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Geometry (all content)

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.
A trapezoid inside a circle. The trapezoid is being rotated around the center of the circle. Two sides of the trapezoid are parallel. The other two side of the trapezoid are congruent.
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, degree around the point.
A concave octagon. The outline of the concave octagon is rotated twenty-two degrees around one of the points of the octagon.
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.
Point A is rotated about point P in a counterclockwise direction to form A prime.
For example, we can tell that start color #ca337c, A, prime, end color #ca337c is the result of rotating start color #11accd, A, end color #11accd 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 start overline, P, A, end overline and start overline, P, A, prime, end overline.
Segment PA is rotated about point P by forty-five degrees in a counterclockwise direction to form Segment PA prime.
This way, we can say that start color #ca337c, A, prime, end color #ca337c is the result of rotating start color #11accd, A, end color #11accd by 45, degree about P.

Clockwise and counterclockwise rotations

This is how we number the quadrants of the coordinate plane.
Blank coordinate plane with the horizontal axis labeled, x and the vertical axis labeled, y. The top right quadrant is labeled quadrant one. The top left quadrant is labeled quadrant two. The bottom left quadrant is labeled quadrant three. The bottom right quadrant is labeled quadrant four.
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.
A pre-image line segment where one endpoint is labeled P rotates the other part of the line segment and other endpoint clockwise negative thirty degrees.
For example, here's the result of rotating a point about P by minus, 30, degree.

Pre-images and images

For any transformation, we have the 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 was start color #11accd, A, end color #11accd, and the image point was start color #ca337c, A, prime, end color #ca337c.
Note that we indicated the image by start color #ca337c, A, prime, end color #ca337c—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

Problem 1
• Current
Plot the image of point A after a minus, 120, degree rotation about P.

Challenge problems

Challenge problem 1
• Current
start color #ca337c, R, end color #ca337c, start color #ca337c, S, end color #ca337c, and start color #ca337c, T, end color #ca337c are all images of start color #11accd, Q, end color #11accd under different rotations.
Point P is the center of the image. Point R is at twelve o'clock in relation to point P. Point Q is at three o'clock in relation to Point P. Point T is at six o'clock in relation to point P. Point S is at nine o'clock in relation to Point P.
Match each image with its appropriate rotation.

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!
• I also made exactly this mistake. It says 255, not 225
• I didn't really understand how to do the challenge 1 problem. Perhaps give a tip on how to enter it using the computer.
• 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.
• What would be the default direction for rotation if it does not specify (counter clockwise or clockwise)??
• when it puts - before the degrees it means counter clockwise and when there is no - it means clockwise
• In the options for answers what are those symbols?
• 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.
• what happens if you rotate a dorito, is it still the same cos it's a triangle shape
• As long as you do not take a bite out of it, yes you can rotate a dorito.
• 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.
• 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👍🏽
• I do not under stand Challenge problem #2
WHAT IS THE EXPRESSION!
Please give a answer that I can understand- I'm only in Gr5
• These are letters of the Greek alphabet, known as alpha, beta, and I do not know the third. Like the letters in the standard alphabet used in math, these are also variables, and you will see them in later years.
• I need more help understanding rotations a little bit more.
• I overall don't get the expression part. Like, I don't understand where it comes from or where to gather the info. I did try to use some common sense, but I want to be able how to do it without doubting myself and guessing.
• Hi Debora,

Few things you need to keep in mind when doing a rotation are-
1. The point of rotation (in most of the above examples, it is marked as P).
2. The angle of rotation.
3. Positive rotations are anti-clockwise and negative rotations are clockwise.

When you start practicing rotations in the above examples, have a look at the line below the "Rotation Tab". This will tell you if you are on the right track or not.

A game that you can play with your friends is by taking a rope, one of you can hold one end of it and the other one can hold the other end. Bring that cord on the floor/playground. One of you can be the center of rotation (point P) and the other person can move the other end of the cord clockwise or anti-clockwise around that center of rotation. You can move 90 degree (anti-clockwise) or minus 180 (clockwise)- these just two examples!

Try to visualize it in your head and have fun with it- you can see how the shape is the same but its orientation has changed :-)

Please feel free to ask any specific questions, if you have any.

Happy learning.

Aiena.