Monday, December 14, 2015

Lesson 11-5: The Midpoint Connector Theorem (Day 71)

Lesson 11-5 of the U of Chicago text is on the Midpoint Connector Theorem -- a result that is used to prove both the Glide Reflection Theorem and the Centroid Concurrency Theorem. Last year I only briefly mentioned the Midpoint Connector Theorem on the way to those higher theorems, and then when we actually reached Chapter 11 I only covered it up to Lesson 11-4, as I knew that I'd already incorporated 11-5 into the other lessons. But this year, I'm giving 11-5 its own worksheet.

Midpoint Connector Theorem:
The segment connecting the midpoints of two sides of a triangle is parallel to and half the

As I mentioned last week, our discussion of Lesson 11-5 varies greatly from the way that it's given in the U of Chicago text. The text places this in Chapter 11 -- the chapter on coordinate proof -- and so students are expected to prove Midpoint Connector using coordinates.

It also appears that one could use similar triangles to prove Midpoint Connector -- to that end, this theorem appears to be related to both the Side-Splitting Theorem and its converse. Yet we're going to prove it a third way -- using parallelograms instead. Why is that?

It's because in Dr. Hung-Hsi Wu's lessons, the Midpoint Connector Theorem is used to prove the Fundamental Theorem of Similarity and the properties of coordinates, so in order to avoid circularity, the Midpoint Connector Theorem must be proved first. In many ways, the Midpoint Connector Theorem is case of the Fundamental Theorem of Similarity when the scale factor k = 2. Induction -- just like the induction that we saw last week -- can be used to prove the case k = n for every natural number n, and then Dr. Wu uses other tricks to extend this first to rational k and ultimately to real k.

I've decided that I won't use Wu's Fundamental Theorem of Similarity this year because the proof that he gives is much too complex. Instead, we'll have an extra postulate -- either a Dilation Postulate that gives the properties of dilations, or just one of the main similarity postulates like SAS. I won't make a decision on that until the second semester.

Nonetheless, I still want to give this parallelogram-based proof of the Midpoint Connector Theorem, since this is a proof that students can understand, and we haven't taught them yet about coordinate proofs or similarity.

On this worksheet, I include several interesting bonus problems. First, notice that the first thing we see in the U of Chicago text is a picture of Sierpinksi's Triangle -- one of the fractals that we learned about two months ago when reading Benoit Mandelbrot's book. It's included here since it's closely related to midpoints. Then there is a problem from the text about Midpoint Quadrilaterals. Wu's proof is added as the last bonus question.

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