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# Rotations

## Transformations by which a figure is turned around a fixed point to create an image.

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Practice Rotations
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Recognizing Rotation Transformations

Have you ever seen something rotate? Do you know what it takes to classify a rotation transformation? How about rotational symmetry? Take a look at this dilemma.

Does this figure have rotational symmetry?

To answer this question, you will need to know about rotations and rotational symmetry. This Concept will teach you all you need to know to answer this question successfully.

### Guidance

A transformation is the movement of a geometric figure on the coordinate plane.

A rotation is a turn. A figure can be turned clockwise or counterclockwise on the coordinate plane. In both transformations the size and shape of the figure stays exactly the same.

Let's look at rotations or turns.

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

We can turn a figure 90\begin{align*}90^\circ\end{align*}, a quarter turn, either clockwise or counterclockwise. When we spin the figure exactly halfway, we have rotated it 180\begin{align*}180^\circ\end{align*}. Turning it all the way around rotates the figure 360\begin{align*}360^\circ\end{align*}.

Now if you look at these two triangles, you can see that one has been turned a quarter turn clockwise. If we talk about that turn or rotation in mathematical language, we can describe the turn as 90\begin{align*}90^\circ\end{align*} clockwise. We could also turn it 180\begin{align*}180^\circ\end{align*}, which would show a triangle completed upside down.

Next, let’s look at rotating figures on the coordinate plane.

Rotate this figure 90\begin{align*}90^\circ\end{align*} clockwise on the coordinate plane.

First, let’s write down the coordinates for each of the points of this pentagon.

A(3,5)B(4,4)C(3,3)D(1,2)E(1,4)

Now we have the points. The easiest way to think about rotating any figure is to think about it moving around a fixed point. In the case of graphing figures on the coordinate plane, we will be rotating the figures around the center point or origin.

If we rotate a figure clockwise 90\begin{align*}90^\circ\end{align*}, then we are going to be shifting the whole figure along the x\begin{align*}x-\end{align*}axis. To figure out the coordinates of the new rotated figure, we switch the coordinates and then, we need to multiply the second coordinate by -1. This will make perfect sense given that the entire figure is going to shift.

Let’s apply this to the coordinates above.

Now we can graph this rotated figure. Notice that we use \begin{align*}A^\prime\end{align*} to represent the rotated figure. Here is the graph of this rotation.

That is a great question. Let’s think about what would happen to the figure if we were to rotate it counterclockwise. To do this, the figure would move across the \begin{align*}y-\end{align*}axis in fact, the \begin{align*}x-\end{align*}coordinates would change completely. In actuality, we would switch the original coordinates around. The \begin{align*}x-\end{align*}coordinate would become the \begin{align*}y-\end{align*}coordinate and the \begin{align*}y-\end{align*}coordinate would become the \begin{align*}x-\end{align*}coordinate. Then, we need to multiply the new \begin{align*}x-\end{align*}coordinate by -1.

Here is what that would look like.

Now we can graph this new rotation.

We can also graph figures that have been rotated \begin{align*}180^\circ\end{align*} too. To do this, we multiply both coordinates of the original figure by -1.

Let’s see what this looks like.

Now we can graph this image.

Write the three ways to figure out new coordinates for rotating \begin{align*}90^\circ\end{align*} clockwise and counterclockwise and for rotating a figure \begin{align*}180^\circ\end{align*}. Put these notes in your notebook.

Now let’s think about symmetry and rotations.

We can call this rotational symmetry.

A figure has rotational symmetry if, when we rotate it, the figure appears to stay the same. The outlines do not change even as the figure turns.

Look at the figure below.

Look at this image. The star will look the same even if we rotate it. We could turn it \begin{align*}72^\circ\end{align*} or \begin{align*}144^\circ\end{align*} clockwise or counterclockwise. It won’t matter. The star will still appear the same.

Do the following figure have rotational symmetry?

#### Example A

A square.

