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Rigid Transformations

Identify transformations, translations, reflections and rotations.

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Rigid Transformations

Have you ever built a clubhouse? Take a look at this dilemma.

Cody and his Dad are working on building a clubhouse in his backyard. Cody is excited about the project and can’t wait to start building. He wrote to his penpal Trevor, in New Zealand, and told him all about the clubhouse. Trevor asked Cody to email him some pictures, and Cody said that he would. In fact, Cody told him that he would email them right away.

The only problem is that they haven’t started yet. Cody’s Dad has insisted that he draw a complete plan of the clubhouse. Cody did that and thought that they would be able to start, but then Cody’s Dad said that he wanted Cody to draw a plan of the backyard and where the clubhouse was going to be built. This made Cody a bit frustrated, but he decided to do it anyway in hopes that they could start building the clubhouse on the weekend. Then he could take pictures and email them to Trevor.

Cody decided to use a coordinate grid to map out the backyard. He drew the following plan and went to show his Dad.

Cody has created a scale where each unit on the coordinate grid represents 5 feet. His clubhouse dimensions are \begin{align*}14.1 \ ft \times 14.1 \ ft\end{align*}. The point on the grid at (4, -3) represents the back door of Cody’s house. His family has a large backyard, so the clubhouse is the perfect size.

Cody brings his drawing to his Dad and shows him the plan for the clubhouse.

“That looks great, except the clubhouse needs to move to the right 10 feet. Your Mom wants to plant her garden right where the clubhouse is now. I think if you move it over 10 feet the drawing will work,” Dad explains.

Cody gets back to work. How can he draw the clubhouse if he moves it over 10 feet to the right? This Concept is all about transformation. Using a transformation, Cody can redraw the clubhouse. We’ll come back to this problem at the end of this Concept to help Cody with his work.

Guidance

Figures can be transformed three different ways on the coordinate plane. Remember that the coordinate plane is represented by the coordinate grid. So when you are transforming figures on the coordinate plane, you are moving them on the coordinate grid.

What is a transformation?

A transformation is the movement of a figure on the coordinate grid.

Figures can be transformed in three different ways: a translation, a reflection or a rotation.

Translations

A translation is a slide of a figure. When a figure stays in the same position, and it is simply slid from one part of the coordinate grid to another part of the coordinate grid, we call it a translation.

Here is a translation.

You can see here that the figure was not changed at all. It simply slid from one point to another point.

Reflection

A reflection is a flip of a figure. We think of reflections when we think about a mirror. One half is like the other half, but they are reflected. When we reflect a figure on the coordinate grid, we flip it. Figures can be flipped over the \begin{align*}x\end{align*} axis or over the \begin{align*}y\end{align*} axis. Here is a reflection.

Rotations

A rotation is a turn. When you turn a figure on the coordinate grid, you are rotating the figure. We can rotate figures in terms of degrees. Let’s look at one.

Notice that the triangle was turned in each quadrant. Each turn is a rotation of the figure.

We can also graph transformations by using given vertices. Once you have graphed the figures, then you can identify whether you have a reflection, a rotation or a translation. Let’s begin.

Graph Figure \begin{align*}ABC, A(-1, 5) \ B(-1, 1) \ C(-3, 3)\end{align*}

Then graph figure \begin{align*}DEF, D(1, 5) \ E(1, 1) \ F(3, 3)\end{align*}

First, graph figure \begin{align*}ABC\end{align*}, then graph \begin{align*}DEF\end{align*} and compare the two figures. Here is a graph of the two figures.

In looking at these two figures, you can see that they represent a reflection.

We can describe translations by looking at how the \begin{align*}x\end{align*} and the \begin{align*}y\end{align*}coordinate have changed from one figure to the other corresponding figure.

Here we have two figures. We have triangle \begin{align*}ABC\end{align*} and we have triangle \begin{align*}A'B'C'\end{align*}. Triangle \begin{align*}ABC\end{align*} is the figure that we started with. We translated or slid the figure and created \begin{align*}A'B'C'\end{align*}.

We can describe this translation as the change in the \begin{align*}x\end{align*} value and as the change in the \begin{align*}y\end{align*} value.

We do that by writing the change in the coordinates of each vertex. This becomes an ordered pair of integers that expresses the translation.

Looking at the two figures, you can see that each vertex was moved +3 on the 'x' axis and -2 on the 'y' axis.

The ordered pair that expresses the change is (3, -2).

You can describe other transformations according to the change in \begin{align*}x\end{align*} and \begin{align*}y\end{align*} too.

In the last few problems, we identified equivalent or equal transformations using graphing and using the change of the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} coordinates. We can also identify equivalent transformations without using graphing. We can look at the coordinate changes and determine whether or not two figures are equivalent.

How can we do this?

That is a good question. Let’s look at the coordinates of two figures and determine whether or not the two figures are equivalent by examining the coordinate changes.

Figure \begin{align*}XYZ\end{align*} has the following vertices.

Figure \begin{align*}ABC\end{align*} has the following vertices.

Are these two figures equivalent?

