Lesson 7-8: The SAS Inequality (Day 78)

Today I subbed in a middle school special ed history class. Like most special ed classes, there are aides plus co-teaching today, and thus there's no "A Day in the Life."

This is my second visit to this classroom -- I mention the first visit in my May 20th post. Once again, my last visit was during the previous school year, and so there are some changes in the regular teacher's schedule. He no longer has a reading class -- it's all history now, with two seventh grade and two eighth grade courses. The co-teaching class is also seventh grade history.

The eighth grade U.S. History classes have a worksheet on the National Guard, while the seventh grade World History classes have a similar worksheet on the Code of Hammurabi (which is technically a sixth grade topic, reviewed this year for comparison). For each respective grade, the aide during the earlier period leaves to make copies of the worksheet -- while waiting for her to return, I sing both "One Billion Is Big" and "Ghost of a Chance" -- my recent go-to songs. (Hmm, perhaps I should have sung "The Packet Rap" instead since there are several worksheets that can be stapled in a packet.) In each grade, the aide makes enough copies for the later class.

Even though all classes have aides, in one period the aide must go home early, so let's take a look at classroom management during that class (which was seventh grade). Even though most students work hard on the packet, they are a bit loud. I must write the name of two students -- a boy and a girl -- when they start arguing over which one stole a pencil from the other. (According to the Code of Hammurabi, the thief would have to give the victim either ten or thirty pencils in retribution!)

The co-teaching class, though also seventh grade, is doing something completely different. They are learning about Africa, including the three great medieval kingdoms (Ghana, Mali, and Songhai).

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

If a, b, c, d are integers, ax^3 + bx^2 + cx + d = 0. The solutions for a certain cubic equation are -1, -4, and -6. What is the coefficient of the x^2-term of this equation?

You might notice that this question has nothing to do with Geometry. Yet I'm posting it today anyway, for two reasons. First, it's because it contains a slight error. Pappas intends to use the result that the sum of the roots equals -b. Since the three solutions are -1, -4, -6, their sum is -11. Therefore b = 11, and of course, today's date is the eleventh.

But unfortunately, this only works if a = 1 -- and nowhere in this problem does she indicate that the leading coefficient is 1. If a isn't 1, then we can always divide by a, so in general, -b/a is the actual sum of the roots. Then again, there's no way to know what a is in this problem -- Pappas assumes that we'll assume that a = 1.

Now I have a second reason for bringing this problem up -- the Putnam. I wish to continue briefly discussing some of the Putnam problems. And Problem A3 is related to polynomials:

https://artofproblemsolving.com/community/c7h1966308_2019_putnam_a3

I won't cut-and-paste the problem here, since as I found out, it might not appear well here. Instead, I'll just mention a few things about the problem.

Today's Pappas problem is a degree-3 (cubic) polynomial. Meanwhile, the Putnam problem has a degree-2019 polynomial. That's right -- the degree is the current year. It's typical for one of the Putnam problems to mention the current year. In fact, I recommend spending the night before the Putnam factoring the current year, since the factorization is often important. This year, though, 2019 is just the degree of a polynomial, so its factorization isn't needed.

(For next year's Putnam, notice that 2020 = 2^2 * 5 * 101, which might come up in a problem.)

It turns out that the polynomial that solves the problem has its coefficients in a geometric sequence -- the constant is 1, the leading coefficient is 2019, and ratio of successive terms is 1 over the 2019th root of 2019 (that is, b/a = (1/2019)^(1/2019)).

The roots of this polynomial equal this common ratio b/a multiplied by each of the complex 2020th roots of unity (omitting 1 itself). The sum of these roots is indeed -b/a. Notice that 2018 of the 2019 values cancel out -- all except -b/a (since we conveniently omitted b/a itself).

Once again, I save the details of the proof for the link. There's one interesting step in the proof, which two of the posters call "AM-GM." This means "Arithmetic Mean Geometric Mean Inequality" -- the arithmetic mean (average) of a set of positive numbers always exceeds their geometric mean. We learn about the geometric mean of a pair of real numbers in Lesson 14-2 of the text.

In fact, it's easy to use a right triangle to verify AM-GM for two numbers. If we find that the altitude to the hypotenuse of a triangle divides that hypotenuse into two segments a and b, then the length of that altitude is the geometric mean sqrt(ab). The arithmetic mean (a + b)/2 isn't just half of the hypotenuse -- it's also the length of the median to the hypotenuse. The altitude and median are themselves two sides of a smaller right triangle -- the altitude (GM) becomes a leg, while the median (AM) becomes the new hypotenuse. Since the hypotenuse is always the longest side, this implies that AM always exceeds GM.

