ .

Reflecting P’ Q’ R’ S’ across
line m gives P’’ Q’’ R’’ S’’ .

The result is a rotation
around point T.

To find the angle of rotation,
draw a line from S to T

and S’’ to T. The angle

between these two lines

is 180˚.

The transformation is
a rotation with center of
rotation T and angle of

rotation 180˚.

273

SYMMETRY

If a figure is reflected across a line and the new figure

is unchanged, then the figure has LINE SYMMETRY .
The line of reflection is called the LINE OF SYMMETRY .

A line of symmetry divides a figure into two mirror images.

line of
symmetry

Sometimes a figure can have more than one line of
symmetry.

There are six different lines along which you can reflect
the figure of the flower, and it will still look the same.

274

If a figure is rotated between 0˚ and 360˚ about

its center and the figure remains the same, then it has

ROTATIONAL SYMMETRY . The point of rotation is
called the CENTER OF ROTATION .

This figure has rotational
symmetry because it
still looks the same after

a rotation of 180˚, which is
less than one full turn (360˚).

This figure has rotational symmetry because when rotated

90˚, 180˚, or 270˚, it still looks the same. It maps to itself.

275

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1. Graph the composition of point P(4, 1), rotated 270˚

counterclockwise about the origin to P’, and then
reflected across y = -1 to P’’.

2. True or False: If figure A

is translated to figure B

and figure B is translated

to figure C, then figure A

to figure C is a translation.

3. GHI is translated along vector (7, -3) to G’ H’ I’.
G’ H’ I’ is translated along vector (-2, 13) to G’’ H’’ I’’.

Describe the transformation from GHI to G’’ H’’ I’’.

4. What is the composition of two reflections across two
parallel lines?

5. Complete the sentence.

A composition of two reflections across two lines

forms a rotation about the point of .

276

6. Graph the reflection of PQR across l and then m.

Describe the single transformation that maps PQR to

P’’ Q’’ R’’.

7. Does this figure have line symmetry? If so, how many
lines of symmetry does it have?

answers 277

1.

2. True

3. Translation along vector (5, 10) (Hint: Add the
4. A translation coordinates of
translation vectors.)

5. intersecting, intersection

6. A rotation with center of rotation O and angle of
rotation 270˚ counterclockwise (or 90˚ clockwise).

7. Yes, 5 lines of symmetry

278

Chapter 24

CONGRUENCE

Two figures are CONGRUENT if there is a sequence of

rigid motions that maps one figure directly onto the other.

Not a rigid motion A rigid motion
(Side lengths get larger) (Ref lection)
➜ Congruent
➜ Not Congruent

A rigid motion Not a rigid motion
(Tr a n s l a ti o n) (Angles and side
➜ Congruent lengths change size)
➜ Not Congruent

279

EXAMPLE: Is ABC

congruent to DEF?

If there is a rigid motion that
takes ABC to DEF, then the
triangles are congruent.

A rotation of 270˚ counterclockwise maps ABC to DEF.

Therefore, ABC is congruent to DEF.

ABC ≅ DEF

Order is important when writing congruence statements.

ABC ≅ DEF means that A maps to D, B maps to E, and C

maps to F.

ABC ≅ EFD is incorrect because A does not map to E.

Equivalent to ABC ≅ DEF:

ACB ≅ DFE DEF ≅ ABC
BAC ≅ EDF EDF ≅ BAC

280

EXAMPLE: Determine

whether the two figures in
the graph are congruent.
If they are, write a
congruence statement.

JKLM maps to NOPQ by a
reflection across the y-axis,
followed by a translation
along vector.

Because a sequence of rigid
motions maps JKLM to NOPQ,
the figures are congruent.

The congruence statement is JKLM ≅ NOPQ.

EXAMPLE: Determine if XYZ is congruent to GHI.

If you trace XYZ, and rotate,
reflect, and/or translate it, you’ll
see it is not possible to map
to GHI. Because there is no
sequence of rigid motions that
maps XYZ to GHI, these
triangles are not congruent.

281

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1. True or False: Two figures are congruent if there is a

sequence of transformations that maps one figure onto
another.
For questions 2 and 3, determine if the following images
are congruent.
2. 3.

4. Write a congruence statement for the following
congruent figures.

282

For questions 5-7, determine if the figures are congruent.
If they are, write a congruence statement.

5.

6.

7.

answers 283

1. False. Two figures are congruent if there is
a sequence of rigid motions that maps one
figure onto another. (Not all transformations
are rigid motions.)

2. Yes. There is a rigid motion (translation) that maps one
figure onto the other.

3. No. There is not a sequence of rigid motions that maps
one line segment onto the other.

4. ABC ≅ FED (or ACB ≅ FDE, BAC ≅ EFD,
BCA ≅ EDF, CAB ≅ DFE)

5. Yes, GH ≅ IJ or HG ≅ JI

6. Yes, PQRS ≅ UVWT (or QRSP ≅ VWTU, RSPQ ≅ WTUV,
SPQR ≅ TUVW, SRQP ≅ TWVU, RQPS ≅ WVUT,
QPSR ≅ VUTW, PSRQ ≅ UTWV)

7. No. There is not a sequence of rigid motions that maps
one figure onto the other.

284

Unit

6

Similarity

285

Chapter 25

RATIO AND
PROPORTION

RATIO

A RATIO is a comparison of two or more quantities. It can

be written in different ways.

When comparing a to b, we can write:

a to b or a:b or a
b

a represents the first quantity.
b represents the second quantity.

The ratio 4 to 8 can be writ ten 4 to 8 or 4:8 or 4 .
8

286