TRANSFORMATIONS
Date
10/8
Lesson
1
Topic
Introduction to Transformations
HMWK
HW 3-1
10/9
2
Isometries (Rigid Motions) in the Coordinate Plane
HW 3-2
10/10
3
Transformations and Symmetry
Half Day
HW 3-3
10/13
----
No School
------
10/14
4
Examining Translations, Rotations and Reflections
HW 3-4
10/15
5
Properties of Isometries
HW 3-5
10/16
6
Point Reflections
HW 3-6
Quiz
10/17
7
Compositions of Transformations
HW 3-7
10/20
8
Isometries and Congruence
HW 3-8
10/21
9
Review
STUDY!
10/23
10
Test
1
Day 1 Notes Goal: Connect student knowledge about functions (input/output process)
and equations to transformations and coordinate rules in Geometry.
Recall from Algebra: What is a function?
______________________________________________________________________________
Types of Functions:
How do we represent them?
Finding the input/output of a function
𝑓(𝑥) = 𝑥 2 − 3
𝑓(𝑥) = 𝑥 + 7
𝑓(𝑥) = 2𝑥 + 3
What is the difference between the first 2 and the last example above in terms
of the input and output?
Connecting Geometry and Transformations to Functions
Objects and shapes in the coordinate plane can be moved according to a specific
coordinate rule. These rules can alter the shape in many different ways that we will
talk about in this unit.
All transformations are rules that take points in the plane as inputs and assign
other points as outputs.
With transformations, the input and output are also known as the
________________________and the __________________________. We denote the
pre-image as a capital letter, A for example, and the image as A’ which is read
as A prime.
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Fundamental Transformations : There are two categories of Transformations: Rigid
Motions (Isometries) and Non-Rigid Motions (Non-Isometries)
NON-RIGID MOTIONS:
A NON-ISOMETRIC TRANSFORMATION (NON-RIGID MOTION) is a
transformation that does not necessarily preserve the distances and/or
angles between the pre-image and image.
(In other words, the shape is _____________________ in some way)
There are many ways to distort a shape but these are the most popular:
Dilation – Where both dimension’s
scale factors is the _____________. The
shape is proportional, not identical. A
dilation changes the size of the shape
making it a NON-ISOMETRIC
transformation.
Stretch – Where one dimension’s scale
factor is ________________ than the
other dimension’s scale factor . A
stretch definitely __________________ the
shape making it a NON-ISOMETRIC
transformation.
Label each of the following examples as a stretch of the pre-image or a dilation.
RIGID MOTIONS:
An ISOMETRIC TRANSFORMATION (RIGID MOTION) is a transformation that
preserves distance and angle measure.
There are many examples of this:
In this unit, we are going to focus
on RIGID MOTIONS
(or isometries)
1) ______________________________
2) ______________________________
3) ______________________________
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Translation: _______________ or pushes an object while
maintaining the size.
This is a shifting of an object in at least one of two directions
left/right (_____________) and up/down (_____________)
For example, a “rule” for a translation could look like this:
Example 1: Given the points A(-1,5) and B(3,-2) find the coordinates of A’ and
B’ under the translation given above – sometimes written as T3, -6.
Sometimes, you are given a _____________________ to tell you how to move a figure
Example 2: Translate the quadrilateral
along the given vector.
To use a translation vector:
Step 1: Draw a line parallel to the vector
through each vertex of the quadrilateral.
Step 2: Measure the length of the vector
and mark off that distance from each
vertex on each of the parallel lines.
Step 3: Connect the vertices of the image.
Translations (T) change the inputs to outputs using addition or subtraction and the
size is always preserved. A transformation that preserves size is an _________________
or_____________ _______________.
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Since translations are _____________, the image and pre-image are always ___________.
Example 2: Are the transformations below translations?
A)
B)
C)
D)
Example 3: Complete the boxes below
Notes 2: Transformations in the coordinate plane.
