Transformations in the Plane
A point or set of points in the plane
moves to relocate or form figures in the plane. These movements are called transformations.
The original point or set of points is called the preimage, and what
emerges after the transformation is called the image. The four
transformations with which we are concerned are translations, reflections,
rotations, and dilations.
Translation
A translation moves a point or
set of points in a given direction for a given distance.
Figure 1
Figure 1 illustrates the action
of the translation.
Now let’s see what other attributes of a translation we can discover.
Figure 2
What do you
notice about the preimage and its image? What you should see is that segment CD
and segment C’D’ are equal in measure, 2.51 cm. And, m/ECD = m/E’C’D’
or 64.82o, which means that a translation is an isometry. The
formal definition for an isometry is a transformation that preserves distance
(length) and angle measure. (Notice that Geometer’s Sketchpad uses
primes on points/segments that compose the image of the translation. This
standard is used in most texts, also.) Also, we say that one figure “slides”
away from the other. There is no symmetry associated with a translation.
Reflection
A reflection is a transformation
that moves a point or set of points with respect to a line l such that a
segment drawn between a preimage and image point is bisected by this line.
Figure 3
In Figure 3 we see that the segments joining the image and preimage points (These are the thin line segments in the figure.) in the flag have the same length on either side of line l, so line l is the perpendicular bisector of those segments. What else do you notice about the flag?
Figure 4
What do you notice about the preimage and image in Figure 4? You should see that angle measure and distance/length is preserved in a reflected figure and that the orientation has “flipped” from the original figure. What does that tell you about the transformation that is a reflection?
Figure 5
In Figure 5 there is another line of reflection line m. Which figures are preimages and which are images? The symmetry associated with reflections is line symmetry. Where is/are the line(s) of symmetry in the shapes in Figure 6?
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Figure 6
Rotation
A rotation is a transformation that moves a point or set of points through a certain angle about a fixed point. See the illustration in Figure 7.
Figure 7
In the figure the bold flag is the preimage while the dotted flags are the images under a 45o rotation. The fixed point in the rotation is the point at the tip of the flag pole. What would you suspect about the rotation transformation? The symmetry associated with the rotation transformation is, oddly enough, rotational symmetry.
If we rotate the flag through each position, how many rotations are necessary to bring the bold flag back to its original position? ____ This is called 8 – fold symmetry. Now go back to the figures in Figure 6 with line symmetry. Which ones will have rotational symmetry? ____ What are the symmetries? This type of symmetry is generally known as N – fold symmetry.
Dilation
The last transformation we will consider is dilation. A dilation is a shrinking or expansion of a given figure about a fixed point. See the illustration in Figure 8.
Figure 8
The bold flag is the preimage for each of the others. The figure shows one set of points has shrunk while the second set of points has expanded. What would you suspect about this transformation?
Application of Dilation
Transformation
An important application of the dilation is similar figures. Similar figures are figures with equal angle measures, correspondingly, and proportional sides, correspondingly. Figure 9 shows an example.
Figure 9
How does the fact that the figures are similar help us find lengths in the figures?