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? After completing this Concept, you'll be able to perform a series of transformations on a figure like this one in the coordinate plane.

### Watch This

Composing Transformations CK-12

### Guidance

##### 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:

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

2) The composition of two reflections over parallel lines that are \begin{align*}h\end{align*}**Reflections over Parallel Lines Theorem**).

3) If you compose two reflections over each axis, then the final image is a rotation of \begin{align*}180^\circ\end{align*}**Reflection over the Axes Theorem**).

4) A composition of two reflections over lines that intersect at \begin{align*}x^\circ\end{align*}**Reflection over Intersecting Lines Theorem**).

#### Example A

Reflect \begin{align*}\triangle ABC\end{align*}

The green image to the right is the final answer.

\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*}

#### Example B

Write a single rule for \begin{align*}\triangle ABC\end{align*}

Looking at the coordinates of \begin{align*}A\end{align*}

#### Example C

Reflect \begin{align*}\triangle ABC\end{align*}

Order matters, so you would reflect over \begin{align*}y = 3\end{align*}

#### Example D

A square is reflected over two lines that intersect at a \begin{align*}79^\circ\end{align*}

From the Reflection over Intersecting Lines Theorem, this is the same as a rotation of \begin{align*}2 \cdot 79^\circ=178^\circ\end{align*}

Composing Transformations CK-12

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### Guided Practice

1. Write a single rule for \begin{align*}\triangle ABC\end{align*}

2. \begin{align*}\triangle DEF\end{align*}

3. Reflect \begin{align*}\triangle DEF\end{align*}

4. Copy the figure below and reflect the triangle over \begin{align*}l\end{align*}

**Answers:**

1. In the graph, the two lines are 8 units apart \begin{align*}(3-(-5)=8)\end{align*}** double** reflection is the same as a translation that is

**the distance between the parallel lines. \begin{align*}(x,y) \rightarrow (x,y-16)\end{align*}**

*double*2. From the Reflections over Parallel Lines Theorem, we know that this double reflection is going to be the same as a single translation of \begin{align*}2(1 -(-5))\end{align*}

3. \begin{align*}\triangle D''E''F''\end{align*}

\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*}

4. The easiest way to reflect the triangle is to fold your paper on each line of reflection and draw the image. The final result should look like this (the green triangle is the final answer):

### Explore More

*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 \begin{align*}x-\end{align*}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 \begin{align*}x-\end{align*}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 \begin{align*}y-\end{align*}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 \begin{align*}y = -1\end{align*} followed by \begin{align*}y = -7\end{align*}. 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 \begin{align*}y = -7\end{align*}
*followed*by \begin{align*}y = -1\end{align*}. 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 \begin{align*}x-\end{align*}axis then the \begin{align*}y-\end{align*}axis. Draw the new trapezoid.
- Now, start over. Reflect the trapezoid over the \begin{align*}y-\end{align*}axis then the \begin{align*}x-\end{align*}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 \begin{align*}165^\circ\end{align*} 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 \begin{align*}83^\circ\end{align*} 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*} 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*} 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?

### Answers for Explore More Problems

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