Warm Up:
Define rigid motion: _______________________________________________________________
____________________________________________________________________________________
Another term for a geometric rigid motion is _____________________
1. Given the translation T-5, -1, find the
image of A(-1, 5)
2. Given the rule (x, y)  (x – 2, y + 1),
determine the pre-image of A`(-4, 11)
5
Reflections:
Reflections are a flip of a figure over some line – called the “line of reflection.” Since
reflections are also ________________, the image and pre-image are always congruent.
They are denoted with a lower case r with the line of reflection listed as a subscript.
(For example ry=x is a reflection in the line y=x)
Example 1: Tell whether each transformation appears to be a reflection. If so, draw
in the line of reflection.
“Rules of reflection”
In the coordinate plane, there is a pattern for the changes
to a pre-image for reflections over certain lines.
Try some example points to come up with the rules for the
reflections listed below.
(x,y) reflected over x-axis
(
,
)
(x,y) reflected over y-axis
(
,
)
(x,y) reflected over the line y = x
(
,
)
Example 2: Find the indicated image.
1. r
x-axis
(5,7)___________
2. r
y=x
(-1,0)___________ 3. r
Example 3: Reflect the rectangle with vertices S(3,4),
T(3,1), U(-2,1) and V(-2,4) across the x-axis.
Graph both the pre-image and the image. Be sure to list
the new coordinates below.
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y-axis
(5,7)
Rotations:
Rotations are the turning of a figure around a fixed point. Since rotations are
___________________, the image and pre-image are always congruent. They are
denoted with a capital R with the degree of rotation listed as a subscript. (For
example R90 would be a rotation of 90 degrees)
Example 4: Tell whether each transformation appears to be a rotation. If so, draw a
dot on the point of rotation.
Rotation on the coordinate plane:
Unless otherwise stated, all rotations are always ______________ - _________________
“about” the ______________.
Example 5:
Which way would you go
for a rotation of 80o?
Which way would you go for a
rotation of -50o?
Rotation of 90o about the origin.
Rotation of 180o about the origin.
7
Rotate ∆JKL with vertices J(2,2), K(4,-5)
and L(-1,6) by 180o about the origin.
Rotate ΔABC with vertices A(2,-1),
B(4,1) and C(3,3) by -90o about the
origin.
*Trick: Rotate the graph on your paper by the appropriate amount, and “replot” the
original points on the “new” coordinate axes.
Notes 3 - Given a shape describe the rotations and reflections that carry it onto itself.
Symmetry
A figure has symmetry if there is a transformation of the figure such that the
__________________ coincides with the ____________________.
Line Symmetry
A figure has ____________________________ (or reflection symmetry) if it can be
reflected across a line so that the image coincides with the preimage.
The ________________________ (also called the axis of symmetry) divides the
figure into two congruent halves.
Example 1: Tell whether each figure has line symmetry. If so, draw all lines of
symmetry.
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Rotational Symmetry
A figure has Rotational Symmetry (or radial symmetry) if it can be rotated about a
point by an angle greater than _______ and less than _______ so that the image
coincides with the pre-image.
The _______________ of rotational symmetry is the smallest angle through which
a figure can be rotated to coincide with itself.
The number of times an image coincides with the _____________________ as it
rotates through the 360o is called the _____________ of the rotational symmetry.
Example 2: Tell whether each figure has rotational symmetry. If so, give the angle of
rotational symmetry and the order of the symmetry.
Try these! Fill in the chart below.
Figure
Line Symmetry
Diagram
Number of
Lines of
Symmetry
Parallelogram
Rectangle
Trapezoid
Regular
Polygon
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Degree of
Rotational
Symmetry
Order of
Rotational
symmetry
Notes 4: Examining Translations, Rotations, and Reflections
Warm Up:
Perform the following rotations. Then, use your results
to come up with a “rule” for the rotations listed.
1. R90 (3, 1) = ___________
R90 (-2, 4) = ___________
2. R180 (3, 1) = ___________
R180 (-2, 4) = ___________
3. R270 (3, 1) = ___________
R270 (-2, 4) = ___________
Rules:
R90 (x, y) = ___________
R180 (x, y) = ___________ R270 (x, y) = ___________
Let’s Review Rotations
Remember: A rotation is an isometric transformation that _____________ a figure
about a fixed point called the center of rotation. Rays drawn from the center of
rotation to a point and its image form an angle called the angle of rotation (notation
Rcenter, degree).
