Monday, April 3, 2017

Lesson 12-10: The Side-Splitting Theorem (Day 130)

This is what Theoni Pappas writes on page 93 of her Magic of Mathematics:

"Engineers, architects, and other designers have not hesitated to embrace the computer in their creations. With just a few clicks of the mouse a building could easily be modified, a plane could be rotated to show all possible perspectives, cross-sections could be added, parts added or removed effortlessly."

This section is all about computer art. On this page, Pappas provides a picture of an "oil painting" that was really generated from a photograph using a computer. We see that creating art on the computer ultimately goes back to geometry -- that is, to mathematics.

Meanwhile, Barnes and Noble has announced a third Educator Week -- and just like the first two discount weeks, I plan on purchasing one science book and one math book. For the first weekend, I decided to choose my science book: Ian Stewart's Calculating the Cosmos.

This is the second Ian Stewart book I purchased during an Educator Week -- the first was Incredible Numbers six months ago. Stewart is a British mathematician, and as the title implies, Calculating the Cosmos attempts to connect mathematics to science.

Here is the Table of Contents:

1. Attraction at a Distance
2. Collapse of the Solar Nebula
3. Inconstant Moon
4, The Clockwork Cosmos
5. Celestial Police
6. The Planet that Swallowed its Children
7. Cosimo's Stars
8. Off on a Comet
9. Chaos in the Cosmos
10. The Interplanetary Superhighway
11. Great Balls of Fire
12. Great Sky River
13. Alien Worlds
14. Dark Stars
15. Skeins and Voids
16. The Cosmic Egg
17. The Big Blow-Up
18. The Dark Side
19. Outside the Universe

Notice that unlike the last few science books that I purchased during Educator Weeks, this text isn't directly related to middle school science at all. In middle school, there are a few simple topics related to the universe, but that is all.

On the other hand, both Pappas and Stewart tie math to science and technology, and provide students with reasons for learning advanced math -- especially graphing. Computer-based art, after all, is prominent in video games, especially the modern 3D games. The ideas of perspective help make the 3D images appear lifelike. The conic sections of Algebra II, meanwhile, are the orbital shapes of several bodies in outer space.

I won't feature Stewart's book as a side-along reading book, since I don't have time -- instead, I'll continue to cover one page a day from Pappas.

There's one more thing to mention before I get to the next lesson in the U of Chicago, I know that I'm not writing about what I'm teaching in middle school much these days, but I can't help but mention two geometry review worksheets that I gave to middle schoolers today. The first homework sheet is for seventh grade (special angles and triangles) and the other is eighth grade (transformations). There is also a Warm-Up worksheet for eighth grade (plus an answer sheet), from the website:

All worksheets are review of material that I taught to my students around December and January. These aren't Illinois State worksheets, so I post them today, right below the Lesson 12-10 worksheets. (Note that today is coding Monday.)

This is what I wrote last year about today's lesson:

The U of Chicago version of the theorem is:

Side-Splitting Theorem:
If a line is parallel to a side of a triangle and intersects the other two sides in distinct points, it "splits" these sides into proportional segments.

And here's Dr. Wu's version of the theorem:

Theorem 24. Let triangle OPQ be given, and let P' be a point on the ray OP not equal toO. Suppose a line parallel to PQ and passing through P' intersects OQ at Q'. ThenOP'/OP = OQ'/OQ = P'Q'/PQ.

Notice that while the U of Chicago theorem only states that the two sides are split proportionally, Wu's version states that all three corresponding sides of both sides are proportional.

Moreover, the two proofs are very different. The U of Chicago proof appears to be a straightforward application of the Corresponding Angles Parallel Consequence and AA Similarity. But on this blog, we have yet to give the AA Similarity Theorem. So how can Wu prove his theorem?

We've seen several examples during the first semester -- a theorem may be proved in a traditionalist text using SSS, SAS, or ASA Congruence, but these three in Common Core Geometry are theorems whose proofs go back to reflections, rotations, and translations. Instead, here we skip the middle man and prove the original high-level theorem directly from the transformations. We saw this both with the Isosceles Triangle Theorem (proved from reflections in the U of Chicago) and the Parallelogram Consequences (proved from rotations in Wu).

So we shouldn't be surprised that Wu proves his version of the Side-Splitting Theorem using transformations as well. Naturally, Wu uses dilations. In fact, the names that Wu gives the points gives the game away -- O will be the center of the dilation, and P and Q are the preimage points, while P' and Q' are the images.

Here is Wu's proof: He considers the case where point P' lies on OP -- that is, the ratio OP'/OP, which he labels r, is less than one. This is mainly because this case is the easiest to draw, but the proof works even if r is greater than unity. Let's write what follows as a two-column proof:

Given: P' on OPQ' on OQPQ | | P'Q'r = OP'/OP
Prove: OP'/OP = OQ'/OQ = P'Q'/PQ

Statements                                           Reasons
1. P' on OPQ' on OQPQ | | P'Q'      1. Given
2. OP' = r * OP                                    2. Multiplication Property of Equality
3. Exists Q0 such that OQ0 = r * OQ  3. Point-Line/Ruler Postulate
4. For D dilation with scale factor r,    4. Definition of dilation
    D(Q) = Q0, D(P) = P'
5. P'Q0 | | PQP'Q0 = r * PQ               5. Fundamental Theorem of Similarity
6. Lines P'Q0 and P'Q' are identical      6. Uniqueness of Parallels Theorem (Playfair)
7. Points Q0 and Q' are identical          7. Line Intersection Theorem
8. OQ' = r * OQOP' = r * OP,            8. Substitution (Q' for Q0)
    P'Q' = r * PQ
9. OP'/OP = OQ'/OQ = P'Q'/PQ = r    9. Division Property of Equality

Now the U of Chicago text also provides a converse to its Side-Splitting Theorem:

Side-Splitting Converse:
If a line intersects rays OP and OQ in distinct points X and Y so that OX/XP = OY/YQ, then XY | | PQ.

The Side-Splitting Converse isn't used that often, but it can be used to prove yet another possible construction for parallel lines:

To draw a line through P parallel to line l:
1. Let XY be any two points on line l.
2. Draw line XP.
3. Use compass to locate O on line XP such that OX = XP.
4. Draw line OY.
5. Use compass to locate Q on line OY such that OY = YQ.
6. Draw line PQ, the line through P parallel to line l.

This works because OX = XP and OY = YQ, so OX/XP = OY/YQ = 1.

The U of Chicago uses SAS Similarity to prove the Side-Splitting Converse, but Wu doesn't prove any sort of converse to his Theorem 24 at all. Notice that many of our previous theorems for which we used transformations to skip the middle-man, yet the proofs of their converses revert to the traditionalist proof -- once again, the Parallelogram Consequences.

Another difference between U of Chicago and Wu is that the former focuses on the two segments into which the side of the larger triangle is split, while Wu looks at the entire sides of the larger and smaller triangles. This is often tricky for students solving similarity problems!

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