Section 12-6: The Fundamental Theorem of Similarity (Day 86)

Section 12-6 of the U of Chicago text covers the Fundamental Theorem of Similarity. As its name implies, it is the most important theorem related to dilations and similarity. Here is how this theorem is stated in the U of Chicago:

Fundamental Theorem of Similarity (U of Chicago):
If G ~ G' and k is the ratio of similitude, then
(a) Perimeter(G') = k * Perimeter(G) or ...
(b) Area(G') = k^2 * Area(G) or ...
(c) Volume(G') = k^3 * Volume(G) or ...

Notice how I had to rewrite this theorem so that it fits into ASCII. Here the * and ^ symbols denote multiplication and exponentiation, respectively -- these symbols should be recognizable as they appear on TI graphing calculators. The "or ..." sections refer to the text rewriting each equation as a ratio, so that the ratio of the perimeters is k, the ratio of the areas is k^2, and so on, but that is rather awkward to write in ASCII.

David Joyce describes this theorem in his criticism of the Prentice-Hall text:

The proofs are omitted for the theorems which say similar plane figures have areas in duplicate ratios, and similar solid figures have areas in duplicate ratios and volumes in triplicate rations. At least there should be a proof that similar triangles have areas in duplicate ratios; that's easy since the areas of triangles are already known.

But we won't be proving the U of Chicago or Prentice-Hall versions of this theorem. Instead, we will be proving Dr. Wu's version of the Fundamental Theorem of Similarity. In his document on Common Core Geometry, Wu numbers this as Theorem 22:

Theorem 22. (Fundamental Theorem of Similarity (FTS)) Let D be a dilation
with center O and scale factor r > 0. Let P and Q be two points so that line PQ does
not contain O. If D(P) = P' and D(Q) = Q', then P'Q' | | PQ and P'Q' = r * PQ

Let's observe the differences between the U of Chicago's FTS and Wu's FTS. First of all, the U of Chicago's FTS refers to area and volume, but once again, we won't discuss area or volume in today's lesson because we skipped Chapters 8 through 10.

More importantly, notice that the U of Chicago's FTS refers to two similar figures G and G'. But we haven't even defined similarity yet -- Wu's FTS describes a property of the dilations themselves. So in some ways, Wu's Fundamental Theorem of Similarity is a misnomer. Wu's theorem discusses how a dilation D acts on a segment PQ. It states that its image, P'Q', is parallel to the preimage and exactly r times as long as the preimage, where r is the scale factor of the dilation D.

As I warned yesterday, Wu's FTS is one of the most difficult to prove. He does so by dividing the FTS into various cases, which he labels Lemmas 13 and 14:

Lemma 13. FTS is valid when the scale factor r is a positive integer.

Lemma 14. FTS is valid for all unit fractions 1/n, where n is a positive integer.

Recall that a lemma is just a very low-level theorem. I use the term low-level theorem to refer to theorems that are mainly used to prove other theorems. A lemma is even lower than that -- a lemma is used only once, to prove one very specific theorem -- in this case, the FTS. The two lemmas refer to two specific cases of the FTS that are used to prove the general case.

Now notice that some of the proof techniques that Wu uses to prove these lemmas are some of those that we discussed here last month. He proves Lemma 13 using induction -- the same technique that the U of Chicago uses to prove the Center of a Regular Polygon Theorem and the college Putnam exam uses to prove Question B-1 on "overexpansions." And the initial case, where r = 2, corresponds to the Midpoint Connector Theorem.

My original plan was for these theorems, plus the Median Concurrency Theorem, to make up the last lesson before the first semester final. But then I scrapped that because I had subbed in a geometry class that day and wanted to give the review questions on triangle congruence, since the point of a blog is to post what I see in an actual classroom, not from a book or theorem list. I posted proofs of these theorems on this blog -- but not on any printed worksheet -- because I'd already looked ahead in Wu and saw the dependence of his FTS on these theorems. This would have provided a nice bridge from the first to the second semester.

I haven't decided how I might remedy this when I redo my pacing guide for next year. I might lump some of the early Chapter 7 material along with Chapter 6, even if this would put Thanksgiving (early start) or Christmas (Labor Day start) between the early and latter parts of Chapter 7. That way, I can get the proofs of the medium-level theorems SSS, SAS, and ASA out of the way, and focus more on using SSS, SAS, and ASA to prove high-level theorems (such as the Midpoint Connector Theorem), which is more typical of a traditionalist geometry class.

Well, let's look at how Wu proves his Lemma 13. I've stated that he proves this theorem using induction on the scale factor r. (Once again, I can't stress enough that whenever a mathematician sees the statement "prove for all natural numbers (or positive integers) n that...," the first thing that should come to his or her mind is "induction.")

The base case is r = 2. (Actually, the base case is r = 1. But a dilation with scale factor 1 is what the U of Chicago calls the identity transformation, so P' is just P and Q' is just Q. Then the proof is trivial because every segment has the same length as itself and is -- according to the U of Chicago definition -- parallel to itself.) This is the already-proved Midpoint Connector Theorem.

