# Precisely defining rotations

CCSS Math: HSG.CO.A.4
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 $\overline{PA}$ and $\overline{PB}$ 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 $\angle P$ that is equal to $\theta\,$?
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 $m\angle APB=\theta$.
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 $\theta\,$?
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 $m\angle APB=\theta$.
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 $m\angle APB=\theta$.
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.