How Do You Rotate a Figure?

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When you rotate a figure, you move it along an arc.

It is often said that something rotates “about an axis.” That means that it rotates around something fixed. In the picture you can see a monkey sitting on a swing. When the monkey swings, it moves in an arc—and not in a straight line. Curved movement like this is called rotation. In this case, the axis the monkey is rotating about is the pole the swing is attached to.

Monkey on a swing is rotated

This monkey has been rotated. The monkey is the same size as before, but it has moved to a different point on the arc of the swing. When you rotate a figure, it’s important that the figure itself stays the same through the rotation. The only difference should be that the figure is tilted relative to the original figure. Below, you will take a closer look at rotation, both on its own, and combined with translation.

Example 1

Rotate a phone 90 clockwise about an axis 3cm from the phone. You can decide where to put the axis.

Phone rotated 90 degree

Draw the axis on a sheet of paper. Draw a 3 cm long line from the axis in any direction. Place the phone where the line ends, measure 90 with your protractor, and mark the point where the protractor shows 90. Imagine that there is an invisible and taut piece of string tied between each corner of the phone and the axis (like a swing). The strings will help you keep the orientation of the phone correct. Move your phone to the point you marked.

Example 2

Rotate a phone 180 clockwise about its own axis, and translate it 4cm to the right

Phone rotated 180 degree

You begin by rotating the phone as in Example 1, except this time you rotate it 180. Then you translate the rotated phone 4 cm to the right.

When you rotate a figure about a point P, you need to know how many degrees the figure should be rotated. There are two ways to do this: one for rotating 180, and one for all rotations that are not 180. Below are instructions for both.

The figure shows a triangle rotated 180.

Triangle rotated 180 degree

Rule

Instructions for Rotating 180

1.
Draw straight lines from the corners of the figure through the point P. Draw the lines well past P.
2.
Put the point of the drafting compass on P and measure the distance from A to P with the compass.
3.
Keep the point of the compass on P, and make an arc on the line from A, but on the other side of P. Call that point A.
4.
Repeat this until all the points have their counterpart on the other side of P.
5.
Draw the lines between A, B, C, and so on to complete the rotation.

This figure shows a point A rotated 100 about a point P:

Point rotated 100 degree

Rule

Instructions for Rotations That Are Not 180

1.
Name all the corners. Use letters in alphabetical order: first A, then B, and so on.
2.
Draw straight lines from the corners through the point P, and far past it.
3.
Use a protractor to measure v. Make sure the hole in the protractor is right on P, and draw lines from P along the angle v for all the lines that pass through P.
4.
Measure the distance from P to A. Measure the same distance from P along the new line. Call the new point A.
5.
Repeat this for all the corners.
6.
Draw the lines between A, B, C, and so on.

Think About This

Why do you think 180 rotation about a point is the same as reflection about a line?

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