Now the inductive case takes us from r to r + 1. Wu shows us how the case r = 2 leads to a proof of the case r = 3. It is similar to the proof of the Midpoint Connector Theorem, and uses that theorem in its own proof. Let me convert the proof given by Wu into two-column form:

Given: OP' = 3OP, OQ' = 3OQ

Prove: P'Q' | | PQ, P'Q' = 3PQ

Proof:

Statements                                    Reasons

1. OP' = 3OP, OQ' = 3OQ           1. Given

2. W on ray PQ with PW = 3PQ,  2. Ruler/Point-Line Postulate

    U on ray QQ' with OQ = QU, 

    V on ray QW with PQ = QV

3. UV | | Q'W, Q'W = 2UV             3. Definition of Midpoint and Midpoint Connector Theorem

4. QQ' = 2OQ, QW = 2PQ            4. Betweenness/Segment Addition Theorem

5. OPUV is a parallelogram          5. Diagonals Bisect Pgram Test

6. UV | | OP, UV = OP                  6. Pgram Consequences

7. OP (same as P'P) | | Q'W           7. Transitivity of Parallels Theorem

8. PP' = Q'W                                  8. Algebra and Substitution

9. PP'Q'W is a parallelogram        9. One Pair Parallel & Congruent Pgram Test

10. P'Q' | | PW, P'Q' = PW            10. Pgram Consequences

11. P'Q' | | PQ, P'Q' = 3PQ           11. Substitution

To make this into a complete proof by induction, we must induct on r -- that is, show how the case for r leads to the case for r + 1. To do this in a two-column proof, since the case for r is already proved, we can actually include it with the given statements! So we write:

Given: OP' = (r + 1) * OP, OQ' = (r + 1) * OQ, FTS valid for scale factor r

Prove: P'Q' | | PQ, P'Q' = (r + 1) * PQ

Proof:

Statements                                                    Reasons

1. OP' = (r + 1) * OP, OQ' = (r + 1) * OQ  1. Given

2. W on ray PQ with PW = (r + 1) * PQ,     2. Ruler/Point-Line Postulate

    U on ray QQ' with OQ = QU, 

    V on ray QW with PQ = QV

3. QQ' = r * OQ, QW = r * PQ                     3. Betweenness/Segment Addition Theorem

4. UV | | Q'W, Q'W = r *UV                          4. Given (FTS for scale factor r)

5. OPUV is a parallelogram                          5. Diagonals Bisect Pgram Test

6. UV | | OP, UV = OP                                  6. Pgram Consequences

7. OP (same as P'P) | | Q'W                          7. Transitivity of Parallels Theorem

8. PP' = Q'W                                                 8. Algebra and Substitution

9. PP'Q'W is a parallelogram                        9. One Pair Parallel & Congruent Pgram Test

10. P'Q' | | PW, P'Q' = PW                           10. Pgram Consequences

11. P'Q' | | PQ, P'Q' = (r + 1) * PQ              11. Substitution

Indeed, strictly speaking, this proof still works for r = 2 using r = 1 as a base case. In that case, the points U and Q' become identical, as do V and W, and so step 4 becomes, as I mentioned earlier, the trivial statement that Q'W is both congruent and parallel to itself.

But, as Wu points out, proofs by induction are very confusing for most high school students. This is why Wu suggests proving r = 2 (as the Midpoint Connector Theorem) and r = 3 separately.

This concludes Lemma 13. For Lemma 14, Wu notes that if D is the dilation with center O and scale factor 1/n, then its inverse is the dilation with center O and scale factor n. So if D maps P and Q to P' and Q' respectively, then the dilation with scale factor n maps P' and Q' to P and Q respectively. I point out that this inverse dilation, with scale factor n a positive integer, is a case for which FTS has already been proved as Lemma 13. And so from PQ = n * P'Q', we divide both sides by n to obtain what we wish to prove, P'Q' = (1/n) * PQ.

Now for general fractions m/n, Wu considers the dilation with center O and scale factor m/n to be the composite of two other dilations, one with scale factor 1/n, the other with scale factor m. And these are cases for which FTS has already been proved, as Lemmas 13 and 14.

This is the last case that Wu considers, but there are still many values of r missing. For example, if we take an isosceles right triangle and cut it in half, then the two smaller triangles are similar to the original triangle -- but the scale factor from the smaller to the original triangle is exactly the square root of two, which is irrational. But Wu only proves FTS for rational scale factors!

Wu invokes what he calls the "Fundamental Assumption of School Mathematics" -- which states that knowing that FTS is valid for all positive rational scale factors is enough to prove that it is valid for all positive real scale factors. This "Fundamental Assumption" is often invoked in Algebra II or more likely Pre-Calculus classes, where the laws of exponents may be proved for rational exponents but assumed for irrational exponents.

David Joyce alludes to this "Fundamental Assumption" when he writes:

Appropriately for this level, the difficulties of proportions are buried in the implicit assumptions of real numbers.

These assumptions are studied beyond calculus, in college-level real analysis courses. This is why Joyce, Wu, and all teachers should hand-wave over this technicality.

For my worksheet, I've decided to replace Wu's variables D for the dilation and r for the scale factor with S and k, respectively, so that they agree with the symbols that appeared in yesterday's lesson.