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.
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 \(s\text{?}\)
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?
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.
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.
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.
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.
Why are absolute values necessary when we say that the constant of proportionality \(k=|s|\text{,}\) the absolute value of the scale factor \(s\text{?}\)
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.
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.
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?
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.
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?
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.
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.
