Geometry
Name_________________________
Transformations II: Exploration of Isometric Transformations
3/21 or 3/22
Isometric Transformations & Sketchpad:
Let’s explore transformations on Sketchpad, which allows for rotations, translations, and
reflections to be dynamic!
Exploration 1: Rotations
1. Draw a scalene figure (not to exceed 4-6 sides for practical reasons) and label vertices A,B,C etc
(try to make each segment a distinct size)
2. Draw a point outside the shape
3. With the point still highlighted, go to the TRANSFORM menu and MARK CENTER or simply
double-click this point (you should see it “light-up”)
4. Highlight shape (click and drag over entire shape)
5. Go to TRANSFORM menu and ROTATE (set angle as you want it to be) – to show labels for
vertices of this image, highlight only the vertices, go to DISPLAY menu and SHOW LABELS
(these labels should A’, B’, C’, etc)
6. Rotate this new image by a different angle and show labels (these labels should A’’, B’’, C’’, etc)
TASK: State, in words, what specific single transformation would take A to A”?
Exploration 2: Translations
1. Draw a shape and label vertices A,B,C etc
2. Draw a line segment outside the shape with endpoints labeled PQ.
3. Highlight P then Q, go to TRANSFORM menu and MARK VECTOR
4. Highlight shape
5. Go to TRANSFORM menu and TRANSLATE. Make sure vertices are labeled A’B’C’
6. Translate this new image by a different vector, new image will be A”B”C” etc
TASK: State, in words, what specific single transformation would take A to A”?
Geometry
Exploration 3a: Reflections (over parallel lines)
1. Draw a shape.
2. Draw two parallel lines somewhere in the plane.
3. Reflect your shape twice, once over each line.
TASK: A single translation maps to the same image. What is the translation – what pattern do
you notice? Can you establish a “rule” for accomplishing a translation of c units using
reflections over parallel lines?
Exploration 3b: Reflections over intersecting lines
1. Draw a shape.
2. Draw two intersecting lines somewhere in the plane.
3. Reflect your shape twice, once over each line.
TASK: A single rotation maps to the same image. Where is the center of the rotation? What is
the angle of rotation and where is the center point of rotation – what pattern do you notice? Can
you establish a “rule” for accomplishing a rotation of  degrees from a center point of rotation
using reflections over intersecting lines?
Geometry
Summary of Exploration: An Elegant Theorem
All the basic transformations (reflections, translations, and rotations) can be expressed as a
composition of reflections. This means they can all be expressed by performing one reflection
following another.
1. Write a procedure for finding the reflections needed to accomplish an arbitrary translation
of c units.
2. Write a procedure for finding the reflections needed to accomplish an arbitrary rotation of
of  degrees about a center point P.
Glide Reflections
Another special isometric transformation is called a glide reflection. An example of a glide
reflection is shown below. Actually, the diagram below shows the same glide reflection applied
four times to the leftmost hexagon.
Think about footprints and you will understand everything about glide reflections.
1. In your own words describe how a glide reflection is formed.
2. Could the glide reflection illustrated above be generated using only reflections? If so, how
many reflections would it take?
Geometry
Transformations of the Plane
A transformation of the plane is a function whose domain and range are the xy-plane (or sets of
points within the plane). You are already familiar with geometric descriptions of certain kinds of
transformations, such as translations, reflections, rotations, and dilations.
A pair of transformation equations defines the mapping of the coordinates (x,y) to the
coordinates of the image (x’,y’).
Here is an example of a pair of transformation equations:
x  x  3
y  y 1
These equations say that given any point (x, y), the image point ( x , y ) is found by adding
3 to the x-coordinate and subtracting 1 from the y-coordinate. In other words, these
equations define a horizontal translation by 3 units to the right and a vertical translation by 1 unit
down.
 
Many geometrically interesting transformations have fairly simple equations. In fact, all
translations, reflections, rotations, and dilations can be described using equations of the
form:
x  ax  by  e
y  cx  dy  f
An important fact about transformation equations having this form is that the image of a
line segment is always a line segment. Therefore, if you want to see the effect of a
transformation on a polygon, you just need to find the images of the vertices.
Let’s look at some exercises using points and their images, after undergoing a
transformation…
Geometry
Exercise
1. Give a geometric description of the transformation defined by each pair of equations. Be as
specific as possible—for example, for a reflection, identify the reflection line. If you’re not
sure about an answer, try graphing a few pairs (not just a pair) of input and output points.
a. x = x – 2
y = y + 5
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b. x = 2x
y = y

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c.
x = x
y = 13 y
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d. x = x
y = –y

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e. x = –x
y = –y

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Geometry
f.
x = 6 – x
y = y
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g. x = y
y = x


h. x = –y
y = x


i.
x = x
y = 0
j.
x = kx
y = y + c

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