# 8.17: Composite Transformations

- Page ID
- 6154

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Learn how to compose transformations of a figure on a coordinate plane, and understand the order in which to apply them.

### 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, rotations and reflections. 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 when more than one transformation is performed on a figure. Compositions can always be written as one rule. You can compose any transformations, but here are some of the most common compositions:

- A
**glide reflection**is a composition of a reflection and a translation. The translation is in a direction parallel to the line of reflection.

- The composition of two reflections over parallel lines that are \(h\) units apart is the same as a translation of \(2h\) units (
**Reflections over Parallel Lines Theorem**).

If you compose two reflections over each axis, then the final image is a rotation of \(180^{\circ}\) around the origin of the original (**Reflection over the Axes Theorem**).

A composition of two reflections over lines that intersect at \(x^{\circ}\) is the same as a rotation of \(2x^{\circ}\). The center of rotation is the point of intersection of the two lines of reflection (**Reflection over Intersecting Lines Theorem**).

What if you were given the coordinates of a quadrilateral and you were asked to reflect the quadrilateral and then translate it? What would its new coordinates be?

Example \(\PageIndex{1}\)

Reflect \(\Delta ABC\) over the \(y\)-axis and then translate the image 8 units down.

**Solution**

The green image to the left is the final answer.

\(\begin{aligned} A(8,8)&\rightarrow A′′(−8,0) \\ B(2,4)&\rightarrow B′′(−2,−4) \\ C(10,2)&\rightarrow C′′(−10,−6) \end{aligned}\)

Example \(\PageIndex{2}\)

Write a single rule for \(\Delta ABC\) to \(\Delta A′′B′′C′′\) from Example 1.

**Solution**

Looking at the coordinates of \(A\) to \(A′′\), the \(x\)−value is the opposite sign and the \(y\)−value is \(y−8\). Therefore the rule would be \((x,y)\rightarrow (−x,y−8)\).

Example \(\PageIndex{3}\)

Reflect \(\Delta ABC\) over \(y=3\) and then reflect the image over \(y=−5\).

**Solution**

Order matters, so you would reflect over \(y=3\) first, (red triangle) then reflect it over \(y=−5\) (green triangle).

Example \(\PageIndex{4}\)

A square is reflected over two lines that intersect at a \(79^{\circ}\) angle. What one transformation will this be the same as?

**Solution**

From the Reflection over Intersecting Lines Theorem, this is the same as a rotation of \(2\cdot 79^{\circ}\)=178^{\circ}\).

Example \(\PageIndex{5}\)

\(\Delta DEF\) has vertices \(D(3,−1)\), \(E(8,−3)\), and \(F(6,4)\). Reflect \(\Delta DEF\) over \(x=−5\) and then \(x=1\). Determine which one translation this double reflection would be the same as.

**Solution**

From the Reflections over Parallel Lines Theorem, we know that this double reflection is going to be the same as a single translation of \(2(1−(−5))\) or 12 units.

Example \(\PageIndex{6}\)

Reflect \(\Delta DEF\) from Question 2 over the \(x\)-axis, followed by the \(y\)-axis. Find the coordinates of \(\Delta D′′E′′F′′\) and the one transformation this double reflection is the same as.

**Solution**

\(\Delta D′′E′′F′′\) is the green triangle in the graph to the left. If we compare the coordinates of it to \(\Delta DEF\), we have:

\(\begin{aligned}D(3,−1)&\rightarrow D′′(−3,1) \\ E(8,−3)&\rightarrow E′′(−8,3) \\ F(6,4)&\rightarrow F′′(−6,−4)\end{aligned}\)

## Review

why the composition of two or more isometries must also be an isometry.*Explain*- What one transformation is the same as a reflection over two parallel lines?
- What one transformation is the same as a reflection over two intersecting lines?

Use the graph of the square to the left to answer questions 4-6.

- Perform a glide reflection over the \(x\)-axis and to the right 6 units. Write the new coordinates.
- What is the rule for this glide reflection?
- What glide reflection would move the image back to the preimage?

Use the graph of the square to the left to answer questions 7-9.

- Perform a glide reflection to the right 6 units, then over the \(x\)-axis. Write the new coordinates.
- What is the rule for this glide reflection?
- Is the rule in #8 different than the rule in #5? Why or why not?

Use the graph of the triangle to the left to answer questions 10-12.

- Perform a glide reflection over the \(y\)-axis and down 5 units. Write the new coordinates.
- What is the rule for this glide reflection?
- What glide reflection would move the image back to the preimage?

Use the graph of the triangle to the left to answer questions 13-15.

- Reflect the preimage over \(y=−1\) followed by \(y=−7\). Draw the new triangle.
- What one transformation is this double reflection the same as?
- Write the rule.

Use the graph of the triangle to the left to answer questions 16-18.

- Reflect the preimage over \(y=−7\)
by \(y=−1\). Draw the new triangle.*followed* - What one transformation is this double reflection the same as?
- Write the rule.
- How do the final triangles in #13 and #16 differ?

Use the trapezoid in the graph to the left to answer questions 20-22.

- Reflect the preimage over the \(x\)-axis then the \(y\)-axis. Draw the new trapezoid.
- Now, start over. Reflect the trapezoid over the \(y\)-axis then the \(x\)-axis. Draw this trapezoid.
- Are the final trapezoids from #20 and #21 different? Why do you think that is?

Answer the questions below. Be as specific as you can.

- Two parallel lines are 7 units apart. If you reflect a figure over both how far apart with the preimage and final image be?
- After a double reflection over parallel lines, a preimage and its image are 28 units apart. How far apart are the parallel lines?
- Two lines intersect at a \(165^{\circ}\) angle. If a figure is reflected over both lines, how far apart will the preimage and image be?
- What is the center of rotation for #25?
- Two lines intersect at an \(83^{\circ}\) angle. If a figure is reflected over both lines, how far apart will the preimage and image be?
- A preimage and its image are \(244^{\circ}\) apart. If the preimage was reflected over two intersecting lines, at what angle did they intersect?
- A preimage and its image are \(98^{\circ}\) apart. If the preimage was reflected over two intersecting lines, at what angle did they intersect?
- After a double reflection over parallel lines, a preimage and its image are 62 units apart. How far apart are the parallel lines?

## Review (Answers)

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

## Vocabulary

Term | Definition |
---|---|

composition (of transformations) |
When more than one transformation is performed on a 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. |

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. |

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. |

## Additional Resources

Interactive Element

Video: Composing Transformations Principles - Basic

Activities: Composition of Transformations Discussion Questions

Study Aids: Composition of Transformations Study Guide

Practice: Composite Transformations