# Rotations intro

CCSS Math: HSG.CO.A.2, HSG.CO.A.4, HSG.CO.A.5

Learn what rotations are and how to perform them in our interactive widget.

To see what a

**rotation**is, grab the point on the slider and move it from side to side. This will cause the other point to rotate about point $P$.Nice! You

*rotated*a point. In geometry, rotations make things turn in a cycle around a definite center point. Notice that the distance of the rotated point from the center remains the same, only the relative position changes.Move the point across this slider to see how a square is rotated about one of its vertices.

Notice how the square's sides change direction, but the general shape remains the same. Rotations don't distort shapes, they just whirl them around. Furthermore, note that the vertex that is the center of the rotation does not move at all.

Now that we've got a basic understanding of what rotations are, let's learn how to use them in a more exact manner.

## The angle of rotation

Every rotation is defined by two important parameters: the center of the rotation—we already went over that—and the

**angle of the rotation**. The angle determines by how much we rotate the plane about the center.For example, we can tell that $\maroonD{A'}$ is the result of rotating $\blueD{A}$ about $P$, but that's not exact enough.

In order to define the measure of the rotation, we look at the angle that's created between the segments $\overline{PA}$ and $\overline{PA'}$.

This way, we can say that $\maroonD{A'}$ is the result of rotating $\blueD{A}$ by 45$^\circ$ about $P$.

### Clockwise and counterclockwise rotations

Conventionally, positive angle measures describe

*counterclockwise*rotations. If we want to describe a clockwise rotation, we use*negative*angle measures.For example, here's the result of rotating a point about $P$ by –30$^\circ$.

There isn't a very compelling reason why we define clockwise and counterclockwise rotations this way, but the important thing is that we have an agreed system for describing rotations.

### Sources and images

For any transformation, we have the

*figure, which is the figure we are performing the transformation upon, and the***source***figure, which is the result of the transformation. For example, in our rotation, the source point was $\blueD{A}$, and the image point was $\maroonD{A'}$.***image**Note that we indicated the image by $\maroonD{A'}$—pronounced

*A prime*. It is common, when working with transformations, to use the same letter for the image and the source; simply add the prime suffix to the image.## Let's try some practice problems

### Problem 1

### Problem 2

### Problem 3

### Challenge problem 1

$\maroonD{R}$, $\maroonD{S}$, and $\maroonD{T}$ are all images of $\blueD{Q}$ under different rotations.

### Challenge problem 2

Segment $\maroonD{\overline{C'D'}}$ is the result of rotating $\blueD{\overline{CD}}$ in a counterclockwise direction about $P$.