Chapter 11 Review (Day 119)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

Find x.

(Once again, all the info is given in an unlabeled diagram, so let me invent the labels. We see AE and CD intersecting at B. Angle C is right, BAC, BED are congruent. AB = 5, AC = 4, EB = 45, DB = x.)

It's easy to see that Triangles ABC and EBD are similar by AA Similarity. One pair of congruent angles needed for AA~ is given, while the other is the vertical pair ABC and EBD. (This allows us to conclude that D is a right angle -- without AA~, we can't determine whether D is right or not.)

We might try to write a proportion:

DB/CB = EB/AB
x/CB = 45/5

But side CB is unknown. Since ABC is a right triangle, we must use the Pythagorean Theorem:

a^2 + b^2 = c^2
a^2 + 4^2 = 5^2
a^2 + 16 = 25
a^2 = 9
a = 3 = CB

x/3 = 45/5
5x = 135
x = 27

Therefore the desired length is 27 -- and of course, today's date is the 27th.

Notice that we could have used the Pythagorean Theorem first to find ED = 36 and then set up a different proportion. I assume that you'd rather square/find the square root of numbers like 4, 5, 9 rather than 36 or 45 -- and besides, we should be able to recognize the 3-4-5 Pythagorean triple without using the formula.

This is a similarity problem that we could solve in Chapter 12 -- and we still haven't quite reached Chapter 12, although we're tantalizingly close.  (In the Glencoe text, this problem can be solved after Lesson 7-4 on the Pythagorean Theorem.)

But this question underlies the need to avoid writing a proportion before writing a similarity. A student might just say "the triangles are similar because we're in the similarity chapter" and then set up the proportion to find x = 36, when this is actually ED, not DB as desired. Instead, the students should actually write Triangle ABC ~ EBD. Only then will they be more likely to set up a proportion correctly (and see the need for the Pythagorean Theorem.)

Today is the review for the Chapter 11 Test. This is what I wrote last year about today's lesson:

In particular, this test is based on the SPUR objectives for Chapter 11. As usual, I will discuss which items that I have decided to include and exclude, and the rationale for each:

Naturally, I had to exclude Objective G: equations for circles, which I take to be an Algebra II topic, not a Geometry topic. (If this had been an Integrated Math course, I would have delved more into graphing linear equations, as we covered this week.) Actually, equations of circles really is a Common Core topic, so you might want to cover some circle problems of your own. On the other hand, we may still want to leave out three-dimensional coordinates, since I posted the leprechaun graphing worksheet instead.

One major topic that I had to include is coordinate proof, as this appears in Common Core. I did squeeze in some coordinate proofs involving the Distance or Midpoint Formulas, but not slope. So therefore, the coordinate proofs included on this review worksheet all involve either distance or midpoint, not slope. The only proofs involving parallel lines had these lines either both vertical or both horizontal. Once again, a good coordinate proof would often set it up so that the parallel lines that matter are either horizontal or vertical.

What good are coordinate proofs, anyway? Well, a coordinate proof transforms a geometry problem into an algebra problem. Sometimes I can't see how to begin a synthetic geometry proof, so instead I just start labeling the points with coordinates and see what develops.

So coordinate geometry reduces an unknown problem (in geometry) to one whose answer is solved (in algebra, in this case). Mathematicians reduce problems to previously-solved ones all the time -- enough that some people make jokes about it:

http://jokes.siliconindia.com/recent-jokes/Reducing-the-problem-nid-62964158.html

I ended up including six straight problems -- Questions 8 through 13 from U of Chicago. Most of these questions are from Objective C -- the Midpoint Connector Theorem. The text covers this here in Chapter 11, but we actually covered it early, in our Similarity Unit, because we actually used the Midpoint Connector Theorem to start the proof of the basic properties of similarity. Still, this was recent enough to justify including it on the test.

Next are a few center of gravity problems. This is straightforward, since all we have to do is average the coordinates. Afterwards are a few midpoint problems, including two-step questions where one must calculate the distance or slope from one point to the midpoint of another segment.

Then there are a few more coordinate proofs where one has to set up the vertices -- notice that some hints are given in earlier questions.