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Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Teresa D. Magnus

Section 4.1 An Introduction to Reflections

Subsection 4.1.1 Defining a Reflection

A reflection is the first transformation we will explore.

Definition 4.1.1.

A reflection is a transformation of a plane which is specified by a line \(\ell\text{,}\) called its mirror or reflecting line. Each point \(P\) not on the mirror is mapped by the reflection to a point \(P'\) on the opposite side of \(\ell\text{.}\) The location of each \(P'\) is determined by the fact that \(P'\) would fall directly on the corresponding \(P\) when we fold the plane along the mirror line \(\ell\text{.}\) We call \(P'\) the image of \(P\) and call \(P\) the preimage of \(P'\text{.}\)

Example 4.1.2.

In Figure 4.1.3, the reflection across line \(\ell\) maps points \(A\text{,}\) \(B\text{,}\) and \(C\) to images \(A'\text{,}\) \(B'\text{,}\) and \(C'\text{,}\) respectively.
Reflection of triangle ABC across line l
Figure 4.1.3.
Additional properties of reflections will be discovered through exploration.

Subsection 4.1.2 Exploring Reflections by Sketching

As we begin our study of reflections, it is advantageous to use paper and colored pencils. Using these tactile tools will help us develop spacial skills, give us a broader understanding of reflections, and recognize relationships between geometry and algebra. In addition, working on paper allows us to check our work by folding.

Note 4.1.4.

In future sections, we will use GeoGebra tools for finding reflections. Although GeoGebra interactives are provided at the end of the current section and accessible by links in the margin notes, these GeoGebra applets are designed to mimic tasks you would normally do with paper and pencils. They do not exploit the power of GeoGebra and require additional steps to rename and recolor objects. In fact, the reflection tool in this section’s applets has been turned off. You are encouraged to use paper and pencils or your device’s drawing tools instead of the applets.

Exploration 4.1.1. Reflections on Graph Paper.

(a)
Print out the image in Figure 4.1.5 or carefully copy it onto graph paper.
On graph paper, line GH passes through the points G(0,0) and H(4,4). A backwards P-pentomino is formed using the following sequence of points: A(0,6), B(2,6), C(2,3), D(1,3), E(1,4), and F(0,4). Axes are not shown.
Figure 4.1.5. A pentomino to reflect across \(\overline{GH}\)
(i)
On your copy of Figure 4.1.5, use the coordinate grid to reflect each vertex of pentomino \(ABCDEF\) across line \(\overleftrightarrow{GH}\text{.}\) Give the reflected image of \(A\) the label \(A'\text{,}\) the reflected image of \(B\) the label \(B'\text{,}\) etc. Connect the points you drew to form pentomino \(A'B'C'D'E'F'\text{.}\)
Hint.
If you are using paper, you can check your answer by folding across line \(\overleftrightarrow{GH}\text{.}\) Otherwise, imagine what would happen if you were to fold across this line. Do your figures line up?
(ii)
In what ways are the reflected image \(A'B'C'D'E'F'\) and the original figure \(ABCDEF\) the same? In what ways are they different?
(iii)
Where does \(\overline{BB'}\) intersect \(\overleftrightarrow{GH}\text{?}\) How is this point related to segment \(\overline{BB'}\text{?}\)
(iv)
What type of angle is formed by the intersection of \(\overleftrightarrow{BB'}\) and \(\overleftrightarrow{GH}\text{?}\)
(v)
Using the 1 cm graph paper as a guide, what is the shortest distance from point \(B\) to line \(\overleftrightarrow{GH}\text{?}\) What is the distance from \(B\) to \(B'\text{?}\)
(vi)
What happens to the other points in the reflection? Suppose \(P\) is any point in the plane and let \(P'\) be its image created by reflecting it across any line in the plane. What do you expect to be true about the relationship between the segment \(\overline{PP'}\) and the reflecting line?
Hint.
The letter \(P\) is being used as a variable that can represent any point on the plane, including \(A\text{,}\) \(B\text{,}\) and so forth. For example, when \(P=A\text{,}\) \(P'=A'\text{.}\) When \(P=B\text{,}\) \(P'=B'\text{.}\) This allows us to discuss properties that hold for all points in a single statement.
(vii)
What happens if you reflect point \(G\) across line \(\overleftrightarrow{GH}\text{?}\) Is your claim true for any point on \(\overleftrightarrow{GH}\text{?}\)
(viii)
Revise the definition of reflection in Definition 4.1.1 to include the action on points that lie on the mirror. Add this definition to your toolbox in words and pictures that are meaningful to you.
(b)
Pentomino \(ABCDEFGHIJ\) and its reflection \(A'B'C'D'E'F'G'H'I'J'\) are shown in Figure 4.1.6. Use what you discovered in Task 4.1.1.a.iii to sketch the reflecting line. Describe your technique.
Hint.
What properties should the reflecting line have? Browse the GeoGebra tools to find tool(s) that will draw this line.

