### 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 of2h units (**Reflections over Parallel Lines Theorem**).

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

- A composition of two reflections over lines that intersect at
x∘ is the same as a rotation of2x∘ . 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?

### Examples

#### Example 1

Reflect

The green image to the left is the final answer.

#### Example 2

Write a single rule for

Looking at the coordinates of

#### Example 3

Reflect

Order matters, so you would reflect over

#### Example 4

A square is reflected over two lines that intersect at a

From the Reflection over Intersecting Lines Theorem, this is the same as a rotation of

#### Example 5

From the Reflections over Parallel Lines Theorem, we know that this double reflection is going to be the same as a single translation of

#### Example 6

Reflect

### Review

*Explain*why the composition of two or more isometries must also be an isometry.- 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 byy=−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 *followed*byy=−1 . Draw the new triangle. - 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 they− axis. Draw the new trapezoid. - Now, start over. Reflect the trapezoid over the
y− axis then thex− 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∘ 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∘ angle. If a figure is reflected over both lines, how far apart will the preimage and image be? - A preimage and its image are \begin{align*}244^\circ\end{align*}
244∘ apart. If the preimage was reflected over two intersecting lines, at what angle did they intersect? - A preimage and its image are \begin{align*}98^\circ\end{align*}
98∘ 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.