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 start color maroonD, A, prime, end color maroonD is the result of rotating start color blueD, A, end color blueD 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 start overline, P, A, end overline and start overline, P, A, prime, end overline.
This way, we can say that start color maroonD, A, prime, end color maroonD is the result of rotating start color blueD, A, end color blueD by 45degree 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 –30degree.
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 source figure, which is the figure we are performing the transformation upon, and the image figure, which is the result of the transformation. For example, in our rotation, the source point was start color blueD, A, end color blueD, and the image point was start color maroonD, A, prime, end color maroonD.
Note that we indicated the image by start color maroonD, A, prime, end color maroonD—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

Rotate the line segment about P by 90degree.

First, click on the rotate button. The rotation tool will appear.
Drag the center of the rotation tool and place it on point P. Then, click on the tool's arrow and drag it in the counterclockwise direction.
Once you start rotating, a label will indicate the angle of rotation according to the current position of the tool. Rotate until the label indicates a 90degree rotation.

Problem 2

Rotate the triangle about P by –60degree.

First, click on the rotate button. The rotation tool will appear.
Drag the center of the rotation tool and place it on point P. Then, click on the tool's arrow and drag it in the clockwise direction.
Once you start rotating, a label will indicate the angle of rotation according to the current position of the tool. Rotate until the label indicates a –60degree rotation.

Problem 3

Rotate the circle about P by 255degree.

First, click on the rotate button. The rotation tool will appear.
Drag the center of the rotation tool and place it on point P. Then, click on the tool's arrow and drag it in the counterclockwise direction.
Once you start rotating, a label will indicate the angle of rotation according to the current position of the tool. Rotate until the label indicates a 255degree rotation.

Challenge problem 1

start color maroonD, R, end color maroonD, start color maroonD, S, end color maroonD, and start color maroonD, T, end color maroonD are all images of start color blueD, Q, end color blueD under different rotations.
Match each image with its appropriate rotation.
Image
Rotation
  • start color maroonD, R, end color maroonD
  • start color maroonD, S, end color maroonD
  • start color maroonD, T, end color maroonD
  • A rotation about P by 180degree
  • A rotation about P by 90degree
  • A rotation about P by –90degree

start color maroonD, R, end color maroonD is the result of rotating start color blueD, Q, end color blueD by 90degree in the counterclockwise direction, which is a rotation by 90degree.
In contrast, start color maroonD, T, end color maroonD is the result of rotating start color blueD, Q, end color blueD by 90degree in the clockwise direction, which is a rotation by –90degree.
start color maroonD, S, end color maroonD is exactly on the other side of P relative to start color blueD, Q, end color blueD, which is a rotation by 180degree.

Challenge problem 2

Segment start color maroonD, start overline, C, prime, D, prime, end overline, end color maroonD is the result of rotating start color blueD, start overline, C, D, end overline, end color blueD in a counterclockwise direction about P.
Which expression represents the angle of the rotation?
Please choose from one of the following options.

The angle of rotation is the angle that is formed between the line segments that connect each pair of corresponding source and image points to the center of rotation.
For example, the image of start color blueD, C, end color blueD is start color maroonD, C, prime, end color maroonD. The angle that is formed between the line segments that connect these points to P is the angle alpha, plus, beta. Therefore, this is the angle of rotation.
Similarly, the angle of rotation is also the angle that is formed by the line segments that connect start color blueD, D, end color blueD and start color maroonD, D, prime, end color maroonD to P, which is beta, plus, gamma, but this is not an option in our question.