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Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Teresa D. Magnus

Section 5.2 Dilations and Similar Figures

In the last section, we used our familiarity with the Cartesian coordinate system to understand the actions of dilations. In this section, we will explore more general examples of dilations, study the role of the scale factor, and look at the relationship between dilations and similar figures.

Subsection 5.2.1 A Dynamic Look at Dilations

Exploration 5.2.1. Role of the Scale Factor.

Use Figure 5.2.1 to complete the tasks in this exploration. This GeoGebra applet allows us to see how changing the scale factor affects the dilation.
(a)
With the slider for the scale factor set at \(s=3\text{,}\) how is quadrilateral \(A'B'C'D'\) related to \(ABCD\text{?}\)
Figure 5.2.1. GeoGebra applet for Exploration 5.2.1
(b)
What happens when the scale factor \(s=1\text{?}\) Why?
(c)
What happens when the slider is moved to \(s=0\text{?}\) Why do you think that happens?
(d)
What happens when the slider is moved to a number between 0 and 1?
(e)
What happens when the slider is moved to \(s=-\frac{1}{2}\) ? \(s=-1\) ? \(s=-2\) ? Describe the effect of a negative scale factor on a dilation.
(f)
Construct the four lines, \(\overleftrightarrow{EA}\text{,}\) \(\overleftrightarrow{EB}\text{,}\) \(\overleftrightarrow{EC}\text{,}\) and \(\overleftrightarrow{EA}\text{.}\) What do you notice about these lines and the image \(A'B'C'D'\) ? Is this true regardless of the value of \(s\text{?}\)
(g)
Use the reset tool (circle with arrows in upper right) to move the slider back to \(s=3\) and delete the lines. Use GeoGebra tools to measure the lengths \(EA\) and \(EA'\text{,}\) \(EB\) and \(EB'\text{,}\) \(EC\) and \(EC'\text{,}\) and \(ED\) and \(ED'\text{.}\) What do you notice about these pairs of measurements? Experiment with moving center \(E\) or some of the vertices of \(ABCD\text{.}\) Does your conjecture still hold?
(h)
Use the reset tool again to move the slider back to \(s=3\) and delete the measurements. Use the GeoGebra tools to measure the four sides of the original quadrilateral, \(AB\text{,}\) \(BC\text{,}\) \(CD\text{,}\) and \(DA\text{.}\) Compare these to the measures of the four sides, \(A'B'\text{,}\) \(B'C'\text{,}\) \(C'D'\) and \(D'A'\) of the dilated quadrilateral.
Hint.
Some measurements may be approximated by rounding. This could lead to small discrepancies in values.
(i)
Measure the area of quadrilateral \(ABCD\) and its image \(A'B'C'D'\text{.}\) Move the slider as helpful to determine how the area of a figure is affected by the scale factor.

Subsection 5.2.2 Dilations and Similar Triangles

Our work in Section 5.1 and Exploration 5.2.1 suggests that the image of a polygonal figure under a dilation will have the same shape as the original figure. In addition the ratio of the lengths of the image’s sides to their corresponding lengths in the original, \(\frac{P'Q'}{PQ}\text{,}\) will be preserved.

Definition 5.2.2.

When two figures are similar, the ratio of corresponding sides is constant. This common ratio, \(k\text{,}\) such that \(k=\frac{P'Q'}{PQ}\) for all sides \(PQ\) is called the constant of proportionality.
In Figure 5.1.9, we see that \(\Delta ABC\sim\Delta DEC\) with constant of proportionality \(k=\frac{1}{3}\text{.}\) Also, \(A'B'C'D'\sim ABCD\) in Figure 5.2.1 has \(k=|s|\text{,}\) the absolute value of the variable scale factor of the dilation.

Checkpoint 5.2.3. Understanding Proportionality, Congruence, and Similarity.

