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

Lesson 7: Properties and definitions of transformations# 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!

**Teacher:**

Today we will try to describe what rotations do in a general way.

Suppose we have a rotation by $\theta $ degrees about the point $P$ . How would you describe the effect of this rotation on another point $A$ ?

**Student:**

What do you mean? How can I know what the rotation does to $A$ when I don't know anything about it?

**Teacher:**

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 $A$ ?

**Student:**

Hmmmm... Let me think... Well, I guess that $A$ moves to a different position in relation to $P$ . For example, if $A$ was to the right of $P$ , maybe it's now above $P$ or something like that. This depends on how big $\theta $ is.

**Teacher:**

Neat. We can describe what you just said as follows:

**Suppose the rotation maps**$A$ to the point $B$ , then the angle between the line segments $\stackrel{\u2015}{PA}$ and $\stackrel{\u2015}{PB}$ is $\theta $ .

**Student:**

Yes, I agree with this definition.

**Teacher:**

Remember, however, that in mathematics we should be very precise. Is there just one way to create an angle $\mathrm{\angle}P$ that is equal to $\theta {\textstyle \phantom{\rule{0.167em}{0ex}}}$ ?

**Student:**

Let me see... No, there are two ways to create such an angle: clockwise and counterclockwise.

**Teacher:**

Right! Rotations are performed counterclockwise, and our definition should recognize that:

**A rotation by**$\theta $ degrees about point $P$ moves any point $A$ counterclockwise to a point $B$ where $m\mathrm{\angle}APB=\theta $ .

Of course, if $\theta $ is given as a negative measure, the rotation is in the opposite direction, which is clockwise.

**Student:**

Cool. Are we done?

**Teacher:**

You tell me. The definition should make it absolutely clear where $A$ is mapped to. In other words, there should only be one point that matches the description of $B$ .

Is there only one point that creates a counterclockwise angle that is equal to $\theta {\textstyle \phantom{\rule{0.167em}{0ex}}}$ ?

**Student:**

I think so... Wait! No! There are many points that create this angle! Any point on the ray coming from $P$ towards $B$ has an angle of $\theta $ with $A$ .

**Teacher:**

Good observation! So, can you think of a way to make our definition better?

**Student:**

Yes, in addition to the angle being equal to $\theta $ , the distance from $P$ should stay the same. I think you can define this mathematically as $PA=PB$ .

**Teacher:**

Well done! We can summarize all of our work in the following definition:

**A rotation by**$\theta $ degrees about point $P$ moves any point $A$ counterclockwise to a point $B$ where $PA=PB$ and $m\mathrm{\angle}APB=\theta $ .

**Student:**

Wow, this is very precise!

**Teacher:**

Indeed. As a bonus, let me show you another way to define rotations:

**A rotation by**$\theta $ degrees about point $P$ moves any point $A$ counterclockwise to a point $B$ such that both $A$ and $B$ are on the same circle centered at $P$ , and $m\mathrm{\angle}APB=\theta $ .

**Student:**

Yes, this also works because all the points on a circle have the same distance from the center.

**Teacher:**

That's right! The main difference between the two definitions is that the first uses line segments and the second uses a circle.

**Student:**

Cool. So is that it?

**Teacher:**

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?(49 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.(127 votes)

- my brain just alt f4(65 votes)
- to reopen it try using Ctrl + Shift + T(8 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?(17 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.(54 votes)

- omg i thought the 'θ' was a zero for the longest time and i was so confused as to why 0 was equaling like 35 degrees(35 votes)
- just so you know that 0 is theta a Greek letter

now some theta jokes

Why was epsilon afraid of zeta?

Because zeta eta theta.

I have another one

Where do angles go for fun on the weekends?

To watch movies in the THETA(28 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?(7 votes)
- Reflections are across a line, and rotations are around a point, so this is a very different process. So if you have a figure in the first quadrant, rotating it about the origin 180 degrees either clockwise or counterclockwise would switch (x,y) to (-x,-y). Reflections for the same figure has to be reflected across some line, so most reflections would not even be close (across x axis, y axis, any horizontal or vertical line, y=x, etc.). If you choose the line y = - x, then the point would switch from (x,y) to (-y,-x). This would create a different point except if the values of x and y were the same. (2,2) would rotate 180 to (-2,-2) and reflect across y = -x to (-2,-2), but if you had (2,5) a rotation of 180 would end at (-2, -5) but a reflection across y=-x would end at (-5,-2).(14 votes)

- It's funny how the student keeps asking if they are finished!

Cool, Are we done?

Cool, So is that it?(12 votes) - i aint reading all of that(10 votes)
- brain is not braining(9 votes)
- lol no problem! so you know how we rotate a point about another point. When you are given the degree of rotation, you want to describe it in mathematical terms. For example, if there is a point A being rotated 60 degrees about another point P and the transformation results to another point B (it is like A'). If you have to write that, you would say something like A rotation by 60 degrees "about point P moves any point A counterclockwise to a point B" where the angle measure of angle apb (it forms an angle like shown in the picture) = 60 degrees. The reason why counterclockwise is added to the description is to make it more accurate because a positive rotation is counterclockwise. However, it is not completely accurate because is means it can be any point in the line segment. To make it more precise and make sure it matches the visual description, one needs to add that PA is equal to PB. Here you would be talking about line segments but if you want to use a circle to help describe it you could say so as well and remove the PA=PB.(4 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.(5 votes)