High school geometry
Precisely defining rotations
Read a dialog where a student and a teacher work towards defining rotations as precisely as possible.
The dialog below is between a teacher and a student. Their goal is to describe rotations in general using precise mathematical language. As you'll see, the student must revise their definition several times to make it more and more precise. Enjoy!
Today we will try to describe what rotations do in a general way.
Suppose we have a rotation by degrees about the point . How would you describe the effect of this rotation on another point ?
What do you mean? How can I know what the rotation does to when I don't know anything about it?
It's true that you don't know anything about this specific rotation, but all rotations behave in a similar way. Can you think of any way to describe what the rotation does to ?
Hmmmm... Let me think... Well, I guess that moves to a different position in relation to . For example, if was to the right of , maybe it's now above or something like that. This depends on how big is.
A point P with a point A directly to the right of it. A green arrow curves from point A to a point at the top right of point P the same distance away from point P as Point A is.
Neat. We can describe what you just said as follows:
A horizontal line segment P A. A line segment P B is up and to the right A green arrow curves from point A to a point B. An unknown angle measure is with angle A P B.
Suppose the rotation maps to the point , then the angle between the line segments and is .
Yes, I agree with this definition.
Remember, however, that in mathematics we should be very precise. Is there just one way to create an angle that is equal to ?
Let me see... No, there are two ways to create such an angle: clockwise and counterclockwise.
A horizontal line segment P A. A line segment P B is up and to the right A green arrow curves from point A to a point B. An unknown angle measure is with angle A P B. Another line segment P B slants down to the right forming the same unknown angle measure and a green arrow curving from point A down to point B.
Right! Rotations are performed counterclockwise, and our definition should recognize that:
A rotation by degrees about point moves any point counterclockwise to a point where .
Of course, if is given as a negative measure, the rotation is in the opposite direction, which is clockwise.
Cool. Are we done?
You tell me. The definition should make it absolutely clear where is mapped to. In other words, there should only be one point that matches the description of .
Is there only one point that creates a counterclockwise angle that is equal to ?
I think so... Wait! No! There are many points that create this angle! Any point on the ray coming from towards has an angle of with .
Angle A P B where A P is a line segment and P B is a ray. The angle measure of P is theta. The ray coming from P towards B has multiple plotted points on it with question marks to represent possible answers.
Good observation! So, can you think of a way to make our definition better?
Yes, in addition to the angle being equal to , the distance from should stay the same. I think you can define this mathematically as .
Well done! We can summarize all of our work in the following definition:
A horizontal line segment P A. A line segment P B is up and to the right A green arrow curves from point A to a point B. An unknown angle measure is with angle A P B. Both line segments have a congruent line.
A rotation by degrees about point moves any point counterclockwise to a point where and .
Wow, this is very precise!
Indeed. As a bonus, let me show you another way to define rotations:
A circle with a center labeled point P. A horizontal line segment P A forms the radius of the circle. A line segment P B is up and to the right forms another radius on the circle. A green arrow curves on the arc of the circle from point A to a point B. An unknown angle measure is with angle A P B.
A rotation by degrees about point moves any point counterclockwise to a point such that both and are on the same circle centered at , and .
Yes, this also works because all the points on a circle have the same distance from the center.
That's right! The main difference between the two definitions is that the first uses line segments and the second uses a circle.
Cool. So is that it?
Yes. I think we've defined rotations as precisely as we can.
Want to join the conversation?
- Hi, What does the 0 with a slash through it mean?(36 votes)
- I believe in this context it refers to the Greek letter "theta", which is commonly used in planar geometry to designate/represent the measure of an angle.(81 votes)
- Can somebody please clarify where the lowercase m comes from in m∠APB=θ please?
After finishing the 8th grade curriculum I was led to this geometry class and I don't remember the m notation. What is it?(14 votes)
- The notation m in front of the name of an angle means the measure of that angle. So m∠APB means the measure of ∠APB.(36 votes)
- I noticed that the article didn't mention the concept of a "vertical stretch". I found this concept in a question and was wondering what it meant. Does anyone know?(7 votes)
- Vertical stretch is to double up the y axis value of a figure.(3 votes)
- I am confused. When it says the transformation is a reflection I think it could also be a 180 degree rotation. How do you differentiate?(6 votes)
- Draw the quadrilateral with vertices at (-3, 0), (0, 4), (4, 0), (0, 2). It should look like an arrowhead pointing up, but lopsided.
If you imagine an ant standing on the arrow, facing the tip, the shorter end of the arrow will be to the ants right. And you can imagine that no rotations will change this, for the same reason that driving a car in circles will never change which side the driver is on.
But when we reflect this shape, and make sure the ant stays on top and facing the tip, the shortened side is now on the ants left. You can see that this will always happen, because in order for the ant to stay on top, it must get flipped over and onto the underside of the quadrilateral. So the two transformations are not the same.(2 votes)
- wnhat does the ´m´ before the ´<´ symbol mean?(3 votes)
- 'm∠A' refers to the measure of angle A. The angle is a geometric object, consisting of two rays. The measure is a real number, describing how open the angle is, e.g. 90º.(6 votes)
- How do I know how far to rotate the angles(3 votes)
- in this example the angle is a fixed hypothetical angle and any fixed angle will do for instance take 30 degree instead of that 0 with a slash through it. don worry if u got confuse here in the example.(2 votes)
- From 'Practice: Defining transformations', what do the arrows and the lines that are on top of these arrows mean? (I'm not sure if it was on the video, but I can't understand..(2 votes)
- If the arrow is pointing to the left or right, it just means it points in that direction at no end. If it points in both directions, it means it points in both directions endlessly. No arrow means it stops at some point.(3 votes)
- I am not understanding the necesity to say PA = PB. If you rotate, you rotate, no dilate. It seems like a wild jump to assume you would have to state PA = PB. This hits my ear like a proof of a^2 + b^2 = c^2 that takes time to state that line segments aren't curved. When do you just stick with the definition of something, when do you (needlessly?) restate definitions and why?(3 votes)
- I believe it's because some may think "rotating" point A is directing it on a segment, even though it is a rotary path points a, b, etc. are taking.(1 vote)
- should all rotation be understood as positive? "Right! Rotations are performed counterclockwise, and our definition should recognize that:"(2 votes)
- All counterclockwise rotations should be understood as positive. We obviously can perform clockwise rotations, and these will be understood as negative.(3 votes)
- Why are there 2 letters? Why not one? And why is the second letter not italicized?(2 votes)