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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

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. Why do you think this is?

#### Guidance

A composite transformation (or composition of transformations) is two or more transformations performed one after the other. Sometimes, a composition of transformations is equivalent to a single transformation. The following is an example of a translation followed by a reflection. The original triangle is the brown triangle and the image is the blue striped triangle. The brown striped triangle shows the intermediate step after the translation has taken place.

There is no single transformation that could have replaced the composite transformation above.

Example A

Rotate the rectangle 90\begin{align*}90^\circ\end{align*} counterclockwise about the origin and then translate it along vector v\begin{align*}\vec{v}\end{align*}.

Solution:

Example B

Perform the transformations from Example A in the other order (translation then rotation). How do the final images compare?

Solution: Here are the transformations performed in the opposite order:

The final images are NOT in the same place. This means that transformations are not commutative. The order that transformations are performed matters.

Example C

Describe a possible sequence of transformations that would carry ΔABC\begin{align*}\Delta ABC\end{align*} to ΔDEF\begin{align*}\Delta DEF\end{align*}.

Solution: There is more than one possible answer. This could be a 180\begin{align*}180^\circ\end{align*} rotation about a point directly in between points C\begin{align*}C\end{align*} and D\begin{align*}D\end{align*}

This could also be a reflection across AC\begin{align*}AC\end{align*} followed by reflection across a vertical line.

Another possibility is that ΔABC\begin{align*}\Delta ABC\end{align*} was rotated 180\begin{align*}180^\circ\end{align*} about point C\begin{align*}C\end{align*} and then translated to ΔDEF\begin{align*}\Delta DEF\end{align*}.

These are only three possible descriptions of the transformation. Can you think of another?

Concept Problem Revisited

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.

This is because it's as if the shape was reflected and then glided over to a new location. When done repeatedly, the shapes look like footsteps walking.

#### Vocabulary

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.

A composite transformation or composition of transformations is multiple transformations performed one after the other.

#### Guided Practice

1. Copy the triangle onto graph paper or into geometry software such as Geogebra.

2. Reflect the triangle across AC\begin{align*}AC\end{align*} and then across the line perpendicular to AC¯¯¯¯¯\begin{align*}\overline{AC}\end{align*} through C\begin{align*}C\end{align*}.

3. What one transformation could you have performed to get the same result?

1.

2.

3. A 180\begin{align*}180^\circ\end{align*} rotation about point C\begin{align*}C\end{align*} would have produced the same result.

#### Practice

1. What is a composite transformation?

2. When doing a composite transformation, does the order in which you perform the transformations matter?

3. Describe a possible sequence of transformations that would carry ΔABC\begin{align*}\Delta ABC\end{align*} to ΔDEF\begin{align*}\Delta DEF\end{align*}.

4. Describe another possible sequence of transformations that would carry ΔABC\begin{align*}\Delta ABC\end{align*} to ΔDEF\begin{align*}\Delta DEF\end{align*}.

5. Describe a possible sequence of transformations that would carry ΔGHI\begin{align*}\Delta GHI\end{align*} to ΔJKL\begin{align*}\Delta JKL\end{align*}.

6. Describe another possible sequence of transformations that would carry ΔGHI\begin{align*}\Delta GHI\end{align*} to ΔJKL\begin{align*}\Delta JKL\end{align*}.

7. Describe a possible sequence of transformations that would carry ΔMNO\begin{align*}\Delta MNO\end{align*} to ΔPQR\begin{align*}\Delta PQR\end{align*}.

8. Describe another possible sequence of transformations that would carry ΔMNO\begin{align*}\Delta MNO\end{align*} to ΔPQR\begin{align*}\Delta PQR\end{align*}.

9. Construct a polygon on graph paper or with Geogebra.

10. Reflect the polygon twice across parallel lines. What one transformation could you have performed to get the same result?

11. Reflect the polygon twice across another set of parallel lines. What one transformation could you have performed to get the same result?

12. Make a conjecture by completing the sentence. Two reflections across parallel lines is the same as a ______________.

13. Reflect the polygon twice across intersecting lines (not necessarily perpendicular). What one transformation could you have performed to get the same result?

14. Reflect the polygon twice across intersecting lines (not necessarily perpendicular). What one transformation could you have performed to get the same result?

15. Make a conjecture by completing the sentence. Two reflections across intersecting lines is the same as a ______________.

### 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.