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Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Teresa D. Magnus

Section 4.3 Identifying Isometries

Subsection 4.3.1 What do we know? What do we wonder?

As we continue to build on our knowledge of isometries, it is wise to summarize what we know so far. We know that a transformation of a plane is a mapping or function on points. A transformation sends each point \(P\) on the plane to a unique point \(P'\) called its image under the transformation. Some transformations are collineations, meaning that if a set of points all lie on the same line, then their images will also lie on a single line. We have observed that some collineations preserve shape and angle measure and that some collineations that preserve angle measure will also preserve distance. The transformations that preserve all of these properties (collinearity, angle measure, and distance) are called isometries.

Exploration 4.3.1. Reflect and Wonder.

(a)
Before reading this section, respond to the following questions. There are no right or wrong answers!
  1. Write down three things you know about transformations. In particular, what do you know about reflections, translations, and rotations?
  2. Is there anything that was discussed in the first sections on transformations that you need clarity on?
  3. What do you wonder about transformations? What questions remain?

Subsection 4.3.2 Isometry Between Congruent Figures

We are familiar with three types of isometries: reflections, translations, and rotations. Yet many questions remain: Are there other types of isometries? Are there transformations that are not isometries? We have seen that the image of a polygon under an isometry is congruent to the original polygon. But what about the converse? If two figures are congruent, is there always an isometry mapping one to the other? If so, how do we determine what type of isometry it is?
Although we have not proved this fact, it is true that if we have two congruent figures, there will be an isometry that maps one onto the other. Our goal in this section and next will be to identify the type of isometry that takes one figure onto a congruent copy. Is it a reflection, a translation, a rotation, or something else? Then having determined the type, can we specify the exact action that performs the movement? A good place to start is to identify whether the isometry maintains orientation.

Exploration 4.3.2. Distinguishing Between Types of Isometries by Orientation.

(a)
Consider the following and be prepared to share your thoughts.
(i)
Imagine that \(\Delta ABC\) is a triangular puzzle piece with a beach scene on the front and gray cardboard on the back. We can think of the action of an isometry as moving \(\Delta ABC\) so that it aligns perfectly with its image \(\Delta A'B'C'\text{.}\) Which isometries--reflections, translations, and rotations--studied so far will keep \(\Delta ABC\) beach side up? Which will turn \(\Delta ABC\) over so that it is cardboard side up? What property does this represent?
(ii)
Translations and rotations both preserve orientation. How might you tell the difference between a translation and a rotation? Assume that you are given a picture of two congruent triangles \(\Delta ABC\) and \(\Delta A'B'C'\) where the orientation is the same. How do you know whether a translation takes \(\Delta ABC\) to \(\Delta A'B'C'\text{?}\) How do you know whether the isometry is a rotation? Could it be both? Or could it be neither?

Subsection 4.3.3 Recognizing Isometries by Fixed Points

Another way to distinguish between isometries is to note which points, if any, are unmoved by the isometry. Such a point is called a fixed point of the isometry. Fixed points may occur when the transformation is not an isometry as well, so we will state the definition more generally.

Definition 4.3.1.

If \(f\) is any transformation, then \(P\) is said to be a fixed point of transformation \(f\text{,}\) if \(f(P)=P\text{.}\) Following our convention of denoting the image of point \(P\) with the symbol \(P'\text{,}\) \(P\) is a fixed point if \(P'=P\text{.}\)

Exploration 4.3.3. Determining the Fixed Points of an Isometry.

Figures for Task 4.3.3.a and Task 4.3.3.b are in Section 4.2. Click on the hyperlinks to view each figure locally.
(a)
In Figure 4.2.4, which point(s) are not moved by the reflection across \(\overleftrightarrow{HI}\text{?}\) Are there unlabeled points that are also fixed by this reflection? Explain.
(b)
In Figure 4.2.11, which point(s) are not moved by the rotation about point \(F\text{?}\) Are there any unlabeled points that are also fixed by this reflection? Explain.
(c)
In Figure 4.3.2, which point(s) are not moved by the translation by vector \(\overrightarrow{FG}\text{?}\) Are there any unlabeled points that are fixed by this translation? Explain.
Concave pentagon ABCDE is translated to the left and slightly down by vector FG.
Figure 4.3.2. Translation by vector \(\overrightarrow{FG}\)
Like the function \(f(x)=x\) you may have encountered in another course, there is a transformation of the plane that maps every point to itself. We call this transformation, the identity transformation. Under the action of the identity transformation, every point is a fixed point. Clearly, the identity function also preserves distance, angle measure, and orientation since it leaves everything unchanged.

