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

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 PP 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 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 A\blueD{A}, and the image point was A\maroonD{A'}.
Note that we indicated the image by A\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

Rotate the line segment about PP by 90^\circ.

First, click on the rotate button. The rotation tool will appear.
Drag the center of the rotation tool and place it on point PP. 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 90^\circ rotation.

Problem 2

Rotate the triangle about PP by –60^\circ.

First, click on the rotate button. The rotation tool will appear.
Drag the center of the rotation tool and place it on point PP. 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 –60^\circ rotation.

Problem 3

Rotate the circle about PP by 255^\circ.

First, click on the rotate button. The rotation tool will appear.
Drag the center of the rotation tool and place it on point PP. 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 255^\circ rotation.

Challenge problem 1

R\maroonD{R}, S\maroonD{S}, and T\maroonD{T} are all images of Q\blueD{Q} under different rotations.
Match each image with its appropriate rotation.
Image
Rotation
  • R\maroonD{R}
  • S\maroonD{S}
  • T\maroonD{T}
  • A rotation about PP by 180^\circ
  • A rotation about PP by 90^\circ
  • A rotation about PP by –90^\circ

R\maroonD{R} is the result of rotating Q\blueD{Q} by 90^\circ in the counterclockwise direction, which is a rotation by 90^\circ.
In contrast, T\maroonD{T} is the result of rotating Q\blueD{Q} by 90^\circ in the clockwise direction, which is a rotation by –90^\circ.
S\maroonD{S} is exactly on the other side of PP relative to Q\blueD{Q}, which is a rotation by 180^\circ.

Challenge problem 2

Segment CD\maroonD{\overline{C'D'}} is the result of rotating CD\blueD{\overline{CD}} in a counterclockwise direction about PP.
Which expression represents the angle of the rotation?
Choose 1 answer:
Choose 1 answer:

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 C\blueD{C} is C\maroonD{C'}. The angle that is formed between the line segments that connect these points to PP is the angle α+β\alpha+\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 D\blueD{D} and D\maroonD{D'} to PP, which is β+γ\beta+\gamma, but this is not an option in our question.
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