Examine the rotation below:
What can you say about the segments from the center of rotation to a preimage point and to an image point?
Do all points of the pre-image move in a rotation?
If points don’t move in a transformation, they are called ________________________
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Let’s Review Translations
Remember: A translation ___________________ an object a fixed distance in a given
direction. The direction and distance of a translation can be show in two ways:
1. With ________________________ on a coordinate plane.
For example Tx,y or (x, y) (x, y)
2. With a vector in a non-coordinate plane
Examine the Translation below. What can you say about the segments that would
join the points of the pre-image with the points of the image?
Are there any invariant points in a translation?
Let’s Review Reflections
Remember: A reflection is a ___________ of an figure over a given line (notation Rm).
Examine the reflection below. What can you say about the
segments joining the pre-image and the image?
Are there any invariant points in a reflection?
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Let’s Practice:
Example 1: Reflect the rectangle with vertices S(3,4),
T(3,1), U(-2,1) and V(-2,4) across the y-axis.
Example 2: Rotate ΔABC with vertices A(2,-1), B(4,1) and
C(3,3) by 90o about the origin.
Example 3: Transform ∆ABC with vertices A(–1, 7),
S(2, 4), and T(3, 6) using the transformation T-3, -4
What type of transformation(s) has no invariant points? _____________________________
In which transformation(s) is AA` = BB`? _____________________________
In which transformation(s) is AA` || BB`? _____________________________
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Notes 5: Properties of Isometries
Warm Up:
1. Name a transformation
where the segments
joining the pre-image
with the image are all
bisected by the same
line. Draw a picture to
support your answer.
2. Name a transformation
where the distance
between A and A` would
be different from B and
B`. Draw a picture to
support your answer.
3. Name a transformation
that could have
invariant points. Draw
a picture to support
your answer.
________________________
________________________
________________________
An Isometry is a transformation that _______________________________________________

Translations, Reflections and Rotations are all Isometric Transformations and
therefore have certain properties that are preserved:
o Distance (lengths of segments are the same)
o Angle measure (angles stay the same)
o Parallelism (things that were parallel are still parallel)
o Collinearity (points on a line, remain on the line)
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Let’s look at some properties that might not be consistent between all of them:
Translations
Rotations
Reflections
Distances
between the preimage points and
image points
Distances are
___________________
depending on how
close they were to the
center of rotation
Distances are
___________________
depending on how
close points were to
the line of reflection
Order of lettering in
pre-image:
Order of lettering in
pre-image:
Order of lettering in
pre-image:
Order of lettering in
image:
Order of lettering in
image:
Order of lettering in
image:
Orientation is
___________________
(______________)
Orientation is
___________________
(______________)
Orientation is
___________________
(______________)
Distances are
____________________
Orientation
(Look at the order
of the lettering in
the figure and the
image – clockwise
vs. counterclockwise)
Special points
(points that
remain invariant)
1. Which of the following properties is not
always preserved under a line
reflection?
A) Distance
B) Slope
C) Angle Measure
D) Collinearity
2. Which of the following does not
preserve angle measure?
A) Translation
B) Rotation
C) Dilation
D) None of the above
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3. Which of the following is preserved for a
rotation?
A) Betweenness and slope
B) Parallelism and angle measure
C) Angle measure and slope
D) None of the above
5. A transformation of a polygon that
always preserves both length and
orientation is
A) dilation
B) translation
C) line reflection
D) glide reflection
6. The vertices of parallelogram ABCD are
,
,
, and
. If ABCD
is reflected over the x-axis, how many
vertices remain invariant?
A) 1
B) 2
C) 3
D) 0
4. Which of the following compositions
preserve slope?
A)
B)
C)
D)
Translations only
Translation and glide reflections
Rotations and translations
Rotations and reflections
Notes 6: Properties of Isometries
Warm Up: Study for Quiz
Point Reflections
Another kind of reflection involves a point. In the figure below,
A’B’C’ is the image of ABC after a reflection in point. P. If a line
segment is drawn connecting any point to its image, then the
point of reflection is the midpoint of that segment.
