G E O M E T R Y
CHAPTER 7
TRANSFORMATIONS
Notes & Study Guide
2
TABLE OF CONTENTS
RIGID MOTION IN A PLANE ............................................................................... 3
REFLECTIONS .................................................................................................... 5
REFLECTIONS IN THE COORDINATE PLANE ................................................. 6
ROTATIONS ........................................................................................................ 7
ROTATIONS IN THE COORDINATE PLANE ..................................................... 8
TRANSLATIONS ................................................................................................. 9
TRANSLATIONS IN THE COORDINATE PLANE ............................................ 10
SUMMARY OF RIGID TRANSFORMATIONS .................................................. 11
VECTORS .......................................................................................................... 12
GLIDE REFLECTIONS AND COMPOSITIONS ................................................ 13
HOMEWORK EXAMPLES ................................................................................ 14
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RIGID MOTION IN A PLANE
3
INTRODUCTION
In Chapter 7, we will examine a different topic than previous chapters. Here we will look
at the behaviors of figures when they are moved around.
 TRANSFORMATIONS
Any action that is done to a figure that results in a new
figure is called a transformation.
Shown: the blue truck has been flipped over to create a
new truck (red). The flip is a simple example of a
transformation.
If the action DOES NOT change the figure, the action is called RIGID.
If the action DOES change the figure, the action is called NON-RIGID.
In this chapter, we will focus our attention on the rigid transformations.
 TYPES OF RIGID TRANSFORMATIONS
There are three types of rigid transformations. Each of them can be done to a
figure, but they will not change the figure’s shape or size.
Reflection  flipping a figure over (using a reference line)
Rotation  turning a figure around (using a fixed reference point)
Translation  sliding a figure in any direction (without turning it)
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RIGID MOTION IN A PLANE
All transformations work the same way: Some original image is transformed
(somehow) and a new image, or copy, is created.
It is important to know which is which so the transformation can be interpreted
correctly.
 PRE-IMAGE vs. IMAGE
pre-image: the starting figure before the transformation
happens. (the original)
image: the resulting figure after the transformation
happens (the copy)
Shown: the pre-image (blue) is flipped to the right creating the image (red)
The colors help us know which direction the flipping occurred.
 LABELING
If different colors aren’t an option, then labeling the vertices of the figures is how
you tell the difference. (most common anyway)
Labels on pre-images will be normal letters
(like A, B, C, etc.)
Labels on images will have apostrophes on
them (like A’, B’, C’, etc.)
The labels A’, B’ and C’ are read as “A prime”, “B prime” and “C prime”.
Verbally you say that “ABC maps onto A’B’C’.”
Symbol notation: you can write is as ABC  A’B’C’
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REFLECTIONS
5
 REFLECTIONS
When a pre-image is flipped over an imaginary
reference line it is said to be a reflection.
The line that is used for the reflection is called a
mirror line or a line of reflection.
Sometimes an arrow is drawn across the mirror line to help show the direction of
the reflection. This helps identify the original image (if no labels or colors).
IMPORTANT! Not only is the figure itself reflected, but so is the empty space
surrounding it.
 REFLECTIONAL SYMMETRY
If a figure can be folded so that the first part matches perfectly to
the other, then that figure has reflectional symmetry.
Shown: the red line (line of symmetry) shows where you’d “fold” this
figure so that it matches perfectly.
Some figures have one line of symmetry, some many and some have none.
How many lines of symmetry do these figures have?
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REFLECTIONS IN A COORDINATE PLANE
When a figure is floating in space, it can sometimes be tough to draw a
reflection.
But when it’s on a grid (coordinate plane) it becomes very simple. All you need to
do is look for a pattern in the coordinates.
Shown: AB is the pre-image
If the y-axis is the mirror line…
A(–3, 1) will reflect to C(___, ___)
B(–1, 3) will reflect to D(___, ___)
If the x-axis is the mirror line…
A(–3, 1) will reflect to G(___, ___)
B(–1, 3) will reflect to H(___, ___)
What patterns do you see the in the coordinates when a reflection happens?
x-axis reflection  the sign of the y-coordinates switch
y-axis reflection  the sign of the x-coordinates switch
IMPORTANT! This only works if an axis is the mirror line!
 TRANSFORMATION NOTATION
We have a notation for this to help you remember what to do when reflecting on
a grid. It’s called transformation notation.
T(x, y)  (–x, y) : means change the sign on the x’s, but leave the y’s alone.
T(x, y)  (x, –y) : means change the sign on the y’s, but leave the x’s alone.
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ROTATIONS
7
When a pre-image is turned over a fixed reference
point, it is said to be a rotation.
The point that is used for the rotation is called a
center of rotation.
An arrow is drawn across the rotation to help identify which image is the original
(if no labels or colors).
 AMOUNT OF ROTATION
How much rotation occurs is found by measuring a reference angle (shown in
black) using a protractor.
The angle is created by drawing 2 reference lines from the center of rotation to
the same spot on each figure.
 ROTATIONAL SYMMETRY
If a figure can be turned so that the resulting figure looks like nothing was done
to it, then that figure has rotational symmetry.
The square above can be rotated onto a corner (45°) and it looks different. But rotate it
more (to 90°) and it looks like nothing happened…that’s rotational symmetry.