Checkpoint 4.3.3.

(a)
For each of the following, answer
  1. reflection
  2. rotation
  3. translation, or
  4. none of the above.
(i)
An isomorphism with exactly one fixed point might be a ___.
Answer.
rotation
(ii)
An isomorphism with exactly two fixed points might be a ___.
Hint.
By exactly two, we mean that there are two and only two.
Answer.
None of the above.
(iii)
An isomorphism with an infinite number of fixed points might be a ___.
Answer.
reflection
(iv)
An isomorphism with no fixed points might be a ___.
Answer.
translation

Subsection 4.3.4 Defining Objects of Isometries

First we determine whether the isometry mapping one figure onto a congruent image is a reflection, translation, or rotation. We then need to specify which reflection, which translation, or which rotation it is. We do this by identifying the defining object(s) of the isometry. With a reflection, for example, we specify which line we are reflecting across. This line is unique; no other line will give the same reflection. Table 4.3.4 lists the defining objects for the three isometries we have studied.
Table 4.3.4. Defining Objects for Three Isometries
Isometry Defining Objects
Reflection Reflecting Line (or mirror)
Translation Translation Vector
Rotation Center and Rotational Angle
How do we find the mirror for a reflection, the vector for a translation, or the center and angle for an rotation? Although the defining objects determine how the isometry acts on all points in the plane, we need just a few points along with their images to find the defining objects. As you work through Exploration 4.3.4 make note of these methods and add them to your toolbox.

Exploration 4.3.4. Finding Defining Objects of Isometries.

