More on Rotations (Day 38)

So far, our work on rotations focuses the composite of two reflections in intersecting lines, for that's how the U of Chicago text defines them. But ordinarily, when we think about rotations, we want to think about its center and magnitude -- not the two reflections. Yet the U of Chicago text doesn't spend enough time on rotations do accomplish this. So we must go to another source.

Recall that for Wu, rotations are of primary importance. But the Wu link that I gave yesterday doesn't give exercises or problem sets, since it mainly focuses on definitions and theorems. Luckily, here's a link to the Common Core Geometry curriculum developed for New York State.

https://www.engageny.org/resource/high-school-geometry

I suspect that much of the Empire State's curriculum is based on Wu. We see that transformations are covered in Topic C, Lessons 12-21. Rotations come first (Lesson 13) -- just like Wu. Reflections come next (Lesson 14) and then translations (Lesson 16). But the giveaway that Topic C is based on Wu is Lesson 18 -- which uses 180-degree rotations to construct parallel lines and ultimately prove the Alternate Interior Angles Theorem!

As it turns out, Lesson 15 covers the relationship between a reflection and a rotation -- so it's just yesterday's lesson. And Lesson 17 is essentially the Perpendicular Bisector Theorem, which we've already covered last week. Our focus today will be on Lesson 13, on rotations.

https://www.engageny.org/resource/geometry-module-1-topic-c-lesson-13

Notice that unlike this blog, this New York lesson prescribes what questions are meant to be the Opening Activity, Exercises, Exit Ticket, and Problem Se, as well as how many minutes are to be devoted to each learning task. I don't do that -- but of course, I did sort of hint that the folding tasks at the beginning of my reflection lessons would make good Opening Activities. Actually, yesterday's task of reflecting the letter F in the y-axis and then the x-axis would also make a good Opening Activity or Anticipatory Set -- I considered posting a separate page to perform the reflections, but I want to conserve paper and avoid excessive visits to the Xerox machine. Of course, nothing is stopping the teacher from having the students take a blank sheet of paper, draw a coordinate plane with a large F in the first quadrant, and then perform the reflections, possibly even by folding.

We observe that the New York lesson also uses positive angles to denote counterclockwise and negative angles to denote clockwise. But it also uses the abbreviation CW for clockwise. Also, the New York lesson uses the letter R to denote a rotation, with two subscripts -- the first being the center and the second being the magnitude (that is, the angle). The U of Chicago uses the letter r for reflection, but no letter for translation. This blog doesn't use subscript function notation at all.

Also, I found it interesting that the New York lesson uses the Greek letter theta to denote the measure of an angle. I rarely see the theta symbol used in math texts until Precalculus, or perhaps an Honors Algebra II class with Trigonometry.

For all our talk about how the Wu and New York lessons don't define rotations as a composition of reflections, it does define rotations as a composition of rotations. In particular, a rotation of 180 degrees is the composition of two 90-degree rotations, and rotations with magnitudes greater than 180 are also the composition of rotations whose magnitudes add up to the given magnitude. In the U of Chicago text, 180 degrees is not a special case -- it's included as part of the Two Reflection Theorem for Rotations (where a "non-obtuse" angle could be a right angle, so twice its measure would be a straight angle, or 180 degrees). For larger angles, the text states that we can add or subtract multiples of 360 until the magnitude is between -180 and 180 degrees.

The New York lesson begins with its Opening Task, where the students cut out an angle and then use it to perform the rotation of a given figure. The first exercise asks students to use a protractor to measure the magnitude of a rotation. The second exercise is basically the same task as the first.

But at Exercise 3, things begin to get interesting. This exercise is similar to the first two, where we're rotating the letter M. But there's a problem -- we don't know the center of rotation! So the lesson gives the following algorithm for finding the center of rotation:

a. Draw a segment connecting points A and A'.
b. Using a compass and straightedge, find the perpendicular bisector of this segment.
c. Draw a segment connecting points B and B'.
d. Find the perpendicular bisector of this segment.
e. The point of intersection of the two perpendicular bisectors is the center of rotation. Label this point P.

Can we prove that this algorithm works? It looks a bit familiar -- because it is similar to the construction of an circle through three noncollinear points. But I threw that construction out when we did Section 4-5 -- because the proof that the construction works requires a Parallel Postulate. In this case, even if we had a Parallel Postulate, it's not obvious how to prove that the construction from a-e above works -- we have four points, not just three, and there's nothing saying that three of the points, or even all four of them, aren't collinear. It could be that the perpendicular bisectors of AA' and BB' do not intersect at all, much less intersect at the center of the rotation.

As it turns out, we don't have to worry about that. We already know -- that is, we are given -- that the points A' and B' are rotation images of A and B respectively. That is, we are given that there exists a point P such that P is the center of said rotation. And so we can prove that P lies on both of the perpendicular bisectors -- that is, we can prove that the bisectors must intersect!

And so here is our proof, in paragraph form:

Given: A', B' are the images of A, B under a rotation centered at C
Prove: C lies on the perpendicular bisectors of AA', BB'.

Proof:
The result follows directly from the proof of the Two Reflections Theorem for Rotations. Indeed, here we label the center C rather than P for emphasis -- result (3) of yesterday's proof tells us that AC = A'C (and similarly, BC = B'C). By definition of equidistant, C is equidistant from A and A' (and likewise, C is equidistant from B and B'). Therefore, by the Converse to the Perpendicular Bisector Theorem, C lies on the perpendicular bisector of AA' (and similarly that of BB' as well). QED (The proof sketch given in the New York lesson mentions chords of a circle, which the U of Chicago text doesn't cover until Chapter 15. But we just used that Converse instead.)

Notice that this lesson is clearly a straightedge and compass lesson. But we can find the perpendicular bisector of a segment by folding -- in this case, we fold A over so that it lies on top of A'. The crease of the fold is the perpendicular bisector of AA'. This may seem a bit tricky -- we used folds earlier in order to reflect points, and yes, the reflection image of A over the first fold line is A', yet A' is supposed to be a rotation image of A, not a reflection image. As it turns out, A' can be the image of A via either a reflection or a rotation. The folding is simply to find the perpendicular bisector, not to perform a reflection.

We notice that finding the center rotation is more difficult than finding a line of reflection. If we are given a point A and its reflection image A' -- as long as A' is not A itself, we immediately know the reflecting line to be the perpendicular bisector of AA'. But to find the center of rotation, it's not enough to know a single point A and its image A' (unless A' is A itself) -- we saw that a second point B and its image B' were necessary. Because of this, we say that a reflection has one degree of freedom, while a rotation has two degrees of freedom. (A translation, like a reflection, also has only one degree of freedom.)

For the problem set, I decided to omit the part about a straightedge and compass. The reason for this is that the New York lesson assumes that the students can construct 45-, 60-, and 120-degree angles using only a straightedge and compass. Technically, the students in our course can do this -- we know how to draw perpendicular lines with their 90-degree angles and bisect them to get 45 degrees, and the First Theorem of Euclid's Elements lesson gives us equilateral triangles, hence 60 degrees -- but we can't prove the angle measures of an equilateral triangle until Chapter 5. So we expect the students just to use a protractor. Problem Set 3 discusses vertical angles. Interestingly enough, this is how Wu proves the Vertical Angles Theorem -- but we already proved it in Chapter 3 another way. The answers to Problem Set 4 and 5 are "rectangle" and "rhombus" respectively, but we don't define either of these terms until Chapter 5 as well. So I threw these out.

The New York lesson also mentions Geogebra as a possible tool to perform the rotations -- I suggested that earlier this month as well.