The following questions encourage you to make connections between the concepts of scale factor, proportion, congruence, and similarity:
(a)
Why are absolute values necessary when we say that the constant of proportionality \(k=|s|\text{,}\) the absolute value of the scale factor \(s\text{?}\)
Hint.
Lengths can never be negative.
Answer.
A scale factor can be any real number. The constant of proportionality is a ratio of lengths and must be nonnegative since lengths can never be negative.
(b)
Will a pair of congruent figures also be similar?
Answer.
Yes, you can consider congruent figures to be similar with a constant of proportionality, \(k=1\text{.}\) Some people may insist that size must change in which case we would not consider congruent figures to also be similar. It depends on the definition being used.
(c)
Will a pair of similar triangles also be congruent?
Answer.
Similar triangles will be congruent only when they are the same size; i.e., when \(k=1\text{.}\) Most pairs of similar triangles are not congruent.
When two figures were congruent, we were able to find an isometry that mapped one figure onto the other. Can the same be said about similar figures and dilations? When two figures are similar, must there be a dilation that maps one onto the other?

Exploration 5.2.2. Mystery Similarity Transformations.

(a) Mystery Similarity 1.
Quadrilaterals \(ABCD\) and \(A'B'C'D'\) are given in Figure 5.2.4. Virtual tools for measuring angles and line segments are provided.
Figure 5.2.4. GeoGebra applet for Mystery Similarity 1.
(i)
Are the two quadrilaterals similar? Be sure to check all criteria in the definition of similar.
(iii)
Is there a dilation taking \(ABCD\) to \(A'B'C'D'\text{?}\) If so, locate the center \(F\) and state the scale factor.
Hint.
What lines must \(F\) lie on?
(b) Mystery Similarity 2.
Pentominos \(ABCDEFGH\) and \(A''B''C''D''E''F''G''H''\) are given in the GeoGebra applet, Figure 5.2.5. Virtual tools for measuring angles and line segments are provided.
Figure 5.2.5. GeoGebra applet for Task 5.2.2.b Mystery Similarity 2
(i)
Are the two pentominos similar? Be sure to check all criteria in the definition of similar.
(ii)
Determine the constant of proportionality if it exists.
(iii)
Is there a dilation taking \(ABCDEFGH\) to \(A''B''C''D''E''F''G''H''\text{?}\) If so, locate the center \(Q\) and state the scale factor.
(c) Mystery Similarity 3.
Concave quadrilaterals \(ABCD\) and \(A'B'C'D'\) are given in the GeoGebra applet, Figure 5.2.6.
Figure 5.2.6. GeoGebra applet for Task 5.2.2.c Mystery Similarity 3.
(i)
Are the two quadrilaterals similar? Be sure to check all criteria in the definition of similar.
(ii)
Determine the constant of proportionality if it exists.
(iii)
Is there a dilation taking \(ABCD\) to \(A'B'C'D'\text{?}\) If so, locate the center \(G\) and state the scale factor for the dilation.
(d)
Describe the technique you used to determine whether two figures were similar. How did you find the constant of proportionality?
(e)
Describe the technique you used to determine whether there was a dilation that took a figure to another figure. How did you find the center of the dilation?
(f)
Based on your explorations, which of the following are true?
  1. If two polygons are similar there is a dilation taking one polygon to the other.
  2. If there is a dilation taking one polygon to another, then the polygons are similar.
  3. Both statements are true.
  4. Neither statement is true.
While any dilation will send each figure to a figure that is similar to the original, we observed in Exploration 5.2.2 that there are pairs of similar figures for which no dilation maps one onto the other. Instead, a composition of a dilation and an isometry is necessary.

Definition 5.2.7.

A similarity transformation or similarity is a geometrical transformation that preserves shape, angle measure, and collinearity, but not necessarily length. A similarity maps each figure onto a figure that is similar to the original. In a similarity transformation, lengths are enlarged or reduced by a constant factor \(k\text{,}\) the constant of proportionality.
A similarity is either a dilation or the composition of a dilation and an isomorphism.

Exercises 5.2.3 Exercises

Building Our Toolbox

1.
Add

Skills and Recall

2.
Coordinates given, center given, find image.

Extending the Concepts

3.
Add

Writing Prompts

4.
Add
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