Of course, in today's Putnam problem, we aren't just finding the AM and GM of two numbers, but of a set of 2019 numbers.

Lesson 7-8 of the U of Chicago text is called "The SAS Inequality." But don't worry -- it doesn't appear in the modern Third Edition of the U of Chicago text at all!

And so this is what I wrote last year about this lesson:

But what is in Section 7-8? This section covers the SAS Inequality -- often known as the Hinge Theorem, according to the Exploration Question at the end of the section. Dr. Franklin Mason also calls this theorem the Hinge Theorem.

I'm of two minds as to whether to include Lesson 7-8 on this blog. Notice that Dr. M includes the Hinge Theorem, Triangle Inequality, and all other inequalities related to triangles in his Chapter 5. But we've been waiting because many of these theorems depend on indirect proof, so we were waiting for Chapter 13. Yet on this blog, I posted the Circumcenter Concurrency Theorem even though it also requires an indirect proof (but, as Dr. Wu pointed out, the part that requires indirect proof can be handwaved over).

In the end, I post Section 7-8 today. If one wants, one can just skip to the review questions.

I wrote more about the significance of the SAS Inequality in my September 11th post. Also, three years ago I included a worksheet with an SSS Inequality -- that's what the "indirect proof" was for. In the end, indirect proof and the SSS Inequality have been dropped. Oh, what the heck -- let me reblog what I wrote about SSS Inequality in this post below, even though it's no longer on the worksheet.

As I mentioned before, this section gives a proof of the SAS Inequality using the Triangle Inequality. As usual, I have decided to convert the proof to two-column format. The figure accompanying the proof gives two triangles, ABC and XYZ.

SAS Inequality Theorem:
If two sides of a triangle are congruent to two sides of a second triangle, and the measure of the included angle of the first triangle is less that the measure of the included angle of the second, then the third side of the first triangle is shorter than the third side of the second.

Given: AB = XY, BC = YZ, angle B < angle Y
Prove: AC < XZ.

Proof:
Statements                                 Reasons
1. AB = XY, etc.                          1. Given
2. exists isometry T s.t. A'B' is   2. Definition of congruent
XY, C' same side of XY as Z
3. C'Y = ZY                                 3. Isometries preserve distance
4. Let m symmetry line C'YZ      4. Isosceles Triangle Symmetry Theorem
(m intersects XZ at Q)     
5. m perp. bis. C'Z                      5. In isosceles triangle, angle bis. = perp. bis.
6. QC' = QZ                                6. Perpendicular Bisector Theorem
7. A'C' < A'Q + QC'                    7. Triangle Inequality
8. AC < XQ + QZ                        8. Substitution
9. XQ + QZ = XZ                        9. Betweenness Theorem (Segment Addition)
10. AC < XZ                                10. Substitution

We see that the proof is similar to that of SAS Congruence Theorem, except that this isometry puts C' on the same side of XY as Z, rather than the opposite side. Dr. M gives two proofs of the SAS Inequality (which he calls "the Hinge Theorem," a name mentioned in Exploration Question 19 in our text) -- his second proof is nearly identical to that given in the U of Chicago. In his first proof, Dr. M uses TASI (Unequal Angles Theorem) directly without invoking the Triangle Inequality -- but we would still be dependent on a theorem not proved until Chapter 13 in the U of Chicago.

Both Dr. M and Glencoe state a converse to SAS Inequality -- Glencoe calls it SSS Inequality. Dr. M hints at this proof -- we can use the same strategy that we used to derive Unequal Angles Theorem from its converse, Unequal Sides Theorem. We prove it indirectly:

SSS Inequality Theorem:
If two sides of a triangle are congruent to two sides of a second triangle, and the third side of the first triangle is shorter than the third side of the second, then the measure of the included angle of the first triangle is less that the measure of the included angle of the second.

Given: AB = XY, BC = YZ, AC < XZ.
Prove: angle B < angle Y

Indirect Proof:
Assume not. Then angle B is either less than or equal to angle Y.

Case 1: angle B = angle Y. Then triangles ABC and XYZ are congruent by SAS Congruence, and so AC = XZ, a contradiction.

Case 2: angle B > angle Y. Then AC > XZ by SAS Inequality, a contradiction.

In either case we have a contradiction of AC < XZ. Therefore angle B < angle Y. QED

For Euclid, the SAS Inequality is his Proposition 24. Dr. M's first proof is based on Euclid:

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI24.html

and the converse, the SSS Inequality, is Euclid's Proposition 25:

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI25.html