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### Course: Geometry (all content) > Unit 10

Lesson 1: Introduction to rigid transformations# Rotations intro

Learn what rotations are and how to perform them in our interactive widget.

## What is a rotation?

In the figure below, one copy of the trapezoid is rotating around the point.

In geometry, rotations make things turn in a cycle around a definite center point. Notice that the distance of each rotated point from the center remains the same. Only the relative position changes.

In the figure below, one copy of the octagon is rotated $22\mathrm{\xb0}$ around the point.

Notice how the octagon's sides change direction, but the general shape remains the same. Rotations don't distort shapes, they just whirl them around. Furthermore, note that the vertex that is the center of the rotation does not move at all.

Now that we've got a basic understanding of what rotations are, let's learn how to use them in a more exact manner.

## The angle of rotation

Every rotation is defined by two important parameters: the center of the rotation—we already went over that—and the

**angle of the rotation**. The angle determines by how much we rotate the plane about the center.For example, we can tell that ${{A}^{\prime}}$ is the result of rotating ${A}$ about $P$ , but that's not exact enough.

In order to define the measure of the rotation, we look at the angle that's created between the segments $\stackrel{\u2015}{PA}$ and $\stackrel{\u2015}{P{A}^{\prime}}$ .

This way, we can say that ${{A}^{\prime}}$ is the result of rotating ${A}$ by $45\mathrm{\xb0}$ about $P$ .

### Clockwise and counterclockwise rotations

This is how we number the quadrants of the coordinate plane.

The quadrant numbers increase as we move counterclockwise. We measure angles the same way to be consistent.

Conventionally,

*positive*angle measures describe*counterclockwise*rotations. If we want to describe a*clockwise*rotation, we use*negative*angle measures.For example, here's the result of rotating a point about $P$ by $-30\mathrm{\xb0}$ .

### Pre-images and images

For any transformation, we have the ${A}$ , and the image point was ${{A}^{\prime}}$ .

**pre-image**figure, which is the figure we are performing the transformation upon, and the**image**figure, which is the result of the transformation. For example, in our rotation, the pre-image point wasNote that we indicated the image by ${{A}^{\prime}}$ —pronounced, "A prime". It is common, when working with transformations, to use the same letter for the image and the pre-image; simply add the prime suffix to the image.

## Let's try some practice problems

### Challenge problems

## Want to join the conversation?

- In problem three I placed the rotation tool on P and rotated it 225 degrees but it is saying it is wrong!(58 votes)
- I also made exactly this mistake. It says 255, not 225(28 votes)

- this stuff mad hard(64 votes)
- i agree with you bro(14 votes)

- I didn't really understand how to do the challenge 1 problem. Perhaps give a tip on how to enter it using the computer.(13 votes)
- STEP 1: Imagine that "orange" dot (that tool that you were playing with) is on top of point P.

STEP 2: Point Q will be the point that will move clockwise or counter clockwise.

STEP 3: When you move point Q to point R, you have moved it by 90 degrees counter clockwise (can you visualize angle QPR as a 90 degree angle?).

STEP 4: When you move point Q to point S, you have moved it by 180 degrees counter clockwise (can you visualize QPS points when joined together as a straight line? Hence, 180 degree?).

STEP 5: Remember that clockwise rotations are negative. So, when you move point Q to point T, you have moved it by 90 degrees clockwise (can you visualize angle QPT as a 90 degree angle?). Hence, you have moved point Q to point T by "negative" 90 degree.

Hope that this helped.(38 votes)

- What would be the default direction for rotation if it does not specify (counter clockwise or clockwise)??(26 votes)
- when it puts - before the degrees it means counter clockwise and when there is no - it means clockwise(6 votes)

- Take notes even if this is easy because when you get to complex geometry problems, you are going to need to remember all the little details about the basics. It builds up!(24 votes)
- Im so confused as an 8th grader doing this. 😭(24 votes)
- what happens if you rotate a dorito, is it still the same cos it's a triangle shape(14 votes)
- As long as you do not take a bite out of it, yes you can rotate a dorito.(17 votes)

- literally so hard for no good reason like(17 votes)
- Then do something you are familiar with and it may be more beneficial to get good at something than to learn something you have no idea how to do.(1 vote)

- In the options for answers what are those symbols?(10 votes)
- those are lower case Greek letters which are often used to indicate measurement of angles. See https://www.ibiblio.org/koine/greek/lessons/alphabet.html. In the answers, you have the "ABCs" of the Greek alphabet, alpha, beta, gamma. You will also see these symbols used in Science at some point.(11 votes)

- I can't really figure out what's being said in the explanation for the final question right. May someone please put it into layman's terms for me? Either that or just explain it to me like I'm a toddler. Thank you in advance.(11 votes)
- So the image of C is C′. you connect these points to point P to find the angle formed by these lines which is α+β. i hope this is clear enough for you👍🏽(8 votes)