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# Composition of Transformations

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

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Composite Transformations

Look at the following diagram. It involves two translations. Identify the two translations of triangle \begin{align*}ABC\end{align*}.

### Watch This

First watch this video to learn about composite transformations.

Then watch this video to see some examples.

### Guidance

In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A composite transformation is when two or more transformations are performed on a figure (called the preimage) to produce a new figure (called the image).

#### Example A

Describe the transformations in the diagram below. The transformations involve a reflection and a rotation.

Solution: First line \begin{align*}AB\end{align*} is rotated about the origin by \begin{align*}90^\circ\end{align*}CCW.

Then the line \begin{align*}A^\prime B^\prime\end{align*} is reflected about the \begin{align*}y\end{align*}-axis to produce line \begin{align*}A^{\prime \prime}B^{\prime \prime}\end{align*}.

#### Example B

Describe the transformations in the diagram below.

Solution: The flag in diagram S is rotated about the origin \begin{align*}180^\circ\end{align*} to produce flag T. You know this because if you look at one point you notice that both \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinate points is multiplied by -1 which is consistent with a \begin{align*}180^\circ\end{align*} rotation about the origin. Flag T is then reflected about the line \begin{align*}x = -8\end{align*} to produce Flag U.

#### Example C

Triangle \begin{align*}ABC\end{align*} where the vertices of \begin{align*}\Delta ABC\end{align*} are \begin{align*}A(-1, -3)\end{align*}, \begin{align*}B(-4, -1)\end{align*}, and \begin{align*}C(-6, -4)\end{align*} undergoes a composition of transformations described as:

a) a translation 10 units to the right, then

b) a reflection in the \begin{align*}x\end{align*}-axis.

Draw the diagram to represent this composition of transformations. What are the vertices of the triangle after both transformations are applied?

Solution:

Triangle \begin{align*}A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}\end{align*} is the final triangle after all transformations are applied. It has vertices of \begin{align*}A^{\prime \prime}(9, 3)\end{align*}, \begin{align*}B^{\prime \prime}(6, 1)\end{align*}, and \begin{align*}C^{\prime \prime}(4, 4)\end{align*}.

#### Concept Problem Revisited

\begin{align*}\Delta ABC\end{align*} moves over 6 to the left and down 5 to produce \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}. Then \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*} moves over 14 to the right and up 3 to produce \begin{align*}\Delta A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}\end{align*}. These translations are represented by the blue arrows in the diagram.

All together \begin{align*}\Delta ABC\end{align*} moves over 8 to the right and down 2 to produce \begin{align*}\Delta A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}\end{align*}. The total translations for this movement are seen by the green arrow in the diagram above.

### Vocabulary

Image
In a transformation, the final figure is called the image.
Preimage
In a transformation, the original figure is called the preimage.
Transformation
A transformation is an operation that is performed on a shape that moves or changes it in some way. There are four types of transformations: translations, reflections, dilations and rotations.
Dilation
A dilation is a transformation that enlarges or reduces the size of a figure.
Translation
A translation is an example of a transformation that moves each point of a shape the same distance and in the same direction. Translations are also known as slides.
Rotation
A rotation is a transformation that rotates (turns) an image a certain amount about a certain point.
Reflection
A reflection is an example of a transformation that flips each point of a shape over the same line.
Composite Transformation
A composite transformation is when two or more transformations are combined to form a new image from the preimage.

### Guided Practice

1. Describe the transformations in the diagram below. The transformations involve a rotation and a reflection.

2. Triangle \begin{align*}XYZ\end{align*} has coordinates \begin{align*}X (1, 2)\end{align*}, \begin{align*}Y (-3, 6)\end{align*} and \begin{align*}Z (4, 5)\end{align*}.The triangle undergoes a rotation of 2 units to the right and 1 unit down to form triangle \begin{align*}X^\prime Y^\prime Z^\prime \end{align*}. Triangle \begin{align*}X^\prime Y^\prime Z^\prime\end{align*} is then reflected about the \begin{align*}y\end{align*}-axis to form triangle \begin{align*}X^{\prime \prime} Y^{\prime \prime} Z^{\prime \prime}\end{align*}. Draw the diagram of this composite transformation and determine the vertices for triangle \begin{align*}X^{\prime \prime} Y^{\prime \prime} Z^{\prime \prime}\end{align*}.

3. The coordinates of the vertices of \begin{align*}\Delta JAK\end{align*} are \begin{align*}J(1, 6)\end{align*}, \begin{align*}B(2, 9)\end{align*}, and \begin{align*}C(7, 10)\end{align*}.

a) Draw and label \begin{align*}\Delta JAK\end{align*}.

b) \begin{align*}\Delta JAK\end{align*} is reflected over the line \begin{align*}y=x\end{align*}. Graph and state the coordinates of \begin{align*}\Delta J^\prime A^\prime K^\prime\end{align*}.

c) \begin{align*}\Delta J^\prime A^\prime K^\prime\end{align*} is then reflected about the \begin{align*}x\end{align*}-axis. Graph and state the coordinates of \begin{align*}\Delta J^{\prime \prime} A^{\prime \prime} K^{\prime \prime}\end{align*}.

d) \begin{align*}\Delta J^{\prime \prime} A^{\prime \prime} K^{\prime \prime}\end{align*}undergoes a translation of 5 units to the left and 3 units up. Graph and state the coordinates of \begin{align*}\Delta J^{\prime \prime \prime} A^{\prime \prime \prime} K^{\prime \prime \prime}\end{align*}.

