Mystery Isometries

Section 4.5 Mystery Isometries

Given two congruent figures, what is the isometry that maps one onto the other? In this section, we will learn how to identify the type of isometry for this mapping and to construct the axis of a reflection, the center and angle of a rotation, and a vector of a translation or glide reflection.

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Subsection 4.5.1 Review of Isometries

We know that an isometry is a transformation (or function) that preserves the measures of lines and angles. The original object and its image under an isometry are always congruent to each other. It turns out if we are given any two congruent figures in the plane there is an isometry that maps one of the figures onto the other.

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In the exploration that follows, you will be given a pair of congruent figures for which you will determine the associated isometry that takes one figure to the other. You will need to use your recently-acquired knowledge about the four types of isometries. To prepare for the challenging detective work that lies ahead, open and complete Checkpoint 4.5.1.

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Checkpoint 4.5.1. Isometry Review and Tools.

(a) Orientation Review.

Orientation is a good way of distinguishing between types of isometries, especially rotations and glide reflections.

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In Figure 4.5.2, which triangle has the same orientation as \(\Delta ABC\text{?}\)

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Triangles ABC, C’A’B’, and C1,B1,A1 are shown. — Figure 4.5.2. Orientation of Images of \(\Delta ABC\)
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Hint.

Orientation was defined in Definition 4.2.2.

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Answer.

Triangle \(\Delta C'A'B' \) has the same orientation as \(\Delta ABC\text{.}\) Triangle \(\Delta C'_1 B'_1 A'_1\) has the opposite orientation.

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(b) Fixed Point Review.

Sometimes we can tell which points on the plane are fixed by the isometry and use this information to identify the type of isometry.

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Of the types of isometries (reflections, rotations, translations, and glide reflections):

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(i)

Which type has no fixed points?

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Answer.

Translations and glide reflections.

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(ii)

Which type has exactly one fixed point?

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Answer.

Rotations.

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(iii)

Which type fixes all points that lie on a single line?

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Answer.

Reflections.

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(c) Review of Reflections.

Use the GeoGebra interactive in Figure 4.5.3 as needed to answer the following:

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Figure 4.5.3. An interactive GeoGebra applet for analyzing a reflection across a single line.

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(i)

A reflection the orientation of the figure.

maintains

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reverses

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Answer.

Reverses

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(ii)

How is the reflecting line related to the the segments connecting each point to its image? Consider both the location of intersection and the angle formed by the intersection.

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Answer.

The reflecting line is the perpendicular bisector of each \(\overline{PP'}\)

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(iii)

Describe how you can create the reflecting line given an illustration of two congruent triangles.

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(d) Review of Translations.

Use the GeoGebra interactive in Figure 4.5.4 as needed to answer the following questions about translations:

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Figure 4.5.4. An interactive GeoGebra applet for analyzing a translation.

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(i)

A translation the orientation of the figure.

maintains

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reverses

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Answer.

maintains

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(ii)

How is the translation vector related to the segment from a point to its image? Once again, address both length and direction in your response.

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Answer.

The translation vector is parallel to and congruent to each \(\overline{PP'}\text{.}\)

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(iii)

Describe how you can create the translation vector from an illustration of two congruent triangles \(\Delta ABC\) and \(\Delta A'B'C'\text{.}\)

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(e) Review of Rotations.

The GeoGebra interactive in Figure 4.5.5 shows the action of a clockwise rotation of \(\Delta ABC\) about point \(D\text{.}\) Use this interactive as needed to answer the following questions about rotations:

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Figure 4.5.5. An interactive GeoGebra applet for analyzing a rotation.

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(i)

A rotation the orientation of the figure.

maintains

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reverses

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Answer.

maintains

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(ii)

Where do the perpendicular bisectors of the segments \(\overline{AA'}\text{,}\) \(\overline{BB'}\text{,}\) and \(\overline{CC'}\) intersect?

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Answer.

At the center of the rotation

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(iii)

What are the measures of angles \(\angle ADA'\text{,}\) \(\angle BDB'\text{,}\) and \(\angle CDC'\text{?}\) What does this tell you about the rotation?

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Answer.

The measures of these angles are equal and correspond to the angle of the rotation.

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(iv)

If you were given a picture of a figure and what you think is its rotated image, what process would you use to determine the center of the rotation and its angle of rotation? Write this so that another student could follow your steps.

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(f) Review of Glide Reflections.

A glide reflection is a two-step isometry. We first reflect across a line and then translate the result by a vector lying on (or parallel to) that line. The GeoGebra interactive Figure 4.5.6 shows the original triangle \(\Delta ABC\) and its final image \(A''B''C''\) after a reflection across line \(\overleftrightarrow{DE}\) and translation by vector \(\overrightarrow{DE}\text{.}\) Answer the following questions:

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Figure 4.5.6. An interactive GeoGebra applet for analyzing a glide reflection.

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(i)

A glide reflection the orientation of the figure.

maintains

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reverses

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Answer.

reverses

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(ii)

The glide reflection image in Figure 4.5.6 was produced by reflecting triangle \(\Delta ABC\) across line \(\overleftrightarrow{DE}\) to get \(\Delta A'B'C'\) and then translating \(\Delta A'B'C'\) by vector \(\overrightarrow{DE}\) to get image \(\Delta A''B''C''\text{.}\) Do you get the same result if you translate by vector \(\overrightarrow{DE}\) first and then reflect across line \(\overleftrightarrow{DE}\text{?}\)

yes

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no

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sometimes but not always

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Answer.

yes

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(iii)

On the sketch construct the midpoints of each of the segments \(\overline{PP'}\) from a vertex \(P\) of \(\Delta ABC\) to its final image \(P''\) on \(\Delta A''B''C''\text{.}\) What information do these midpoints give you?

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Answer.

The reflecting line will pass through these midpoints.

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(iv)

What strategy could you use to find the reflecting line and translation vector for a glide reflection if you only have the original figure and final image?

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Subsection 4.5.2 Solving the Mysteries

Now it is time to apply our knowledge! In the next activity, each task has a pair of congruent figures. In each of the tasks, you will determine the type of isometry that maps the original figure to its image and then to sketch the defining objects (the reflecting line, translation vector, and/or center). In the case of a rotation, also determine the angle of rotation and whether the rotation is clockwise or counterclockwise.

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