Unit A: Geometry

Chapter 2: Transformations

Predicting the Coordinates of an Image after a Rotation

The coordinates of an image after a rotation can be determined by rotating the piece of paper that contains the original shape, as seen in the video. There is, however, another method. Write down the coordinates of the smiley face rectangle and its images in the table below to find a pattern.

The first line of the table has been completed as an example. All rotations are made about the origin.

Original	90° Counterclockwise	180° Counterclockwise	270° Counterclockwise
A(1, –10)	A'(10, 1)	A'(–1, 10)	A'(–10, –1)

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Click here for the populated table.

The following are the rules for finding the coordinates after a rotation:

• When a 90° rotation or a 270° rotation occurs about the origin, use the values from the original point and switch the x and y values. Look at the position of the point on the graph to determine if the x and y values are positive or negative. For example, A(1, –10) is the original point and  A'(10, 1) is the image. The image is in quadrant 1 (top right) where the x and y values are both positive.
• When a 180° rotation about the origin occurs, the x and y values of the image remain the same as the original point, but the signs are the opposite. For example, A(1, –10) is the original point and the image is A'(–1, 10).

Use the Rotations #2 applet to observe how the values of the coordinates change in a rotation.