Wednesday, January 14, 2015

Section 12-8: The SSS Similarity Theorem (Day 92)

Last night I tutored my student in geometry. Today he is taking the test on Chapter 3 of the Glencoe text, which is on parallel and perpendicular lines.

I gave him some review questions from the text, and I noticed that the text made a few errors. First, in one of the review questions for Section 3-4 (Equations of Lines), there is a word problem in which a car is decelerating at a constant rate, and the student is asked to find an equation to give the velocity in terms of the time, in meters per second. But the answer in the back of the text gives an equation that is completely unrelated to the question -- indeed, the fact that the answer from the key has a slope of -32 implies that the intended question is about gravity, where the deceleration is 32 feet per second squared, and not about a car at all!

Another error occurs in a review question for Section 3-6 (Perpendiculars and Distance). The student is given a figure and asked to draw a segment to indicate the distance between a given point and a given line -- that is, to draw the perpendicular to the line from the point. The error is that the answers are already drawn in red on the given figure!

Notice that the subject matter of Glencoe's Section 3-6 -- that is, that the distance between a point and line is defined to be the shortest possible distance, namely the perpendicular distance -- is not mentioned in the U of Chicago text at all. If I wanted to add it to my lessons, it might be appealing to add it to the lesson on Uniqueness of Perpendiculars. But this is tricky, for how can such a theorem possibly be proved? Notice that any non-perpendicular segment would be the hypotenuse of a right triangle, one of whose legs is the perpendicular segment, and since the hypotenuse is always longer than the leg, the theorem would be proved. But how do we know that the hypotenuse of a right triangle is always longer than either leg? The Pythagorean Theorem has not yet been given, and the Unequal Angles Theorem, which would also imply this result, must wait until after indirect proofs are given in Chapter 13 of the U of Chicago text.

After showing my student the text filled with errors, I showed him my own version of the Chapter 5 Test Review from this blog. But because of the differences between the Glencoe text and what I chose to present on the blog, I could really only show him Questions 3-5 from that Review. It took a while, but he was able to figure out the answers. (Oh, and we did find one error of my own on that sheet -- in some problems, the symbol for angle is missing.)

Next week I'll be teaching slope on the blog -- but of course, it's one week too late to be of any help for my student.

Also, for the entire week I'm subbing at a continuation school, for a teacher who is becoming a father for the first time. He normally teaches Algebra I and Botany. The students usually work on their Algebra I work independently, but from time to time I do help the students out -- this week they are solving inequalities, Chapter 7.

The Pacemaker Algebra I text is published by the Pearson Learning Group. Near the back of the classroom I also found a geometry text book published by AGS Publishing, dated 2005. Here are the chapters of the geometry book:

1. Exporing Geometry: Points, Linea, and Angles in the Plane
2. Thinking Geometry: Using Proofs
3. Parallel Lines and Transversals
4. Using Algebra: Lines in the Coordinate Plane
5. Triangles and Quadrilaterals
6. Congruent Lines and Transformations
7. Proportion and Similarity
8. The Pythagorean Theorem
9. Perimeter and Area
10. Circles and Spheres
11. Solid Geometric Figures and Their Measures
12. Geometry and Imagination

Notice that the stest that I showed my student last night resembles the AGS Chapter 3 more than the Glencoe Chapter 3. This is a pre-Common Core text: the transformations of Chapter 6 appear after the SAS and other postulates, not before.

Section 12-8 of the U of Chicago text is on SSS Similarity. This is Wu's Theorem 28. There is not much difference between the U of Chicago and Wu proofs. Let me once again give the U of Chicago proof in its original paragraph form:

SSS Similarity Theorem:
If the three sides of one triangle are proportional to the three sides of a second triangle, then the triangles are similar.

Given: XY/AB = YZ/BC = XZ/AC
Prove: Triangle ABC ~ XYZ

Let k = XY/AB. Then by transitivity, k = YZ/BC and k = XZ/AC. Apply any dilation with scale factor k to triangle ABC. In the image Triangle A'B'C', A'B' = k * AB, B'C' = k * BC, and A'C' = k * AC. But k * AB = XY/AB * AB = XYk * BC = YZ/BC * BC = YZ, and k * AC = XZ/AC * AC = XZ. Thus the three sides of Triangle A'B'C' have the same lengths as the sides of Triangle XYZ. So by the SSS Congruence Theorem, Triangle A'B'C' is congruent to XYZ. The definition of congruence tells us there is an isometry mapping Triangle A'B'C' onto XYZ. So there is a composite of a dilation (the one we started with) and an isometry mapping Triangle ABC onto XYZ. By the definition of similarity, Triangle ABC ~ XYZ. QED

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