For each isometry, we will observe some properties of the isometry and defining object(s) and then use these properties to determine the defining object(s).
(a) Finding the Mirror of a Reflection.
Refer to Figure 4.3.5 which shows a reflection of \(\Delta ABC\) across a hidden line \(m\text{.}\)
Figure 4.3.5. An interactive GeoGebra applet for finding the mirror of a reflection.
(i)
How should the reflecting line \(m\) be related to each segment \(\overline{PP'}\) connecting a vertex of the triangle to its image?
Hint.
Does \(m\) intersect \(\overline{PP'}\text{?}\) If so, where and what angle is formed?
(ii)
Explore the tools available in the first seven menus of the interactive in Figure 4.3.5. Then use some of these tools to sketch the reflecting line for the isometry that takes \(\Delta ABC\) to \(\Delta A'B'C'\text{.}\) Do not use guess and check and do not use the ‘Reflect About Line’ tool! Instead choose GeoGebra tools from the other menus that will construct the exact line. If you move any of the vertices \(A\text{,}\) \(B\text{,}\) or \(C\text{,}\) the reflecting line should move accordingly.
Hint.
The tools across the top of this interactive are select, intersect, draw segment, draw line, draw perpendicular (draw parallel is also under this tab), draw perpendicular bisector, reflect across line, write text, and move graphics (zoom in, zoom out, and hide labels are also under this tab).
(iii)
Use the ‘Reflect about Line’ tool in the above app to check that reflecting \(\Delta ABC\) across the line you created results in a triangle that aligns perfectly with \(\Delta A'B'C'\text{.}\)
(iv)
Write down instructions of how you can create the reflecting line given any triangle \(\Delta ABC\) and its reflected image \(\Delta A'B'C'\text{.}\) The instructions should be detailed enough that others can follow them.
(b) Finding the Vector for a Translation.
In Exploration 4.2.2, you were given a translation vector \(\overrightarrow{FG}\) which indicated the direction and distance that each point should be moved, specifically three units right and one unit down. In the activity \(E\) had the coordinates \((2,4)\) and \(F\) had the coordinates \((5,3)\text{.}\) The difference of the \(x\)-values, \(5-2=3\text{,}\) gives the number of units moved right, and the difference of the \(y\)-values \(3-4=-1\) tells us how far to move up. Since this latter difference is negative, we move down instead of up. Note that we could have chosen a different vector to represent “three units right and one unit down.”
(i)
If the starting point \(F\) of vector \(\overrightarrow{FG}\) has the coordinates \((0,0)\text{,}\) what must the coordinates of \(G\) be so that \(\overrightarrow{FG}\) represents a move of three units right and one unit down?
(ii)
If the ending point \(G\) has the coordinates \((-2,2)\text{,}\) what would the coordinates of the starting point \(F\) be so that \(\overrightarrow{FG}\) would represent three units right and one unit down?
(iii)
If point \(A\) has the coordinates \((-1,5)\) and the picture below shows a translation that moves the figure three units right and one unit down, what are the coordinates of \(A'\text{?}\)
(iv)
Draw, identify, or describe a vector for the translation that takes \(\Delta ABC\) to \(\Delta A'B'C'\) in Figure 4.3.6.
Figure 4.3.6. Triangle \(\Delta ABC\) is translated to \(\Delta A'B'C'\)
(v)
Write instructions that you or someone else could use to find the translation vector when any figure and its image under a translation are given.
(c) Finding the Center and Angle of a Rotation, part 1.
In Figure 4.3.7, an L-pentomino and its image under a rotation are shown.
Figure 4.3.7. An interactive GeoGebra applet for finding the center and angle of a rotation in Task 4.3.4.c.
(i)
Draw or identify the point that is the center of the rotation taking \(ABCDEF\) to \(A'B'C'D'E'F'\text{.}\) How do you know this point is the center?
(ii)
Determine the angle of this rotation and specify whether the rotation is performed clockwise or counterclockwise.
(d) Finding the Center and Angle of a Rotation, part 2.
The center in Figure 4.3.8 is more difficult to identify. Make an initial guess at where the center might be. What are some strategies or tools we could try?
Figure 4.3.8. An interactive GeoGebra applet for finding the center and angle of a rotation.
(i)
Perhaps the segments \(\overline{AA'}\text{,}\) \(\overline{BB'}\text{,}\) and \(\overline{CC'}\) might provide information. Sketch these three line segments. Do they help?
(ii)
GeoGebra tools will allow you to find perpendicular lines, perpendicular bisectors, angle bisectors, circles, angle measures, and lengths. Any ideas of what we might try?
(iii)
Since we have sketched the segments \(\overline{AA'}\text{,}\) \(\overline{BB'}\text{,}\) and \(\overline{CC'}\text{,}\) use the perpendicular bisector tool to find the perpendicular bisector of each segment. What do you notice?
(iv)
Create point \(E\) as a candidate for your center. Verify that \(m\angle AEA'=m\angle BEB' =m\angle CEC'\text{.}\) What is the angle of rotation?
(v)
Finally, use the ‘Rotate around Point’ tool to check your work.
Hint.
The ‘Rotate around Point’ tool has the icon showing the sweep of an angle and three points but no sides. I have placed in on the main ribbon, but it is usually found in the same menu as the ‘Reflect across Line’ tool. After selecting the ‘Rotate around Point’ tool, click on the interior of the figure to be rotated and the point that is the center. You will be prompted to enter an angle measure.
(vi)
Write instructions to yourself and others on how to find the center and angle of a rotation. Base your instructions on the assumption that a picture of a figure and its rotated image is provided. Be sure that your argument addresses the situation in Make your argument general by including the possibility that no vertex of the figure is mapped to itself.

Subsection 4.3.5 What have we learned?

In Table 4.3.4, the defining objects for each transformation were given. We now add new columns to this table. In the column Orientation, write either ‘reverses’ or ‘maintains’ for the entries. The column Fixed Points, is where you will identify the point(s), if any, that are fixed by the transformation. In the column Properties of Object(s), you should write the properties that can assist in locating the defining object(s) for each type of transformation.
Table 4.3.9. Defining Objects for Three Isometries
Isometry Orientation Fixed
Points		 Defining
Object(s)		 Properties of
Def. Object(s)
Reflection Reverses Points on
Mirror		 Reflecting Line
(or mirror)		 Mirror is the
perpendicular
bisector of each \(\overline{PP'}\)
Translation Translation
Vector



Rotation Center and
Angle		 Perp. bisector
of each \(\overline{PP'}\)
passes through
center

Checkpoint 4.3.10. Updating Your Toolbox.

Complete Table 4.3.9 by filling in the column for orientation and the row for translation. Compare the last column in your table with the last item in each task of Exploration 4.3.4. You are encouraged to rewrite each entry in words that are meaningful to you before adding this to your toolbox. You may find this table useful in the next section.

Exercises 4.3.6 Exercises

Building Our Toolbox

1.
add

Skills and Recall

2.
Add some.

Extending the Concepts

3.
Add

Writing Prompts

4.
Add
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