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 $3 cm$ 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 $4 cm$ to the right

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 $P$ :

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 $v^{\circ}$ . 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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Instructions for Rotating 180°

Instructions for Rotations That Are Not 180°

Instructions for Rotating $180 °$

Instructions for Rotations That Are Not $180 °$