Each of the transformations we have studied so far (reflections, translations, and rotations) performs an action on the points of the plane. In fact, a transformation is a function where the inputs and outputs are geometric objects instead of numbers. In algebra, you learned about the “add three” function, written \(f(x)=x+3\text{,}\) and the “multiply by four” function, \(g(x)=4x\text{.}\) You also learned that you could perform two functions in succession. For example, you could perform the “add three” function on the number \(1\) and then perform the “multiply by 4” function on the result.
In function notation, we may write \(g(f(1))=g(1+3)=g(4)=4(4)=16\) where we perform the inner function first. We may also perform these operations on a generic value \(x\text{:}\)
We now explore what happens when we compose geometric transformations. We could compose any two transformations, but we will first focus on reflections. What happens when we perform two or more reflections in succession? Do we get another reflection? Does it matter whether the lines are parallel? Does the order in which we perform the reflections matter?
A composition of two transformations is a sequence of two transformations, \(s\) and \(t\text{,}\) performed in succession. To find the image of a point \(P\) under the composition, \(t(s(P))\text{,}\) first find the image \(P'\) of \(P\) under the action of \(s\text{.}\) Then find the image \(P''\) of \(P'\) under \(t\text{.}\)
To use the ‘Reflect about Line’ tool, select the icon that shows two dots across a diagonal line from each other. Click the interior of the original figure and then click the mirror line.
To help you focus on \(ABCDEF\) and \(A_2B_2C_2D_2E_2F_2\text{,}\) click on the reset button to hide your work. Then check the ‘Show image’ box to reveal the image of the composition.
Draw the segments \(\overline{AA_2}, \overline{BB_2}, \ldots, \overline{FF_2}\) and measure the lengths \(AA_2, BB_2, \ldots, FF_2\text{.}\) What do you notice?
What do you notice about the angle between each segment, \(\overline{AA_2}, \overline{BB_2}, \ldots, \overline{FF_2}\) and the original two lines \(g\) and \(h\text{?}\)
Measure the distance between the original pair of reflecting lines \(g\) and \(h\text{.}\) To do this, reveal segment \(\overline{TU}\) using the checkbox. Then use the ‘Distance or length’tool to measure \(\overline{TU}\text{.}\)
Each type of isometry has a special object or two that helps to determine it precisely. As shown in Table 4.3.13, a reflection has a reflecting line, a translation has a vector, and a rotation has a center and angle. Identify and draw in the specific line, vector, or center and angle for this transformation. How does this object relate to the parallel reflecting lines \(g\) and \(h\text{?}\) Verify your answer using the appropriate transformation tool (Reflect across Line, Translate by Vector, or Rotate around Point) in geogebra.
A second copy of the GeoGebra app for reflecting across parallel lines is provided in Figure 4.4.4. This time reflect \(ABCDEF\) first across line \(h\) and then reflect the result across line \(g\text{.}\) How does the result compare to what happened when we reflected across g before h? Where is \(A''B''C''D''E''F''\text{?}\) Have any of the distances or angles changed?
Refer to the original pentomino \(ABCDEF\) and its image \(A''B''C''D''E''F''\) created by reflecting across intersecting lines \(g\) and \(h\) in Figure 4.4.6. You may choose to hide \(A'B'C'D'E'F'\) (and the duplicated \(A''B''C''D''E''F''\)) after saving a screenshot of your work. Use the reset button and ‘Show image’ checkbox.
What type of isometry takes \(ABCDEF\) to the final image \(A_2 B_2C_2 D_2 E_2 F_2\text{?}\) Is it a reflection, a translation, a rotation, or something else?
Create the perpendicular bisector of each segment \(\overline{AA_2},
\overline{BB_2}, \ldots, \overline{FF_2}\text{.}\) What do you notice about the perpendicular bisectors?
