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# Rotations

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

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Rotations

Which one of the following figures represents a rotation? Explain.

### Rotations

In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A rotation is an example of a transformation where a figure is rotated about a specific point (called the center of rotation), a certain number of degrees. The figure below shows that the Preimage A has been rotated 90\begin{align*}90^\circ\end{align*} about point A\begin{align*}A\end{align*} to form the rotated image. Point A\begin{align*}A\end{align*} is the center of rotation.

In order to describe a rotation, you need to state how many degrees the preimage rotated, the center of rotation, and the direction of the rotation (clockwise or counterclockwise). The most common center of rotation is the origin. The table below shows what happens to points when they have undergone a rotation about the origin. The angles are given as counterclockwise.

Center of Rotation Angle of Rotation Preimage (Point P\begin{align*}P\end{align*}) Rotated Image (Point P\begin{align*}P^\prime\end{align*})
(0, 0) 90\begin{align*}90^\circ\end{align*} (or 270\begin{align*}-270^\circ\end{align*}) (x,y)\begin{align*}(x, y)\end{align*} (y,x)\begin{align*}(-y, x)\end{align*}
(0, 0) 180\begin{align*}180^\circ\end{align*} (or 180\begin{align*}-180^\circ\end{align*}) (x,y)\begin{align*}(x, y)\end{align*} (x,y)\begin{align*}(-x, -y)\end{align*}
(0, 0) 270\begin{align*}270^\circ\end{align*} (or 90\begin{align*}-90^\circ\end{align*}) (x,y)\begin{align*}(x, y)\end{align*} (y,x)\begin{align*}(y, -x)\end{align*}

#### Let's describe the following rotations:

1. The rotation of the blue triangle in the diagram below.

Looking at the angle measures, ABA=90\begin{align*}\angle ABA^\prime=90^\circ\end{align*}. Therefore the preimage, Image A, has been rotated 90\begin{align*}90^\circ\end{align*} counterclockwise about the point B\begin{align*}B\end{align*}.

1. Image A to image B

Looking at the angle measures, CAB+BAC=180\begin{align*}\angle CAB^\prime + \angle B^\prime AC^\prime=180^\circ\end{align*}. The triangle ABC\begin{align*}ABC\end{align*} has been rotated 180\begin{align*}180^\circ\end{align*}CCW about the center of rotation Point A\begin{align*}A\end{align*}.

1. The rotation in the diagram below.

To describe the rotation in this diagram, look at the points indicated on the S shape.

Points BC\begin{align*}BC\end{align*}: B(3,4)\begin{align*}B(-3, 4)\end{align*} C(5,0)\begin{align*}C(-5, 0)\end{align*}

Points BC\begin{align*}B^\prime C^\prime\end{align*}: B(4,3)\begin{align*}B^\prime (4, 3)\end{align*} C(0,5)\begin{align*}C^\prime (0, 5)\end{align*}

These points represent a rotation of 90\begin{align*}90^\circ\end{align*} clockwise about the origin. Each coordinate point (x,y)\begin{align*}(x, y)\end{align*} has become the point (y,x)\begin{align*}(y, -x)\end{align*}.

### Examples

#### Example 1

Earlier, you were shown following three figures and asked which one of the figures represents a rotation.

You know that a rotation is a transformation that turns a figure about a fixed point. This fixed point is the turn center or the center of rotation. In the figures above, Figure 1 and Figure 3 involve turning the heart about a fixed point. Figure 1 rotates the heart about the point A\begin{align*}A\end{align*}. Figure 3 rotates the heart about the point directly to the right of A\begin{align*}A\end{align*}. Figure 2 does a translation, not a rotation.

#### Example 2

Describe the rotation of the pink triangle in the diagram below.

Examine the points of the preimage and the rotated image (the blue triangle).

Points on BCD\begin{align*}BCD\end{align*} B(1,1)\begin{align*}B(1, -1)\end{align*} C(2,6)\begin{align*}C(2, 6)\end{align*} D(5,1)\begin{align*}D(5, 1)\end{align*}
Points on BCD\begin{align*}B^\prime C^\prime D^\prime\end{align*} B(1,1)\begin{align*}B^\prime (1, 1)\end{align*} C(6,2)\begin{align*}C^\prime (-6, 2)\end{align*} D(1,5)\begin{align*}D^\prime (-1, 5)\end{align*}

These points represent a rotation of 90\begin{align*}90^\circ\end{align*}CW about the origin. Each coordinate point (x,y)\begin{align*}(x, y)\end{align*} has become the point (y,x)\begin{align*}(-y, x)\end{align*}.

#### Example 3

Describe the rotation of the blue polygon in the diagram below.

For this image, look at the rotation. It is not rotated about the origin but rather about the point A\begin{align*}A\end{align*}. We can measure the angle of rotation:

The blue polygon is being rotated about the point A 145\begin{align*}A \ 145^\circ\end{align*} clockwise. You would say that the blue polygon is rotated 145\begin{align*}145^\circ\end{align*}CW to form the orange polygon.

#### Example 4

Describe the rotation of the green hexagon in the diagram below.

For this image, look at the rotation. It is not rotated about the origin but rather about the point A\begin{align*}A\end{align*}. We can measure the angle of rotation:

The green polygon is being rotated about the point D 90\begin{align*}D \ 90^\circ\end{align*} clockwise. You would say that the green hexagon is rotated 90\begin{align*}90^\circ\end{align*}CW to form the orange hexagon.

### Review

If the following points were rotated about the origin with a 180\begin{align*}180^\circ\end{align*}CCW rotation, what would be the coordinates of the rotated points?

1. (3, 1)
2. (4, -2)
3. (-5, 3)
4. (-6, 4)
5. (-3, -3)

If the following points were rotated about the origin with a 90\begin{align*}90^\circ\end{align*}CW rotation, what would be the coordinates of the rotated points?

1. (-4, 3)
2. (5, -4)
3. (-5, -4)
4. (3, 3)
5. (-8, -9)

Describe the following rotations:

1. Why is it not necessary to specify the direction when rotating 180\begin{align*}180^\circ\end{align*}?

To see the Review answers, open this PDF file and look for section 10.7.

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### Vocabulary Language: English

TermDefinition
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.
Image The image is the final appearance of a figure after a transformation operation.
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 The pre-image is the original appearance of a figure in a transformation operation.
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.
Rigid Transformation A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.