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Lesson 3: Rotations

# Rotations review

Review the basics of rotations, and then perform some rotations.

### What is a rotation?

A rotation is a type of transformation that takes each point in a figure and rotates it a certain number of degrees around a given point.
For example, this animation shows a rotation of pentagon $IDEAL$ about the point $\left(0,-1\right)$. You can see the angle of rotation at the bottom, which increases the further we rotate the figure from its original position.
The result of a rotation is a new figure, called the image. The image is congruent to the original figure.

## Performing rotations

Although a figure can be rotated any number of degrees, the rotation will usually be a common angle such as ${45}^{\circ }$or ${180}^{\circ }$.
If the number of degrees are positive, the figure will rotate counter-clockwise.
If the number of degrees are negative, the figure will rotate clockwise.
The figure can rotate around any given point.
Example:
Rotate $\mathrm{△}OAR$ ${60}^{\circ }$ about point $\left(-2,-3\right)$.
The center of rotation is $\left(-2,-3\right)$.
Rotation by ${60}^{\circ }$ moves each point about $\left(-2,-3\right)$ in a counter-clockwise direction. The rotation maps $\mathrm{△}OAR$ onto the triangle below.

## Practice

Problem 1
$\mathrm{△}NOW$ is rotated ${90}^{\circ }$ about the origin.
Draw the image of this rotation.

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• Why is positive counter-clockwise? Is that just a rule or...
• It’s because you’re rotating in the order of the coordinate plane’s quadrants, which goes from right to left to down to right (*counterclockwise*)
• So how do you suppose the exact rotation on here? Or do you just guess how much to turn it ....
• I don't understand how to rotate about the origin, please explain.
• @garrettcummings22, I realize the frustration of these geometric principles, but these same principles are the foundations of graphic design, several types of engineering, carpentry, masonry, many forms of art. The reality is no one in grades 7-12 will ever know if they will use any of the math they are required to study. No one knows what their future holds. Reality also tells us that every math principle taught is a math concept actually used somewhere in real life. I have used several concepts, especially writing, solving, and graphing linear equations, Pythagorean Theorem, ratios and percents, and many other aspects of statistics throughout my many years of life and many occupations in life.

Good luck in all you do. I hope you find some purpose in your life for the math. I genuinely mean that.
• I don’t see how it’s possible to rotate any polygon free-hand,without the rotation tool.
• You could rotate each point of the polygon on its own, and then connect each point correctly. Remember that each point will be the same distance from the centre of rotation.
• I have to find the coordinates of a point on a shape after a rotation, but I wasn't given a graph or any image.
(Example):'triangle'NPQ has vertices N(-6,-4), P(-3,4), and Q(1,1). If the triangle is rotated 90 degrees about the origin, what are the coordinates of P'?
Is there a rule or formula to solve this or do I have to actually draw a graph, plot the points, rotate, ect.? Thanks.
• Great question! There are actually several helpful shortcuts for finding rotations. For rotating 90 degrees counterclockwise about the origin, a point (x, y) becomes (-y, x). So for example, your N would become (4, -6).