Play with the applet below to explore the rotation transformation. Try changing the shape and size of the original triangle, then use the slider to change the angle of rotation.

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The size and shape of the image of the object stays the same as it turns around the center of rotation.

The center of rotation does not always have to be a point on the image. Consider the figure below, which shows square A being rotated about the point O.

This image shows a rotation of a square. Ask your teacher for more information.

Square A is rotated 135\degree clockwise, or 225\degree counterclockwise, about O resulting in square B.

We can use a protractor to measure the angle of rotation between the original object and the rotated object. We can also use a protractor to measure the correct angle of rotation so we can draw the transformation.

Examples

Example 1

Which is the correct image after triangle A is rotated 90\degree counterclockwise about the point O?

Worked Solution

Create a strategy

Each quadrant has an angle of 90\degree , all we need to do is rotate the triangle A to the next quadrant in a counterclockwise direction.

Apply the idea

Triangle A is in quadrant 3. The next quadrant in an counterclockwise direction is quadrant 4 which has triangle D in it.

So the answer is triangle D.

Reflect and check

If we were to instead rotate triangle A by 90\degree clockwise, the correct image would then be triangle B.

Example 2

Consider the shape below. Draw the result of a rotation by 180\degree clockwise about point A.

Worked Solution

Create a strategy

Draw horizontal and vertical line using A as the center, then recall that the first quarter of rotation is 90\degree, second quarter is 180\degree, third quarter is 270\degree, and last quarter is 360\degree.

Apply the idea

180\degree is half of the full turn of 360 \degree, so the image should be halfway around a 360 \degree rotation from the original image.

The rotation is shown below.

Idea summary

Rotation of a shape can be done by rotating it about a fixed point or the center of rotation.

Outcomes

8.G.A.1

Verify experimentally the properties of rotations, reflections, and translations.

8.G.A.1.A

Lines are taken to lines, and line segments to line segments of the same length.

8.G.A.1.B

Angles are taken to angles of the same measure.

8.G.A.1.C

Parallel lines are taken to parallel lines.

8.G.A.3

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

8.03 Properties of rotations

Introduction

Ideas

Rotations

Exploration