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Representing Transformations as Compositions

Instructor: Elizabeth Popelka-Brown
In this lesson, you'll learn about four geometric transformations: translations, reflections, rotations and dilations. You'll also find out about the effects of multiple transformations on a shape, known as a composition of transformations.

Compositions of Transformations

I'm not much of a video game fan, but, back in the day, I was pretty crazy about Tetris, the game where you move and turn geometric pieces made of four squares to complete rows with no open spaces.


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Sometimes, I could just shift a piece or rotate it once as it fell. These individual shifts and rotations were examples of a geometric concept called transformations, changes to a shape that can affect its size, location or orientation (the direction the shape faces). Frequently, I had to complete a series of shifts and rotations to get each piece to fit into an open space. When I was performing more than one transformation, back to back, on a single game piece, I was performing a composition of transformations. The original game piece, before I made any changes to it, would be called the preimage, and the shape, following a transformation or a composition of transformations, would be called the image. Let's take a look at some kinds of transformations, and then some compositions of them.

Isometries

Rigid transformation and isometry are both terms which describe transformations that do not change the size of a shape. We will examine three kinds here. A translation changes a shape's location without changing its orientation. In this case, the preimage, hexagon ABCDEF, has been translated 6 units to the right to become the image, hexagon A'B'C'D'E'F'.


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A rotation changes a shape's location and orientation by turning it. To perform a rotation, you need to know the center of rotation and the angle of rotation. In this example, the preimage, triangle ABC, has been rotated, using the origin of the coordinate plane as the center. Imagine the triangle swings around the origin on an imaginary rope, and rotates around onto the image, triangle A'B'C'. Because the imaginary rope in its old location and the rope in its new location form a right angle, the angle of rotation is 90 degrees. It is also important to note that the rotation was done in a clockwise direction.


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A reflection changes a shape's location and orientation by flipping it across a line of reflection. This time, the preimage, triangle ABC, has been flipped over the x axis, the line of reflection, and a copy, image A'B'C', has been made. Notice that each vertex, corner, of the image lies in the opposite direction of the preimage, relative to the x axis. For example, B is 3 units above the x axis, and B' is three units below the x axis.


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Dilations

An example of a transformation that is not an isometry is a dilation. When performing a dilation, the size of a shape is changed. In this case, rectangle ABCD has been dilated to A'B'C'D'. Because the lengths of the sides of the image are half the lengths of the preimage, the scale factor of the dilation is 1/2.


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Composition Theorem

When you perform a composition of isometries, more than one rigid transformation, your end result will also be an isometry. In other words, the final image may have a change in location or orientation, but it will still have the same size. This is known as the Composition Theorem. Often, a composition of transformations can look like one single transformation.

Compositions of Like Transformations

For example, if ABCD is translated 2 units up and 3 to the right, as shown by the green arrow, to A'B'C'D', and is then translated 4 units to the right and 2 down (see blue arrow), it has the same effect as if ABCD had just been translated once, 7 units to the right (shown by the red arrow).


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A translation can also be created by a series of reflections. Here, in Diagram 1, we see that A is reflected twice, once across Line 1, and again over Line 2. These two reflections have the same effect as a single translation 8 units to the left, as seen in Diagram 2.


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