Subsection 4.1.3 Reflections and Coordinates

Every point on the Cartesian plane can be identified by its coordinates \((x,y)\text{.}\) The coordinates \(x\) and \(y\) represent the location of the point as being \(x\) units to the right and \(y\) units above the origin. In the next exploration, we observe how we can use the coordinates of a point \(P\) to determine the coordinates of its image \(P'\) under a reflection across the \(x\)- or \(y\)-coordinate axes.

Exploration 4.1.2. Reflecting Across the Coordinate Axes.

Use a copy of the graph below to complete the following tasks. You will need to be able to draw in four colors as described.
A coordinate grid showing the x-axis and y-axis and points A=(2,5), B=(-1,3), C=(0,-4), and D=(6,-3).
Figure 4.1.7. Four Points on Coordinate Grid for Exploration 4.1.2
(a)
Plot the images \(A'\text{,}\) \(B'\text{,}\) \(C'\text{,}\) and \(D'\) of the points \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) and \(D\) across the horizontal (\(x\)-) axis. Determine the coordinates of each new point. Create the quadrilateral \(A'B'C'D'\) and color it red. What effect does reflecting across the \(x\)-axis have on coordinates?
(b)
Plot the reflections of the original points \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) and \(D\) across the vertical (\(y\)-) axis, labeling these points with double primes, \(A''\text{,}\) \(B''\text{,}\) \(C''\text{,}\) and \(D''\text{.}\) Determine the coordinates \(A''\text{,}\) \(B''\text{,}\) \(C''\text{,}\) and \(D''\text{.}\) Create the quadrilateral \(A'B'C'D'\) and color it green. What effect does reflecting across the \(y\)-axis have on coordinates?
(c)
Reflect the vertices \(A'\text{,}\) \(B'\text{,}\) \(C'\text{,}\) and \(D'\) of the red quadrilateral \(A'B'C'D'\) created in Task 4.1.2.a across the vertical (\(y\))-axis, Create and color this new quadrilateral blue and label the new points with asterisks, \(A^*\text{,}\) \(B^*\text{,}\) \(C^*\text{,}\) and \(D^*\text{.}\) Determine the coordinates of the newest points \(A^*\text{,}\) \(B^*\text{,}\) \(C^*\text{,}\) and \(D^*\text{.}\) What effect does reflecting first across the \(x\)-axis and then across the \(y-axis\) have on the coordinates (from the original \(A\) to the final \(A^*\) for example)?
(d)
Focus on the original points \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) and \(D\) and the final points \(A^*\text{,}\) \(B^*\text{,}\) \(C^*\text{,}\) and \(D^*\text{.}\) Is there a single reflecting line that gives this mapping? What type of transformation describes this mapping?
Hint.
Look at the quadrilaterals \(ABCD\) and \(A^*B^*C^*D^*\text{.}\) If there is a reflecting line, the quadrilaterals should line up when you fold along that line.
(e)
Suppose that you were to reflect the green points from Task 4.1.2.b across the horizontal (\(x\))-axis. What do you notice about their images in relation to other points?

Subsection 4.1.4 Key Ideas Regarding Reflections

In this section, we have drawn reflections by hand using a coordinate grid as an aid and we observed a few of the properties of reflections. Since we will be building on these discoveries, it is wise to summarize the main ideas for future reference.

Definition 4.1.8.

A reflection across a line \(\ell\) is a transformation of the points in the plane which maps each point \(P\) in the plane by flipping it over line \(\ell\text{.}\) The reflecting line \(\ell\) is called the mirror or axis. The image of a point \(P\) under a reflection (or any other transformation) is the point \(P'\) to which the transformation sends \(P\text{.}\) In this case of a reflection, \(P'\) is the reflection of point \(P\text{.}\)
While the notion of reflecting can be intuitively understood as flipping, the properties you discovered in Task 4.1.1.a.iii provide a more specific definition that can be extended to other geometries. These properties are identified in Checkpoint 4.1.9 and should be added to your toolbox.

Checkpoint 4.1.9. Properties of a Reflecting Line.

The reflecting line \(\ell\) meets the segment \(\overline{PP'}\text{,}\) connecting a point \(P\) and its reflected image \(P'\text{,}\) at a -degree angle. The point where \(\ell\) meets \(\overline{PP'}\) is between points \(P\) and \(P'\text{.}\)
Answer.
90; halfway
So far, we have only reflected across lines that were either horizontal, vertical, or which make a 45-degree angle to the horizontal. As we expand our consideration to reflections across other lines, we will drop the coordinate grid and use technology.

Subsection 4.1.5 Optional GeoGebra Interactives

The GeoGebra interactive Figure 4.1.10 may be used instead of Figure 4.1.5 in Exploration 4.1.1. Since the paper version of these activities is recommended, the interactive version has been placed here at the end of the section.
Figure 4.1.10. GeoGebra applet for reflecting across line \(\overleftrightarrow{GH}\text{.}\)
Similarly, the GeoGebra interactive Figure 4.1.11 may be used in the place of Figure 4.1.7.
Figure 4.1.11. GeoGebra interactive for reflecting across axes

Exercises 4.1.6 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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