1. The transformations involve a reflection and a rotation. First line \begin{align*}AB\end{align*} is reflected about the \begin{align*}y\end{align*}-axis to produce line \begin{align*}A^\prime B^\prime\end{align*}.

Then the line \begin{align*}A^\prime B^\prime\end{align*} is rotated about the origin by \begin{align*}90^\circ\end{align*}CCW to produce line \begin{align*}A^{\prime \prime} B^{\prime \prime}\end{align*}.

2.

3.

### Practice

1. A point \begin{align*}X\end{align*} has coordinates (-1, -8). The point is reflected across the \begin{align*}y\end{align*}-axis to form \begin{align*}X^\prime\end{align*}. \begin{align*}X^\prime\end{align*} is translated over 4 to the right and up 6 to form \begin{align*}X^{\prime \prime}\end{align*}. What are the coordinates of \begin{align*}X^\prime\end{align*} and \begin{align*}X^{\prime \prime}\end{align*}?
2. A point \begin{align*}A\end{align*} has coordinates (2, -3). The point is translated over 3 to the left and up 5 to form \begin{align*}A^\prime\end{align*}. \begin{align*}A^\prime\end{align*} is reflected across the \begin{align*}x\end{align*}-axis to form \begin{align*}A^{\prime \prime}\end{align*}. What are the coordinates of \begin{align*}A^\prime\end{align*} and \begin{align*}A^{\prime \prime}\end{align*}?
3. A point \begin{align*}P\end{align*} has coordinates (5, -6). The point is reflected across the line \begin{align*}y = -x\end{align*} to form \begin{align*}P^\prime\end{align*}. \begin{align*}P^\prime\end{align*} is rotated about the origin \begin{align*}90^\circ\end{align*}CW to form \begin{align*}P^{\prime \prime}\end{align*}. What are the coordinates of \begin{align*}P^\prime\end{align*} and \begin{align*}P^{\prime \prime}\end{align*}?
4. Line \begin{align*}JT\end{align*} has coordinates \begin{align*}J(-2, -5)\end{align*} and \begin{align*}T(2, 3)\end{align*}. The segment is rotated about the origin \begin{align*}180^\circ\end{align*} to form \begin{align*}J^\prime T^\prime \end{align*}. \begin{align*}J^\prime T^\prime \end{align*} is translated over 6 to the right and down 3 to form \begin{align*}J^{\prime \prime} T^{\prime \prime} \end{align*}. What are the coordinates of \begin{align*}J^\prime T^\prime \end{align*} and \begin{align*}J^{\prime \prime} T^{\prime \prime} \end{align*}?
5. Line \begin{align*}SK\end{align*} has coordinates \begin{align*}S(-1, -8)\end{align*} and \begin{align*}K(1, 2)\end{align*}. The segment is translated over 3 to the right and up 3 to form \begin{align*}S^\prime K^\prime\end{align*}. \begin{align*}S^\prime K^\prime\end{align*} is rotated about the origin \begin{align*}90^\circ\end{align*}CCW to form \begin{align*}S^{\prime \prime} K^{\prime \prime}\end{align*}. What are the coordinates of \begin{align*}S^\prime K^\prime\end{align*} and \begin{align*}S^{\prime \prime} K^{\prime \prime}\end{align*}?
6. A point \begin{align*}K\end{align*} has coordinates (-1, 4). The point is reflected across the line \begin{align*}y=x\end{align*} to form \begin{align*}K^\prime\end{align*}. \begin{align*}K^\prime\end{align*} is rotated about the origin \begin{align*}270^\circ\end{align*}CW to form \begin{align*}K^{\prime \prime}\end{align*}. What are the coordinates of \begin{align*}K^\prime\end{align*} and \begin{align*}K^{\prime \prime}\end{align*}?

Describe the following composite transformations:

1. Explore what happens when you reflect a shape twice, over a pair of parallel lines. What one transformation could have been performed to achieve the same result?
2. Explore what happens when you reflect a shape twice, over a pair of intersecting lines. What one transformation could have been performed to achieve the same result?
3. Explore what happens when you reflect a shape over the x-axis and then the y-axis. What one transformation could have been performed to achieve the same result?
4. A composition of a reflection and a translation is often called a glide reflection. Make up an example of a glide reflection. Why do you think it's called a glide reflection?

### Vocabulary Language: English

Reflection

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

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

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

Composite Transformation

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