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8.16: Rigid Transformations

Difficulty Level: At Grade Created by: CK-12
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Melvin is an artist. He enjoys drawing translations, reflections and rotations of shapes. He took a picture of a water lily and he has decided that he is going to change the picture when he does the painting. He wants to paint the water lily 6 inches to the right. If the pictures are held next to each other, what type of transformation will his painting appear to be?

In this concept, you will learn about transformations.

Transformations

A transformation is the movement of a geometric figure. There are three different kinds of transformations

In any transformation, the size and shape of the figure stay exactly the same, only its location changes or shifts.

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 a few examples of translations.

Notice when you look at each picture that all that changed for each figure is 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.

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

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

The second type of transformation is a reflection. 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. This is the line of symmetry. 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.

You can also reflect figures across a horizontal line of symmetry. The 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.

The third type of transformation is a rotation. A rotation is a transformation that turns the figure in either a clockwise or counterclockwise direction.

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*}. Let’s see how that might look.

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

Examples

Example 1

Earlier, you were given a problem about Melvin and his painting.

He wants to paint the water lily 6 inches to the right, which slides the original image. If the pictures are held next to each other, what type of transformation will his painting appear to be?

First, reference the original image to see how the shape moved.

The shape appears to have slid to the right.

Next, review the different type of transformations.

Translation, reflection, rotation.

Then, determine the type of transformation that slides a figure.

Translation.

Example 2

Solve the following problem.

If the figure is turned \begin{align*}90^o\end{align*}, what type of transformation will that be?

First, note the description of the movement.

The figure is turned \begin{align*}90^o\end{align*}.

Next, review the different types of transformations.

Translation, reflection, rotation.

Then, determine the type of transformation that turns a figure.

Rotation.

The answer is that it is a rotation.

Example 3

Name the transformation.

First, reference the dots to see how the shape moved.

The shape appears to have flipped over a vertical line.

Next, review the different type of transformations.

Translation, reflection, rotation.

Then, determine the type of transformation that flips a figure.

Reflection.

The answer is that it is a reflection.

Example 4

What type of transformation has a line of symmetry?

First, review the different types of transformation.

Translation, reflection, rotation.

Next, list what each transformation does.

Slide, flip, turn.

Then, determine the type of transformation that flips over a line of symmetry.

Reflection.

Example 5

Identify the transformation shown below as a translation, reflection, or rotation.

First, reference the dots to see how the shape moved.

The shape appears to have turned.

Next, review the different type of transformations.

Translation, reflection, rotation.

Then, determine the type of transformation that turns a figure.

Rotation.

Review

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

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

Draw the second half of each figure.

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.16.

Resources

 

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Vocabulary

Angle

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

Line of Symmetry

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

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

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

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

Transformations

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

Rigid Transformation

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

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Dec 02, 2015
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Sep 08, 2016
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