Rotational symmetry is identified by how many degrees of rotation it takes for it
to occur. Some figures have 1 example, some more, some none.
Normally, 0° doesn’t count and your rotation limit is 360°.
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ROTATIONS IN A COORDINATE PLANE
When a figure is floating in space, it is usually tough to draw a rotation.
But when it’s on a grid (coordinate plane), it becomes simpler. What helps us out
is that there is a pattern in the coordinates.
Shown: AB is the pre-image and we rotate using the origin.
Rotating AB 90° clockwise (CW)…
A(–3, 1) will rotate to D(___, ___)
B(–1, 3) will rotate to C(___, ___)
Rotating AB a little more (180° CW)…
A(–3, 1) will rotate to E(___, ___)
B(–1, 3) will rotate to F(___, ___)
Rotating AB a little more (270° CW)…
A(–3, 1) will rotate to H(___, ___)
B(–1, 3) will rotate to G(___, ___)
What pattern do you see in the coordinates when the rotations happen?
 The coordinates are switched and 1 sign changes OR
 Both signs are changed
IMPORTANT! This only works if the origin is the center or rotation!
 TRANSFORMATION NOTATION
T(x, y)  (–y, x) : switch & change the y
T(x, y)  (–x, –y) : change both signs
T(x, y)  (y, –x) : switch & change the x
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90° CW
180° CW
270° CW
TRANSLATIONS
9
When a pre-image is slid in any direction, without being
turned or flipped, it is said to be a translation.
This is the simplest of all rigid transformations.
An arrow is drawn across the translation to help identify which image is the
original (if no labels or colors).
Translations do not require any fixed reference lines or points.
 PROPERTIES OF TRANSLATIONS
One way to illustrate translations is to draw segments (guidelines) connecting the
endpoints or vertices.
In a translation, these guidelines will be…
(1) congruent
(2) parallel
Translations can be applied in any direction.
The key is to never turn or flip the pre-image while you translate it.
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TRANSLATIONS IN THE COORDINATE PLANE
When a figure is floating in space, it is simple to draw a translation.
When it’s on a grid (coordinate plane), it’s even easier. All you need to do is look
for the pattern in the coordinates.
Shown: figure PQRS to be moved 3 space in each direction
Translation up…
P(–1, 1) will move to P’(___, ___)
Q(2, 4) will move to Q’(___, ___)
Translation down…
R(6, 3) will move to R’(___, ___)
S(3, –1) will move to S’(___, ___)
Translation left…
P(–1, 1) will move to P’(___, ___)
Q(2, 4) will move to Q’(___, ___)
Translation right…
R(6, 3) will move to R’(___, ___)
S(3, –1) will move to S’(___, ___)
What pattern do you see in the coordinates when the translations happen?
 The amount of the translation is either added or subtracted to one of the
coordinates
 TRANSLATION NOTATION
T(x, y)  (x ± a, y) : add or subtract to the x’s
T(x, y)  (x, y ± b) : add or subtract to the y’s
T(x, y)  (x ± a, y ± b) : add or subtract to both
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left or right
up or down
diagonal
SUMMARY OF TRANSFORMATIONS
11
TYPE
NOTATION
EXTRA
Reflection
y–axis: T(x, y)  (–x, y)
x–axis: T(x, y)  (x, –y)
Reflections use sign
changes only!
Rotation
90° CW: T(x, y)  (y, –x)
180° CW: T(x, y)  (–x, –y)
270° CW: T(x, y)  (–y, x)
180° rot. = double reflection
90° CW = 270° CCW
Translation
L/R: T(x, y)  (x ± a, y)
U/D: T(x, y)  (x, y ± b)
Both: T(x, y)  (x ± a, y ± b)
Translations use addition
and subtraction only!
Remember, when describing a transformation you must include specific
information…
Type of transformation (translation, rotation, reflection)
Direction (up, down, left, right, CW, CCW)
Amount (number of spaces, number of degrees, which axis)
EXAMPLES
A translation 3 spaces up and 2 spaces right
A rotation 45° counter-clockwise
A reflection in the y-axis
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VECTORS
Another way to describe a translation is by using
a vector.
A vector is an arrow whose length (magnitude)
and direction matches that of the actual
translation.
The point where the vector start is the initial point (point P).
The point where the vector nds is the terminal point (point Q).
Most vectors have 2 parts (or components); one horizontal and one vertical.
Combining the two gives you a vector in any direction.
 COMPONENT (VECTOR) NOTATION
Component notation is a shortcut notation for vectors. Each number in the
notation represents the horizontal and vertical components.
Vector PQ (shown above) has the component notation <5, 3>.
To apply a vector to a pre-image, simply add the numbers from the component
notation to the x and y coordinates from the pre-image.
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GLIDE REFLECTIONS & COMPOSITIONS
13
There’s no rule that says you can only apply one transformation to a figure.
There’s no limit to how many you can do.
When two or more transformations are
applied to figure that is called a
composition.
Shown: ABCD (blue) is reflected first (red
figure) and then rotated (green figure).
This is a 2-step composition.
Clearly, there are a wide variety of compositions that you can do. There are
many combinations that can be created.
Glide reflection – a composition of
transformations that is specifically
made up of a translation (glide) and
then a reflection.
(must be in that order!)
For the most part, no other composition gets its own name.
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14
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HOMEWORK EXAMPLES