To figure this out, we have to figure out the change in the \begin{align*}x\end{align*} coordinate from one vertex to the other and the change in the \begin{align*}y\end{align*} coordinate from one vertex to the other. If the change is the same for all three vertices, then the two figures are equivalent.

\begin{align*}X(4, 5)\end{align*} to \begin{align*}A(1, 7)\end{align*} From \begin{align*}x\end{align*} to \begin{align*}x\end{align*} is a change of -3, from \begin{align*}y\end{align*} to \begin{align*}y\end{align*} is change of +2. We can write this as ordered pair (-3, 2).

If \begin{align*}Y\end{align*} to \begin{align*}B\end{align*} and \begin{align*}Z\end{align*} to \begin{align*}C\end{align*} also have a change of (-3, 2), then the two figures are equivalent. Each vertex must have the same change or the figures are not equivalent.

\begin{align*}Y(2, 7)\end{align*} to \begin{align*}B (-1, 9)\end{align*} From \begin{align*}x\end{align*} to \begin{align*}x\end{align*} is a change of -3, from \begin{align*}y\end{align*} to \begin{align*}y\end{align*} is a change of +2. This vertex also has a change of (-3, 2).

\begin{align*}Z(3, 6)\end{align*} to \begin{align*}C (0, 8)\end{align*} from \begin{align*}x\end{align*} to \begin{align*}x\end{align*}, 3 to 0 is a change of -3, from \begin{align*}y\end{align*} to \begin{align*}y\end{align*}, 6 to 8 is a change of +2. This vertex also has a change of (-3, 2).

The change in each vertex is the same from the first figure to the second figure. Therefore, the two figures are equivalent.

Identify each example as a rotation, translation or reflection.

Example A

Solution: Rotation

Example B

Solution: Translation

Example C

Solution: Reflection

Now back to the clubhouse.

Cody and his Dad are working on building a clubhouse in his backyard. Cody is excited about the project and can’t wait to start building. He wrote to his penpal Trevor, in New Zealand, and told him all about the clubhouse. Trevor asked Cody to email him some pictures, and Cody said that he would. In fact, Cody told him that he would email them right away.

The only problem is that they haven’t started yet. Cody’s Dad has insisted that he draw a complete plan of the clubhouse. Cody did that and thought that they would be able to start, but then Cody’s Dad said that he wanted Cody to draw a plan of the backyard and where the clubhouse was going to be built. This made Cody a bit frustrated, but he decided to do it anyway in hopes that they could start building the clubhouse on the weekend. Then he could take pictures and email them to Trevor.

Cody decided to use a coordinate grid to map out the backyard. He drew the following plan and went to show his Dad.

Cody has created a scale where each unit on the coordinate grid represents 5 feet. His clubhouse dimensions are \begin{align*}14.1 \ ft \times 14.1 \ ft\end{align*}. The point on the grid at (4, -3) represents the back door of Cody’s house. His family has a large backyard, so the clubhouse is the perfect size.

Cody brings his drawing to his Dad and shows him the plan for the clubhouse.

“That looks great, except the clubhouse needs to move to the right 10 feet. Your Mom wants to plant her garden right where the clubhouse is now. I think if you move it over 10 feet the drawing will work,” Dad explains.

First, let’s think about which type of transformation Cody needs to move the clubhouse. If the clubhouse is going to move 10 feet to the right, Cody needs to slide the clubhouse over. A slide is another name for a translation.

To complete the translation, Cody needs to move each of the vertices of the clubhouse two units to the right. He needs to move them each two units because each unit is worth 5 feet and Cody’s Dad has told him to move the clubhouse 10 feet to the right. Here is Cody’s redesign of the clubhouse location.

The arrows show where Cody moved each vertex.

Cody shows his Dad the drawing. Cody’s Dad is pleased with Cody’s perfect translation. Together, the two of them begin to work on building the clubhouse.

Vocabulary

Here are the vocabulary words in this Concept.

Transformation
when a figure is moved in some way on the coordinate plane, the way that the figure is moved is called a transformation.
Translation
a slide, this when a figure slides on the coordinate plane from one place to the next
Reflection
the flip of a figure. Figures can be reflected over the \begin{align*}x\end{align*} axis or over the \begin{align*}y\end{align*} axis
Rotation
a turn. A figure can be turned in various directions on the coordinate plane.

Guided Practice

Here is one for you to try on your own.

Which type of transformation is shown below?

This is a reflection.

Here you can see that the figure was reflected or flipped over the \begin{align*}x\end{align*} axis.

Practice

Directions: Identify each image as a translation, rotation or reflection.

1.

2.

3.

4.

5.

Directions: For numbers 6 – 12 Draw your own figures, and demonstrate three different translations, three different rotations and three different reflections.

Directions: Complete the following reflections according to the directions.

13. Reflect this image across the \begin{align*}y\end{align*} axis.

14. Rotate the following image.

15 – 20 Go on a scavenger hunt and find examples of each of the different types of transformations in real world contexts.

Vocabulary Language: English

Angle

Angle

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

Translation

A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.
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.