Solution: Yes, because you can turn it and it will appear exactly the same.

#### Example B

The letter U.

Solution: No, it will not appear the same if it is turned.

#### Example C

An octagon.

Solution: Yes, because you can turn it and it will appear exactly the same.

Now let's go back to the dilemma from the beginning of the Concept.

Does this figure have rotational symmetry?

While the outline of this image has rotational symmetry, the design inside prevents it from having rotational symmetry. If we turn the circle, then the design inside will change. Therefore this image does not have rotational symmetry.

### Vocabulary

Transformation
moving a geometric figure on the coordinate plane.
Coordinate Notation
using ordered pairs to represent the vertices of a figure that has been graphed on the coordinate plane.
Reflection
A flip of a figure on the coordinate plane.
Translation
A slide – when a figure moves up, down, left or right on the coordinate plane, but does not change position.
Rotation
A turn – when a figure is turned \begin{align*}90^\circ, 180^\circ\end{align*} on the coordinate plane.
Rotational Symmetry
when a figure can be rotated but appears exactly the same no matter how you rotate it.

### Guided Practice

Here is one for you to try on your own.

Does a hexagon have rotational symmetry?

Solution

It has rotational symmetry. You can see that because we can rotate it \begin{align*}90^\circ\end{align*} and \begin{align*}180^\circ\end{align*} and it will still look exactly the same. We could rotate it less than \begin{align*}90^\circ\end{align*} too and it still has rotational symmetry. We can also look at the angles to determine rotational symmetry. Each time we turn the figure, it has two parallel sides on the top and bottom and four other sides at the same angles. It has rotational symmetry.

### Practice

1. What is a translation?
2. What is a rotation?
3. What is a tessellation?
4. True or false. A figure can be translated up or down only.
5. True or false. A figure can be translated \begin{align*}180^\circ\end{align*}.
6. True or false. A figure can be rotated \begin{align*}90^\circ\end{align*} clockwise or counterclockwise.
7. True or false. A figure can’t be translated \begin{align*}180^\circ\end{align*}.
8. When rotating a figure \begin{align*}90^\circ\end{align*} counterclockwise, we switch the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} coordinates and multiply which one by -1?
9. When rotating a figure \begin{align*}90^\circ\end{align*} clockwise, we multiply which coordinate by -1?
10. True or false. When rotating a figure \begin{align*}180^\circ\end{align*}, we multiply both coordinates by -1.

Directions: Write the new coordinates for each rotation given the directions.

A Triangle with the coordinates (-4, 4) (-4, 2) and (-1, 1)

1. Rotate the figure \begin{align*}90^\circ\end{align*} clockwise
2. Rotate the figure \begin{align*}90^\circ\end{align*} counterclockwise
3. Rotate the figure \begin{align*}180^\circ\end{align*}

A Triangle with the coordinates (1, 3) (5, 1) (5, 3)

1. Rotate the figure clockwise \begin{align*}90^\circ\end{align*}
2. Rotate the figure counterclockwise \begin{align*}90^\circ\end{align*}
3. Rotate the figure \begin{align*}180^\circ\end{align*}

### Vocabulary Language: English

Center of Rotation

Center of Rotation

In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point.
Coordinate Notation

Coordinate Notation

A coordinate point is the description of a location on the coordinate plane. Coordinate points are commonly written in the form (x, y) where x is the horizontal distance from the origin, and y is the vertical distance from the origin.
Image

Image

The image is the final appearance of a figure after a transformation operation.
Origin

Origin

The origin is the point of intersection of the $x$ and $y$ axes on the Cartesian plane. The coordinates of the origin are (0, 0).
Preimage

Preimage

The pre-image is the original appearance of a figure in a transformation operation.
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.
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.
Rotational Symmetry

Rotational Symmetry

A figure has rotational symmetry if it can be rotated less than $360^\circ$ around its center point and look exactly the same as it did before the rotation.
Transformation

Transformation

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