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.
In Figure 4.5.2, which triangle has the same orientation as \(\Delta ABC\text{?}\)
Hint.
Orientation was defined in Definition 4.2.2.
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.
(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.
Of the types of isometries (reflections, rotations, translations, and glide reflections):
(i)
Which type has no fixed points?
Answer.
Translations and glide reflections.
(ii)
Which type has exactly one fixed point?
Answer.
Rotations.
(iii)
Which type fixes all points that lie on a single line?
Answer.
Reflections.
(c) Review of Reflections.
Use the GeoGebra interactive in Figure 4.5.3 as needed to answer the following:
(i)
A reflection the orientation of the figure.
- maintains
- reverses
Answer.
Reverses
(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.
Answer.
The reflecting line is the perpendicular bisector of each \(\overline{PP'}\)
(iii)
Describe how you can create the reflecting line given an illustration of two congruent triangles.
(d) Review of Translations.
Use the GeoGebra interactive in Figure 4.5.4 as needed to answer the following questions about translations:
(i)
A translation the orientation of the figure.
- maintains
- reverses
Answer.
maintains
(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.
Answer.
The translation vector is parallel to and congruent to each \(\overline{PP'}\text{.}\)
(iii)
Describe how you can create the translation vector from an illustration of two congruent triangles \(\Delta ABC\) and \(\Delta A'B'C'\text{.}\)
(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:
(i)
A rotation the orientation of the figure.
- maintains
- reverses
Answer.
maintains
(ii)
Where do the perpendicular bisectors of the segments \(\overline{AA'}\text{,}\) \(\overline{BB'}\text{,}\) and \(\overline{CC'}\) intersect?
Answer.
At the center of the rotation
(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?
Answer.
The measures of these angles are equal and correspond to the angle of the rotation.
(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.
(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:
(i)
A glide reflection the orientation of the figure.
- maintains
- reverses
Answer.
reverses
(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
- no
- sometimes but not always
Answer.
yes
(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?
Answer.
The reflecting line will pass through these midpoints.
(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?
