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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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Rotations

You want your friend to rotate the pentagon 90\begin{align*}90^\circ\end{align*}. What are the two other pieces of information you need to tell your friend to ensure that she rotates the pentagon the way you are imagining?

#### Guidance

A rotation is one example of a rigid transformation. A rotation of t\begin{align*}t^\circ\end{align*} about a given point O\begin{align*}O\end{align*} takes each point on a shape P\begin{align*}P\end{align*} and moves it to P\begin{align*}P^\prime\end{align*} such that P\begin{align*}P^\prime\end{align*} is on the circle with center O\begin{align*}O\end{align*} and radius OP¯¯¯¯¯\begin{align*}\overline{OP}\end{align*} and POP=t\begin{align*}\angle P^\prime OP=t^\circ\end{align*}. Below, the parallelogram has been rotated 90\begin{align*}90^\circ\end{align*} counterclockwise about point O\begin{align*}O\end{align*} to create a new parallelogram (the image).

Keep in mind that the location of point O\begin{align*}O\end{align*} matters. The parallelogram below was also rotated 90\begin{align*}90^\circ\end{align*} counterclockwise about point O\begin{align*}O\end{align*}, but point O\begin{align*}O\end{align*} is in a different location.

There is a circle with center O\begin{align*}O\end{align*} that passes through each pair of corresponding points. You can see the angles are 90\begin{align*}90^\circ\end{align*} between each pair of corresponding points through point O\begin{align*}O\end{align*}. Below, you can see this connection for C\begin{align*}C\end{align*} and C\begin{align*}C^\prime\end{align*}.

And here is that connection for \begin{align*}D\end{align*} and \begin{align*}D^\prime\end{align*}.

When describing a rotation, it is important to consider three pieces of information. First, the center of rotation. This is the point about which each point turns (the center of each circle passing through corresponding points). Second, the direction of rotation. You can rotate clockwise or counterclockwise. Third, the degree of rotation. This tells you how much you are turning. Remember that a full circle \begin{align*}360^\circ\end{align*}.

Example A

Describe the rotation below.

Solution: This is a \begin{align*}135^\circ\end{align*} counterclockwise rotation about point \begin{align*}O\end{align*}. You could also say this is a \begin{align*}225^\circ\end{align*} clockwise rotation about point \begin{align*}O\end{align*}. Once the rotation has taken place, there is no way to know the direction of rotation.

Example B

Is the following transformation a rotation?

Solution: No, while \begin{align*}\angle DOD^\prime\end{align*} is \begin{align*}180^\circ\end{align*}, the angles connecting other pairs of points and point \begin{align*}O\end{align*} are not \begin{align*}180^\circ\end{align*}.

This transformation is a reflection across a line through point \begin{align*}O\end{align*}, perpendicular to \begin{align*}\overline{DD^\prime}\end{align*}.

Example C

Rotate the rectangle \begin{align*}90^\circ\end{align*} counterclockwise about the origin. How are the points of the original rectangle connected to the points of the image?

Solution: Pick one point, such as point \begin{align*}A\end{align*}, and draw a line segment connecting it to the origin. Because perpendicular lines form \begin{align*}90^\circ\end{align*} angles, to rotate \begin{align*}90^\circ\end{align*}, find a line perpendicular to the line segment connecting \begin{align*}A\end{align*} to the origin. Remember that perpendicular lines have opposite reciprocal slopes. \begin{align*}A^\prime\end{align*} will be on the perpendicular line, the same distance from the origin as \begin{align*}A\end{align*} was.

Once you have \begin{align*}A^\prime\end{align*}, you can build the rectangle around it. \begin{align*}A\end{align*} was at point (-3, 1) and \begin{align*}A^\prime\end{align*} is at point (-1, -3). In general, a \begin{align*}90^\circ\end{align*} counterclockwise rotation about the origin takes \begin{align*}(x,y)\end{align*} and maps it to \begin{align*}(-y,x)\end{align*}.

Concept Problem Revisited

To make sure your friend correctly rotates the pentagon \begin{align*}90^\circ\end{align*}, you need to also tell her the center of rotation and the direction of rotation.

#### Vocabulary

A rotation of \begin{align*}t^\circ\end{align*} about a given point \begin{align*}O\end{align*} takes each point on a shape \begin{align*}P\end{align*} and moves it to \begin{align*}P^\prime\end{align*} such that \begin{align*}P^\prime\end{align*} is on the circle with center \begin{align*}O\end{align*} and radius \begin{align*}\overline{OP}\end{align*} and \begin{align*}\angle P^\prime OP=t^\circ\end{align*}. Point \begin{align*}O\end{align*} is the center of rotation.

A rigid transformation is a transformation that preserves distance and angles.

#### Guided Practice

1. Describe the rotation below.

2. Rotate the rectangle \begin{align*}180^\circ\end{align*} counterclockwise about the origin. How do the points of the original rectangle compare to their corresponding points on the image?

3. How would your answer to #2 be different if instead you rotated \begin{align*}180^\circ\end{align*} clockwise about the origin?

1. This is a \begin{align*}135^\circ\end{align*} clockwise rotation about point \begin{align*}B\end{align*}. You could also say this is a \begin{align*}225^\circ\end{align*} counterclockwise rotation about point \begin{align*}B\end{align*}.

2. You can see that there is a straight line \begin{align*}(180^\circ)\end{align*} passing through the origin connecting each pair of corresponding points. \begin{align*}(x, y) \rightarrow (-x, -y)\end{align*}.

3. There would be no difference. A rotation \begin{align*}180^\circ\end{align*} clockwise is the same as a rotation \begin{align*}180^\circ\end{align*} counterclockwise.

#### Practice

Describe each of the following rotations in TWO ways.

1.

2.

3.

4. What does a \begin{align*}360^\circ\end{align*} rotation look like?

5. Rotate the rectangle below \begin{align*}90^\circ\end{align*} clockwise about the origin.

6. Use the previous example to help you describe what happens to a point \begin{align*}(x, y)\end{align*} that is rotated \begin{align*}90^\circ\end{align*} clockwise about the origin.

7. (3, 2) is rotated \begin{align*}90^\circ\end{align*} clockwise about the origin. Where does it end up?

8. (3, 2) is rotated \begin{align*}180^\circ\end{align*} clockwise about the origin. Where does it end up? Hint: Use Guided Practice #2 for help.

9. (3, 2) is rotated \begin{align*}90^\circ\end{align*} counterclockwise about the origin. Where does it end up? Hint: Use Example C.

10. What do circles have to do with rotations?

11. \begin{align*}\Delta ABC\end{align*} is rotated \begin{align*}130^\circ\end{align*} clockwise about point \begin{align*}O\end{align*} to create \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}. Describe two connections between the points \begin{align*}B, O\end{align*}, and \begin{align*}B^\prime\end{align*}.

For 12-15, consider \begin{align*}\Delta ABC\end{align*} below.

12. Rotate \begin{align*}\Delta ABC \ 90^\circ\end{align*} counterclockwise about point \begin{align*}B\end{align*}.

13. Rotate \begin{align*}\Delta ABC \ 90^\circ\end{align*} counterclockwise about point \begin{align*}A\end{align*}.

14. Rotate \begin{align*}\Delta ABC \ 90^\circ\end{align*} counterclockwise about point \begin{align*}C\end{align*}.

15. Could you have used the rule that a \begin{align*}90^\circ\end{align*} counterclockwise rotation takes \begin{align*}(x, y)\end{align*} to \begin{align*}(-y, x)\end{align*} to perform the previous three rotations? Explain.

### 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.
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.
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.
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.