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# Composition of Transformations

## Learn how to compose transformations of a figure on a coordinate plane.

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Composition of Transformations

What if you were told that your own footprint is an example of a glide reflection? The equations to find your average footprint are in the diagram below. Determine your average footprint and write the rule for one stride. You may assume your stride starts at (0, 0).

### Composition of Transformations

##### Transformations Summary

A transformation is an operation that moves, flips, or otherwise changes a figure to create a new figure. A rigid transformation (also known as an isometry or congruence transformation) is a transformation that does not change the size or shape of a figure. The new figure created by a transformation is called the image. The original figure is called the preimage.

There are three rigid transformations: translations, reflections, and rotations. A translation is a transformation that moves every point in a figure the same distance in the same direction. A rotation is a transformation where a figure is turned around a fixed point to create an image. A reflection is a transformation that turns a figure into its mirror image by flipping it over a line.

##### Composition of Transformations

A composition of transformations is to perform more than one rigid transformation on a figure. One of the interesting things about compositions is that they can always be written as one rule. What this means is you don’t necessarily have to perform one transformation followed by the next. You can write a rule and perform them at the same time. You can compose any transformations, but here are some of the most common compositions.

1. Glide Reflection: a composition of a reflection and a translation. The translation is in a direction parallel to the line of reflection.
2. Reflections over Parallel Lines Theorem: If you compose two reflections over parallel lines that are h\begin{align*}h\end{align*} units apart, it is the same as a single translation of 2h\begin{align*}2h\end{align*} units. Be careful with this theorem. Notice, it does not say which direction the translation is in. So, to apply this theorem, you would still need to visualize, or even do, the reflections to see in which direction the translation would be.
3. Reflection over the Axes Theorem: If you compose two reflections over each axis, then the final image is a rotation of 180\begin{align*}180^\circ\end{align*} of the original. With this particular composition, order does not matter. Let’s look at the angle of intersection for these lines. We know that the axes are perpendicular, which means they intersect at a 90\begin{align*}90^\circ\end{align*} angle. The final answer was a rotation of 180\begin{align*}180^\circ\end{align*}, which is double 90\begin{align*}90^\circ\end{align*}. Therefore, we could say that the composition of the reflections over each axis is a rotation of double their angle of intersection.
4. Reflection over Intersecting Lines Theorem: If you compose two reflections over lines that intersect at x\begin{align*}x^\circ\end{align*}, then the resulting image is a rotation of 2x\begin{align*}2x^\circ\end{align*}, where the center of rotation is the point of intersection.

#### Reflecting and Translating

Reflect ABC\begin{align*}\triangle ABC\end{align*} over the y\begin{align*}y-\end{align*}axis and then translate the image 8 units down.

The green image below is the final answer.

A(8,8)B(2,4)C(10,2)A′′(8,0)B′′(2,4)C′′(10,6)\begin{align*}A(8,8) & \rightarrow A''(-8,0)\\ B(2,4) & \rightarrow B''(-2,-4)\\ C(10,2) & \rightarrow C''(-10,-6)\end{align*}

#### Writing a Transformation Rule

Write a single rule for ABC\begin{align*}\triangle ABC\end{align*} to A′′B′′C′′\begin{align*}\triangle A''B''C''\end{align*} from Example A.

Looking at the coordinates of A\begin{align*}A\end{align*} to A′′\begin{align*}A''\end{align*}, the x\begin{align*}x-\end{align*}value is the opposite sign and the y\begin{align*}y-\end{align*}value is y8\begin{align*}y - 8\end{align*}. Therefore the rule would be (x,y)(x,y8)\begin{align*}(x,y) \rightarrow (-x,y-8)\end{align*}.

Notice that this follows the rules we have learned in previous sections about a reflection over the y\begin{align*}y-\end{align*}axis and translations.

#### Reflecting a Triangle over Specific Points

Reflect ABC\begin{align*}\triangle ABC\end{align*} over y=3\begin{align*}y = 3\end{align*} and y=5\begin{align*}y = -5\end{align*}.

Unlike a glide reflection, order matters. Therefore, you would reflect over y=3\begin{align*}y = 3\end{align*} first, followed by a reflection of this image (red triangle) over y=5\begin{align*}y = -5\end{align*}. Your answer would be the green triangle in the graph below.

#### Reflecting Figures

Copy the figure below and reflect it over l\begin{align*}l\end{align*}, followed by m\begin{align*}m\end{align*}.

The easiest way to reflect the triangle is to fold your paper on each line of reflection and draw the image. It should look like this:

The green triangle would be the final answer.

#### Footprint Problem Revisited

The average 6 foot tall man has a 0.415×6=2.5\begin{align*}0.415 \times 6 = 2.5\end{align*} foot stride. Therefore, the transformation rule for this person would be (x,y)(x,y+2.5)\begin{align*}(x,y) \rightarrow (-x,y+2.5)\end{align*}.

### Examples

#### Example 1

DEF\begin{align*}\triangle DEF\end{align*} has vertices D(3,1),E(8,3),\begin{align*}D(3, -1), E(8, -3),\end{align*} and F(6,4)\begin{align*}F(6, 4)\end{align*}. Reflect DEF\begin{align*}\triangle DEF\end{align*} over x=5\begin{align*}x = -5\end{align*} and x=1\begin{align*}x = 1\end{align*}. This double reflection would be the same as which one translation?