Each type of translation has a special object or two that helps to determine it precisely. Identify and draw in the specific line, vector, or center and angle for this transformation, referring to Table 4.3.13 as needed. How does the object(s) relate to the intersecting reflecting lines \(g\) and \(h\text{?}\)
Create point \(P\) where the original two lines intersect. What is the measure of \(\angle A_2 PA\text{?}\)\(\angle B_2 PB\text{?}\)\(\angle C_2 PC\text{?}\)\(\angle D_2 PD\text{?}\)\(\angle E_2PE\text{?}\)\(\angle F_2PF\text{?}\) What does this tell you about the transformation taking \(ABCDEF\) to \(A_2B_2C_2D_2E_2F_2\text{?}\)
A second copy of the GeoGebra app for reflecting across intersecting lines is provided in Figure 4.4.7. This time reflect \(ABCDEF\) first across line \(h\) and then reflect the result across line \(g\text{.}\) How does the result compare to what happened when we reflected across g before h? Where is \(A''B''C''D''E''F''\text{?}\) Have any of the lengths or angles changed?
Next we will explore what happens when we reflect across three lines in succession. Once again, we will consider the case where the lines are parallel and when they intersect. Will we get the same transformations as before or might these be something different?
Each type of translation has a special object or two that helps to determine it precisely. Identify and draw in the specific line, vector, or center and angle for this transformation. How, if at all, is this object related to the three lines \(d\text{,}\)\(e\text{,}\) and \(f\text{?}\)
As of this writing, the author has not found a relationship between this defining object and the three lines. If you discover one, please let her know. You may be recognized in a future edition!
Unlabeled points on lines \(d\text{,}\)\(e\text{,}\) and \(f\) allow the lines to be moved while staying parallel. Experiment to see whether your claims continue to hold. You can also experiment with reflecting across the three lines in a different order; however, the ‘Show image’ will not work.
Finally, we reflect the triangle across three lines, at least some of which intersect. Use the ‘Reflect about Line’ tool to reflect \(\Delta ABC\) across \(d\text{,}\) then reflect its image \(\Delta A'B'C'\) across line \(e\text{,}\) and finally reflect \(\Delta A''B''C''\) across \(f\text{.}\) After creating \(\Delta A'''B'''C'''\text{,}\) check the result using the checkbox ‘Show image’. Record your work.
Hide lines \(d\text{,}\)\(e\text{,}\) and \(f\) by unchecking the ‘Show mirrors’ box. Plot the midpoint of each of the line segments, \(\overline{AA'_1}\text{,}\)\(\overline{BB'_1}\text{,}\) and \(\overline{CC'_1}\text{.}\) What do you notice about these midpoints?
Draw the line that passes through the midpoints of \(\overline{AA'_1}\) and \(\overline{BB'_1}\text{.}\) Then reflect \(\Delta ABC\) across this line. What do you notice about the relationship between this new triangle \(\Delta A'B'C'\) and \(\Delta A'_1 B'_1 C'_1\text{?}\)
In this chapter, we’ve explored the behavior of a particular type of transformation, called an isometry. This term comes from the Greek words “iso-” meaning “equality” and “metria” meaning “measure”.
An isometry is a function taking points on the plane to points on the plane which sends collinear points to collinear points and preserves the distance between points.
There are four types of isometries. Through our explorations, we have seen that the orientation of a figure is reversed each time it is reflected. Thus, orientation is changed under an odd number of reflections and maintained (flipped back) under an even number of reflections. Hence, the composition of two reflections in succession will be either a translation (when the reflecting lines are parallel) or a rotation (when the reflecting lines intersect). When we reflect across three lines, not all of which are parallel, we get an isometry which reverses the orientation of figures but is not a reflection. Instead we obtain what is known as a glide reflection.
A glide reflection is defined as a composition of a reflection across a line and a translation along a vector parallel to (or on) the reflecting line. In Figure 4.4.13, note how the pentomino is first reflected across the line and then translated along vector \(\overrightarrow{GH}\text{.}\) Any of the parallel vectors, like \(\overrightarrow{A'A''}\) or \(\overrightarrow{CC''}\text{,}\) could be used as the translation vector, but the reflecting line here is \(\overleftrightarrow{GH}\text{,}\) not \(\overleftrightarrow{CC''}\text{.}\)
