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# Rigid Transformations

## Identify transformations, translations, reflections and rotations.

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Rigid Transformations

In one of the rooms of the museum, this tile completely covered two of the walls. The students walked inside and began moving automatically because of the pattern on the wall.

“This tile makes me dizzy,” Greg stated sitting on a nearby bench.

“Yes, it seems to move,” Lane commented.

“I think it’s cool. Look at all of the transformations are everywhere,” Emma said smiling.

“Are you sure? You are beginning to sound like Mrs. Gilman,” Lane said jokingly.

“I am sure, and thank you for the compliment,” Emma said nudging her best friend.

Do you think Emma can see the transformations? Well, they are there. While sometimes a pattern like this one can make you dizzy, there are many different transformations in pattern. Your task is to find them. Use what you learn in this lesson to revisit the problem and find the examples of transformations in the tile pattern.

### Guidance

In this Concept, we will examine different kinds of transformations.

A transformation is the movement of a geometric figure.

There are three different kinds of transformations.

In a translation, also called a slide, the figure moves left, right, up, or down.

In a reflection, the figure flips. A reflection is like a mirror image of the original figure.

Finally, in a rotation, the figure turns.

The key thing to remember is that in any transformation, the size and shape of the figure stay exactly the same, only its location changes or shifts.

Now let’s look at each type of translation in more detail.

The first type of transformation is called a translation. It is also known as a slide because the figure in question does exactly that. It moves up, down, to the left or to the right. Nothing about the figure changes except its location.

Here are some translations.

Notice when you look at each picture that all that changed for each figure was its location. There are different colors used to show you the actual translation, but other than the location, each looks exactly the same.

This is how you always know that you are working with a translation or a slide.

How do we perform a translation?

To perform a translation, we measure a distance and then redraw the figure. For example, let’s move this triangle 6 inches.

We measure 6 inches from each point of the triangle and make a new point. This way, every part of the triangle moves 6 inches.

In this way, we can translate any figure in any direction for any distance.

Write down the definition for a translation in your notebook as well as how to perform a translation.

You have heard the word “reflection” all the time. From the reflection in a mirror to the reflection in a pond, reflections are all around us.

How do we apply the term reflection to geometry?

A reflection is a different kind of transformation. In a reflection, the figure flips to make a mirror image of itself. Take a look at the reflection below.

The line in the middle acts like a mirror. We call this the line of symmetry. This is a vertical line of symmetry. Imagine standing in front of a mirror and holding up your left hand. Where is your hand in the mirror’s reflection? A reflected figure works the same way: when we flip it over the line, all of its points are reversed. When reflected, the figure above looks like a backwards \begin{align*}L\end{align*}. Notice that, on both sides of the line, the dot is closest to the line.

We can also reflect figures across a horizontal line of symmetry. Then our reflection would look like this.

In this case, the “top” of the figure becomes the “bottom” in the reflection! Notice, however, that in both cases the figures are symmetrical.

Write down the definition of a reflection and line of symmetry in your notebook.

Now let’s learn about the third kind of transformation. A rotation is a transformation that turns the figure in either a clockwise or counterclockwise direction.

How does the figure below change as it is rotated?

Imagine you could spin the figure around in a circle. It would not change, but might turn upside down. Figures can rotate as much as \begin{align*}360^{\circ}\end{align*}, a full circle. Let’s see how that might look.

When we rotate the figure a full \begin{align*}360^{\circ}\end{align*}, it ends up in the same place it began, unchanged!

Let’s identify some transformations.

#### Example A

Solution: Reflection

#### Example B

A transformation with a line of symmetry must be _______________?

Solution: Reflection

#### Example C

Solution: Reflection

Here is the original problem once again.

In one of the rooms of the museum, this tile completely covered two of the walls. The students walked inside and began moving automatically because of the pattern on the wall.

“This tile makes me dizzy,” Greg stated sitting on a nearby bench.

“Yes, it seems to move,” Lane commented.

“I think it’s cool. Look at all of the transformations are everywhere,” Emma said smiling.

“Are you sure? You are beginning to sound like Mrs. Gilman,” Lane said jokingly.

“I am sure, and thank you for the compliment,” Emma said nudging her best friend.

In this tile pattern, make a note of a reflection. Make a note of a translation. Make a note of a figure that rotates or has rotational symmetry. Make a note of the line of symmetry.

While you won’t find the exact answer here, there are many ways to explore transformations and symmetry by using this pattern. Work with a partner or with your whole class to figure out the math in this tile pattern.

### Vocabulary

Transformation
when a figure moved on a plane. The figure doesn’t change but it’s position does.
Translation
a slide. The figure moved up, down, to the right, or to the left or diagonally.
Reflection
a flip. The figure flips over a line of symmetry, much like a reflection in a mirror.
Rotation
a turn. The figure turns either clockwise or counterclockwise.
Line of Symmetry
the line that a figure reflects over. Also the line that divided a figure in half showing line symmetry.

### Guided Practice

Here is one for you to try on your own.

Could this figure be a rotation? Why or why not?

This figure is congruent in each and every way. This figure could be a rotation.

### Practice

Directions: Identify the transformations shown below as translations, reflections, or rotations.

Directions: Tell whether the figures below have line symmetry, rotational symmetry, both, or neither.

Directions: Draw the second half of each figure.

### Vocabulary Language: English

Angle

Angle

A geometric figure formed by two rays that connect at a single point or vertex.
Line of Symmetry

Line of Symmetry

A line of symmetry is a line that can be drawn to divide a figure into equal halves.
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.
Reflections

Reflections

Reflections are transformations that result in a "mirror image" of a parent function. They are caused by differing signs between parent and child functions.
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.
Transformation

Transformation

A transformation moves a figure in some way on the coordinate plane.
Transformations

Transformations

Transformations are used to change the graph of a parent function into the graph of a more complex function.
Rigid Transformation

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.