From the Reflections over Parallel Lines Theorem, we know that this double reflection is going to be the same as a single translation of 2(1(5))\begin{align*}2(1 -(-5))\end{align*} or 12 units. Now, we need to determine if it is to the right or to the left. Because we first reflect over a line that is further away from DEF\begin{align*}\triangle DEF\end{align*}, to the left, D′′E′′F′′\begin{align*}\triangle D''E''F''\end{align*} will be on the right of DEF\begin{align*}\triangle DEF\end{align*}. So, it would be the same as a translation of 12 units to the right. If the lines of reflection were switched and we reflected the triangle over x=1\begin{align*}x = 1\end{align*} followed by x=5\begin{align*}x = -5\end{align*}, then it would have been the same as a translation of 12 units to the left.'

#### Example 2

Reflect DEF\begin{align*}\triangle DEF\end{align*} from #1 over the x\begin{align*}x-\end{align*}axis, followed by the y\begin{align*}y-\end{align*}axis. Determine the coordinates of D′′E′′F′′\begin{align*}\triangle D''E''F''\end{align*} and what one transformation this double reflection would be the same as.

D′′E′′F′′\begin{align*}\triangle D''E''F''\end{align*} is the green triangle in the graph below. If we compare the coordinates of it to DEF\begin{align*}\triangle DEF\end{align*}, we have:

D(3,1)E(8,3)F(6,4)D(3,1)E(8,3)F(6,4)\begin{align*}D(3,-1) & \rightarrow D'(-3,1)\\ E(8,-3) & \rightarrow E' (-8,3)\\ F(6,4) & \rightarrow F'(-6,-4)\end{align*}

If you recall the rules of rotations from the previous section, this is the same as a rotation of 180\begin{align*}180^\circ\end{align*}.

#### Example 3

Reflect the square over y=x\begin{align*}y = x\end{align*}, followed by a reflection over the \begin{align*}x-\end{align*}axis.

First, reflect the square over \begin{align*}y = x\end{align*}. The answer is the red square in the graph below. Second, reflect the red square over the \begin{align*}x-\end{align*}axis. The answer is the green square below.

#### Example 4

Determine the one rotation that is the same as the double reflection from #3.

Let’s use the theorem above. First, we need to figure out what the angle of intersection is for \begin{align*}y = x\end{align*} and the \begin{align*}x-\end{align*}axis. \begin{align*}y = x\end{align*} is halfway between the two axes, which are perpendicular, so is \begin{align*}45^\circ\end{align*} from the \begin{align*}x-\end{align*}axis. Therefore, the angle of rotation is \begin{align*}90^\circ\end{align*} clockwise or \begin{align*}270^\circ\end{align*} counterclockwise. The correct answer is \begin{align*}270^\circ\end{align*} counterclockwise because we always measure angle of rotation in the coordinate plane in a counterclockwise direction. From the diagram, we could have also said the two lines are \begin{align*}135^\circ\end{align*} apart, which is supplementary to \begin{align*}45^\circ\end{align*}.

### Review

1. What one transformation is equivalent to a reflection over two parallel lines?
2. What one transformation is equivalent to a reflection over two intersecting lines?

Use the graph of the square below to answer questions 3-6.

1. Perform a glide reflection over the \begin{align*}x-\end{align*}axis and to the right 6 units. Write the new coordinates.
2. What is the rule for this glide reflection?
3. What glide reflection would move the image back to the preimage?
4. Start over. Would the coordinates of a glide reflection where you move the square 6 units to the right and then reflect over the \begin{align*}x-\end{align*}axis be any different than #3? Why or why not?

Use the graph of the triangle below to answer questions 7-9.

1. Perform a glide reflection over the \begin{align*}y-\end{align*}axis and down 5 units. Write the new coordinates.
2. What is the rule for this glide reflection?
3. What glide reflection would move the image back to the preimage?

Use the graph of the triangle below to answer questions 10-14.

1. Reflect the preimage over \begin{align*}y = -1\end{align*} followed by \begin{align*}y = -7\end{align*}. Write the new coordinates.
2. What one transformation is this double reflection the same as?
3. What one translation would move the image back to the preimage?
4. Start over. Reflect the preimage over \begin{align*}y = -7\end{align*}, then \begin{align*}y = -1\end{align*}. How is this different from #10?
5. Write the rules for #10 and #13. How do they differ?

Fill in the blanks or answer the questions below.

1. Two parallel lines are 7 units apart. If you reflect a figure over both how far apart with the preimage and final image be?
2. After a double reflection over parallel lines, a preimage and its image are 28 units apart. How far apart are the parallel lines?
3. A double reflection over the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axes is the same as a _________ of ________.
4. What is the center of rotation for #17?
5. Two lines intersect at an \begin{align*}83^\circ\end{align*} angle. If a figure is reflected over both lines, how far apart will the preimage and image be?
6. A preimage and its image are \begin{align*}244^\circ\end{align*} apart. If the preimage was reflected over two intersected lines, at what angle did they intersect?
7. After a double reflection over parallel lines, a preimage and its image are 62 units apart. How far apart are the parallel lines?
8. A figure is to the left of \begin{align*}x = a\end{align*}. If it is reflected over \begin{align*}x = a\end{align*} followed by \begin{align*}x = b\end{align*} and \begin{align*}b > a\end{align*}, then the preimage and image are _________ units apart and the image is to the _________ of the preimage.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 12.6.

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### Vocabulary Language: English

Reflection

A reflection is a transformation that flips a figure on the coordinate plane across a given line without changing the shape or size of the figure.

Rotation

A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure.

Glide Reflection

A reflection followed by a translation where the line of reflection is parallel to the direction of translation is called a glide reflection or a walk.

Composite Transformation

A composite transformation, also known as composition of transformation, is a series of multiple transformations performed one after the other.

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