Monday, January 12, 2015

Section 12-5: Similar Figures (Day 90)

Section 12-5 of the U of Chicago text is on Similar Figures. Here, we finally define the most important concept of the chapter -- similarity.

We begin with the concept of a similarity transformation. It's very similar to isometry. Two figures are congruent if and only if there is an isometry mapping one to the other. The isometry may be a reflection, translation, rotation, or even a glide reflection, but it must be an isometry.

Well, the most general transformation for similar figures is a similarity transformation. Here is the U of Chicago's definition:

A transformation is a similarity transformation if and only if it is the composite of size changes and reflections.

Some texts may add "translations, and rotations" to the end of this definition. But recall that these two are already the composite of reflections, so adding that phrase is redundant. Now we define similar:

Two figures F and G are similar, written F ~ G, if and only if there is a similarity transformation mapping one to the other.

Meanwhile, here is Dr. Wu's definition:

We say S is similar to S', in symbols, S ∼ S', if there is a dilation D so that D(S) is congruent to S'.

So for Wu, only the dilation D is mentioned, since "is congruent" already takes care of the additional reflections that may be needed to map S completely onto S'. Also, Wu uses just the word "similarity" where the U of Chicago would write "similarity transformation."

There is a hierarchy of transformations given in the U of Chicago text. All the transformations that matter in a high school geometry text are similarity transformations. (I actually have seen a high school text that defines something called an affine transformation. These preserve betweenness, collinearity, and parallelism, but not angle measure. If these were included on the hierarchy, all similarity transformations are affine transformations, but an example of an affine transformation that's not a similarity transformation is a transvection or a "shear." The image of a square under a shear is a general rhombus.)

Continuing down the hierarchy, branching off from similarity transformations are both size changes (dilations) and isometries, and of course, we already know that our four isometries are translations, rotations, reflections, and glide reflections. Finally, we have the identity transformation, which is a dilation, a translation, and a rotation. After all, the identity transformation is a dilation of scale factor 1, a translation of the zero vector, and a 0-degree rotation.

Now we reach the main theorem of the section:

Similar Figures Theorem:
If two figures are similar, then:
(a) corresponding angles are congruent;
(b) corresponding lengths are proportional.

This corresponds to Theorem 25 of Wu. It's basically the same theorem, except that Theorem 25 specifically refers to triangles. Technically speaking, the Similar Figures Theorem refers to all similar figures, but of course, the vast majority of figures whose similarity students are asked to consider happen to be triangles.

The U of Chicago text uses the phrase "ratio of similitude." I will instead follow Wu and Common Core by using "scale factor" for both dilations and general similarity transformations.

Over the weekend, I purchased a new geometry book. It is called Girls Get Curves: Geometry Takes Shape, written by Danica McKellar, an actress and former math major. She graduated from UCLA a year before I attended there, and this is the fourth math book that she has written.

As the title implies, McKellar's target audience is female, so why would I, a male teacher, be interested in her books? The answer is that I already know geometry -- the ones who need to learn geometry are my students, many of whom are girls (unless I end up working at some sort of all-boys religious school). When I was a student teacher of Algebra I, many of my female students struggled, and I wonder whether it's because I, as a male, had a subconscious male bias.

So counteract my male bias and help my female students, I decided to buy McKellar's books. I purchased her first book, which appears to be around the level of Common Core Grade 7, right after completing my student teaching, but I didn't purchase any of her others until this weekend. Since this is a geometry blog, I just had to get her geometry text. Maybe if I end up teaching pre-algebra or an Algebra I course, I'll buy her other two books.

Here are the chapters of McKellar's geometry text:

1. Introduction to Logic and Reasoning
2. Geometry Basics
3. Working With Diagrams
4. Introduction to Two-Column Proofs
5. Properties...and More Proofs!
6. Introduction to Triangles
7. Congruence Transformations
8. Congruent Triangles
9. CPCTC and More Proofs
10. Proofs by Contradiction (AKA Indirect Proofs)
11. Right Triangles and the Pythagorean Theorem
12. Intro to Polygons...and Deriving Formulas
13. Parallel Lines and Transversals
14. Proving Lines Are Parallel
15. Introduction to Quadrilaterals
16. Properties of Parallelograms and Other Quadrilaterals
17. Similar Triangles
18. Circles
19. Arcs and Arc Length
20. Finding Perimeter and Area
21. Surface Area and Volume

Notice that McKellar's book was published in 2012 -- after the advent of Common Core. And so we see that McKellar's text is one of the few to use congruence transformations -- that is, isometries (in Chapter 7) -- to show that triangles are congruent.

The topic of today's post -- similar triangles, is McKellar's Chapter 17. And she does briefly mention dilations before introducing AA Similarity -- that's right, she omits SSS and SAS Similarity. Here McKellar does return to transformations from time to time when writing the proofs -- for example, when trying to prove that two overlapping triangles are similar, she reminds the readers that they can rotate one of the triangles, just as they learned in Chapter 7, to make the correspondence clearer.

Because her book is for girls, McKellar is sure to add her feminine touch to every chapter. For Chapter 17, McKellar discusses "Life-Sized Barbie" -- what would a human woman look like if she had the same proportions at the iconic doll? McKellar calculates:

"So, it's pretty widely accepted that Barbie is built at 1/6 scale. In other words, the ratio of 'Barbie to human' would be 1:6. By that scale, because the dolls are about 11 1/2 inches tall, if Barbie were life sized, she'd be around 6 * 11.5 = 69 inches, in other words, 5 feet, 9 inches tall.

"For the waist, w, we'd get: 1/6 = 3.5/w -> w * 1 = 3.5 * 6 -> w = 21 inches. Okay, that's waaaaay too small for any real woman's waist, let alone one who's 5'9" tall.

"However, I'm not sure what to say about the shoe size. Her 1-inch foot, on a full sized scale, becomes just 6 inches, which is a kids' size 8.5, often worn by, um, 2-year olds. [Emphasis hers]

"Although Barbie's proportions don't make total sense on a larger scale, lucky for us, similar polygons' properties do!"

McKellar is hardly the only one to notice that the dolls aren't geometrically similar to real humans. In fact, last summer an artist created a doll that really is similar to the average 19-year-old:

I probably wouldn't ever mention this example in an actual class, unless -- as I said above -- I'm in a class where I, as the teacher, have shown a male bias and need something feminine in order to motivate the girls to learn.

And so here is my worksheet for Day 90 -- the mathematical halfway point of the year, but of course, I'm using a calendar that puts us six days into the new semester.

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