To reflect ABC through point P, follow these steps:
1. For each vertex of ABC, draw a segment through P
such that P is the midpoint of the segment.
2. Connect A’, B’ and C’ to form the image A’B’C’.
Ex 1: Reflect ABC through point P.
A
P
15
B
C
Ex 2: Find the image of the point (-3, -2) after it is
reflected in the origin.
From this, we can generate a rule for reflections in
the origin:
rorigin (x,y)  (_____,______)
** This rule should look familiar! **
A Point reflection in the origin is the same as _____________________________________
Notes 7 - Compositions of Transformations-Performing more than one
transformation to the same figure.
Warm Up: Fill in the following chart with the coordinates of the point (x, y) after the given
transformation. The use of the grid to the right is optional.
Ro,90(x, y)=
rx-axis(x, y)=
Ro,180(x, y)=
ry-axis(x, y)=
Ro, 270(x, y)=
ry=x(x, y)=
Ta,b(x, y)=
rorigin(x, y)=
A Composition of Transformations is _______ transformation followed by
______________ transformation.
A common composition of transformations is the ______________ ____________________;
consisting of a translation and a reflection across a line ___________ to the translation vector.
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Example: Draw the result of the composition of isometries:
Reflect PQRS across line M
and then translate it along
vector v.
Step 1: Draw P’Q’R’S’, the
reflection of PQRS.
Step 2: Translate
P’Q’R’S’ along
vector v
The __________ of each transformation is congruent to the previous __________.
Theroem: A composition of two isometries is an ____________________
Example 1: ΔKLM has vertices K(4,-1), L(5,-2) and M(1,4). Rotate ΔKLM 180o about the origin and then reflect
it across the y-axis.
What single transformation would map the preimage to the final image?
Example 2: ΔJKL has vertices J(1,-2), K(4,-2) and L(3,0).
Reflect ΔJKL across the x-axis and then rotate it 270o
about the origin.
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Theorem: The composition of two reflections across two parallel lines is equivalent
to a translation.
 The translation vector is perpendicular to the lines.
 The length of the translation vector is twice the
distance between the lines.
Theorem: The composition of two reflections across two intersecting lines is
equivalent to a rotation.
 The center of rotation is the intersection of the lines.
 The angle of rotation is twice the measure of the angle
formed by the lines.
Theroem: Any translation or rotation is equivalent to a
composition of two reflections.
Recap:
1) The composition of 2 reflections could be equivalent to a
__________________ or a ____________________
2) If the composition is of 2 reflections across 2 parallel lines, it is
equivalent to a ___________________
3) If the composition is of 2 reflections across 2 intersecting lines, it
is equivalent to a __________________
Notes 8 - Isometries and Congruence
Warm Up: PQR has coordinates P(1,1), Q(1,3) and R(3,4).
a) Draw and label PQR.
b) Graph and state the coordinates of P’Q’R’, the image
of PQR after a reflection in the y-axis.
c) Graph and state the coordinates of P’’Q’’R’’, the image
of P’Q’R’, after a translation which maps (x,y) to (x-4,
y+5).
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Two figures are congruent if and only if one can be mapped onto the other by one or
more rigid motions.
Translations, reflections and rotation are rigid motions and any sequence of
these from one figure onto another would prove congruence.
Examples
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PRACTICE - With a partner, create a preimage
and an image. The image needs to be congruent
to the preimage and located by doing a
composition of isometries. Exchange papers with
a neighbor and try to determine what isometries
were done to transform the preimage to the
image. Clearly show/explain how you know
what isometries were used.
ABC has coordinates A(-5,2), B(-3,6) and C(0,0).
a) Draw and label ABC below.
b) Graph and state the coordinates of A’B’C’,
the image of ABC after a reflection in the
origin (Rotation of 180)
c) Graph and state the coordinates of
A’’B’’C’’, the image of ABC under a
reflection in the y-axis.
What single transformations is this equivalent to?
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