Become a math whiz with AI Tutoring, Practice Questions & more.

HotmathMath Homework. Do It Faster, Learn It Better.

Rotations

As we get further into geometry, we will learn many different techniques for transforming graphs. One of these techniques is "rotation." As we might have guessed, this involves turning a figure around on its axis. But how exactly does this work? Let's find out:

What is a rotation?

We're probably already familiar with the concept of rotation. After all, the wheels on a bicycle or a skateboard rotate. We can even rotate ourselves by spinning around until we get dizzy. As it turns out, the mathematical definition of rotation isn't all that different. Just like the wheel on a bicycle, a figure on a graph rotates around its axis or "center of rotation."

We measure the "amount" of rotation in degrees, and we can do this manually using a protractor. On the other hand, we can also use certain calculations to determine the amount of rotation even without graphing our points.

Visualizing the rules of rotation

There are many important rules when it comes to rotation. Some of the most useful rules to memorize are the transformations of common angles. Let's start with everyone's favorite: The right, 90-degree angle:

As we can see, we have transformed P by rotating it 90 degrees. We might have noticed a pattern: The values are reversed, with the y value on the rotated point becoming negative. In other words:

With a 90-degree rotation around the origin, (x,y) becomes (-y,x)

Now let's consider a 180-degree rotation:

We can see another predictable pattern here. When we rotate a point around the origin by 180 degrees, the rule is as follows:

(x,y) becomes (-x,-y)

Now let's consider a 270-degree rotation:

Can you spot the pattern? The general rule here is as follows:

When rotating a point around the origin by 270 degrees, (x,y) becomes (y,-x).

We don't really need to cover a rotation of 360 degrees since this will bring us right back to our starting point. This means that the (x,y) coordinates will be completely unchanged!

Note that all of the above rotations were counterclockwise.

If we wanted to rotate our points clockwise instead, we simply need to change the negative values. For example, a clockwise rotation of 90 degrees is (y, -x), while a counterclockwise rotation of 90 degrees is (-y,x). This also means that a 270-degree clockwise rotation is equivalent to a counterclockwise rotation of 90 degrees.

Topics related to the Rotations

Dilation

Angle of Rotation

Center of Rotation

Flashcards covering the Rotations

Common Core: High School - Geometry Flashcards

Practice tests covering the Rotations

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

Pair your student with a tutor who understands rotations

Rotations may be difficult for some students to grasp -- especially if they are not visual learners. Tutoring can help students learn via methods that match their learning styles, whether they're visual, verbal, or hands-on learners. Tutors can also personalize your student's sessions in other ways, catering to their ability level, hobbies, and much more. Tutors can also help your student learn at a productive, manageable pace -- whether they want to steam ahead toward new challenges or slow down to revisit past concepts. Your student will also have many opportunities to ask questions during their tutoring sessions. Unlike classroom sessions, students can turn to tutors whenever they feel stuck. Reach out to Varsity Tutors today, and we'll pair your student with a suitable tutor.

;
Download our free learning tools apps and test prep books