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Rotations intro

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

What is a rotation?

In the figure below, one copy of the trapezoid is rotating around the point.
A trapezoid inside a circle. The trapezoid is being rotated around the center of the circle. Two sides of the trapezoid are parallel. The other two side of the trapezoid are congruent.
In geometry, rotations make things turn in a cycle around a definite center point. Notice that the distance of each rotated point from the center remains the same. Only the relative position changes.
In the figure below, one copy of the octagon is rotated 22° around the point.
A concave octagon. The outline of the concave octagon is rotated twenty-two degrees around one of the points of the octagon.
Notice how the octagon'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.
Point A is rotated about point P in a counterclockwise direction to form A prime.
For example, we can tell that A is the result of rotating 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 PA and PA.
Segment PA is rotated about point P by forty-five degrees in a counterclockwise direction to form Segment PA prime.
This way, we can say that A is the result of rotating A by 45° about P.

Clockwise and counterclockwise rotations

This is how we number the quadrants of the coordinate plane.
Blank coordinate plane with the horizontal axis labeled, x and the vertical axis labeled, y. The top right quadrant is labeled quadrant one. The top left quadrant is labeled quadrant two. The bottom left quadrant is labeled quadrant three. The bottom right quadrant is labeled quadrant four.
The quadrant numbers increase as we move counterclockwise. We measure angles the same way to be consistent.
Conventionally, positive angle measures describe counterclockwise rotations. If we want to describe a clockwise rotation, we use negative angle measures.
A pre-image line segment where one endpoint is labeled P rotates the other part of the line segment and other endpoint clockwise negative thirty degrees.
For example, here's the result of rotating a point about P by 30°.

Pre-images and images

For any transformation, we have the pre-image 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 pre-image point was A, and the image point was A.
Note that we indicated the image by A—pronounced, "A prime". It is common, when working with transformations, to use the same letter for the image and the pre-image; simply add the prime suffix to the image.

Let's try some practice problems

Problem 1
Plot the image of point A after a 120° rotation about P.

Challenge problems

Challenge problem 1
R, S, and T are all images of Q under different rotations.
Point P is the center of the image. Point R is at twelve o'clock in relation to point P. Point Q is at three o'clock in relation to Point P. Point T is at six o'clock in relation to point P. Point S is at nine o'clock in relation to Point P.
Match each image with its appropriate rotation.

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