Chapter 7 - Similar Triangles

We began the study of similar triangles with the following questions: "What are the conditions that tell us two triangles are similar? Why do these conditions work and not some other combinations of sides, angles, equality and proportion? How do the conditions that determine similar triangles differ from the conditions that tell us triangles are congruent?"

The students set aside their textbooks, as I requested, and worked together in small groups of 3 or 4 students.

The first thing that I did was to ask them to draw a triangle on a piece of paper. Then I gave each group a small magnifying glass, and asked them to look at their triangle through the magnifying glass and tell me what they saw. They said, of course "it looks bigger". So I then asked for more information - what, specifically, did they mean by bigger? Did they mean bigger like a giraffe is bigger than a snake? Well, maybe a baby giraffe bigger than a python, if the giraffe is standing and the python is lying on the ground. But the python certainly has greater horizontal length when stretched out than the baby giraffe has when standing. Did they mean taller, or wider, or thicker? Of course they thought that was a silly question, but I reminded them that the language of mathematics is a language with great precision, and that "big" has many meanings; we needed to be more precise. After some discussion, we agreed that it would be more precise to say that in our investigation of triangles, we could only say that one was bigger than the other, in the general sense of the word, if the two triangles were similar triangles. (We also discussed the idea that two triangles could have the same area without having the same shape, and that two triangles could not have the same dimensions without having the same shape.) This was an interesting conversation, bordering on argument at some points, but much more alive than our usual math class.

But, I digress. Here is a comment that one of the students wrote, in her reflections on this topic:

"What made similar triangles so easy to understand was when the teacher showed us a magnifying glass and we looked at a drawing of a triangle through the class - the triangle was, of course, the same shape but looked bigger through the magnifying glass, This is what similar means." Sara and Megan.

The students constructed triangles using the Geometer's Sketchpad software, and experimented to determine what combination of sides and angles would form similar triangles. They discussed their findings, compared notes, and came up with the following conclusions: "The first thing we discovered is that two triangles are similar if 2 angles of one triangle are congruent to 2 angles of another. This was surprising at first because it seemed like too little information, just 2 pieces, but that is because once you know 2 angles you automatically know the third, so it's kind of like 3 pieces of information. (We had expected we'd need 3 pieces of information because that's what we needed for proving triangles congruent.) Then we experimented some more and decided that 2 triangles would have to be similar if all 3 sides were proportional. What we mean by this is, for example, the sides of one triangle might be 3, 4 and 5 and the sides of the other triangle might be 6", 8", and 10". So the triangle with sides 6", 8" and 10" would have each side exactly twice as long as the 3", 4"and 5" triangle. What made similar triangles so easy to understand was when the teacher showed us a magnifying glass and we looked at a drawing of a triangle through the class - the triangle was, of course, the same shape but looked bigger through the magnifying glass, This is what similar means." Sara and Megan.

Eventually, after experimentation and discussion, the students agreed on the following: Two triangles are similar if:

1) ... all three sides of one triangle are proportional to all three sides of the other triangle. (This means the ratios of the sides are equal). This is like SSS for congruent triangles but we will call it SSS~ for similarity.

2) ... two angles of one triangle are congruent to any two angles of the other. We'll call that AA~ for similarity.

3) ... a pair of sides are proportional and the included angles are congruent. We'll call that SAS~ for similarity.

The next question was "Why do these conditions work and not other combinations of sides and angles?"

Anya wrote the following essay, in answer to that question: "Only those conditions work because if you try to construct the triangles using SSS, AA, or SAS you will find that there is only one way to put those three measurements together. With SSS, imagine you have 3 straws and you run a piece of string through all 3. When you tie the string tightly, you will have a triangle and you can't change that shape. That's why they use triangles in the construction of bridges; the triangle is a strong shape."

The students did a very interesting project on using similar triangles and "indirect measurement" to find the height of a tree on campus. As we often did with these projects, the students worked in groups of 3 or 4. Sometimes I let them choose their groups, but more often they drew names from a small box containing small cards; each card having a student's name on it. The problem was stated in a "worksheet" as follows:

"Your assignment is to find the height of tree on campus." (or the teacher could ask the students to find the height of a building) "You will calculate the height using two different methods, and then compare the answers that you get using the two methods.

Method 1) Shadow Reckoning: Go to your tree and measure the shadow cast by the tree. Then measure the length of the shadow of a student from your group, who stands by the tree (but not in it's shadow). Measure the student's height. Then draw a sketch of the tree and the student, label the sketch with the measurements, and calculate the height of the assigned object.

Draw the tree, the student, and both shadows on a piece of poster board. Explain every step of your work, and write these explanations clearly and completely, including all of equations and any theorems that you use. This explanation will be part of your "poster".

Method 2) Mirror Method: Use the small hand mirror that has been provided. Place the mirror on the ground about 10 or 15 feet from the base of the tree. Then, looking in the mirror, walk forwards or backwards until you can see the top of the tree centered in the mirror, Measure the distance from where you are standing to the center of the mirror, and the distance from the center of the mirror to the base of the tree. Use similar triangles to calculate the height of the tree.

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In each of the two methods, explain every step of this project. Your explanation must contain labeled diagrams, measurements, and a complete explanation of the steps in this project, including any and all theorems used.

Compare the answers you got. If you found somewhat different heights from you the two methods, explain why this might have happened. Explain why the triangles are similar, in each case. How might these methods be useful in other situations? Do you think this might be how people in earlier times might have measured objects too tall to measure directly?

Later in the school year, we did a similar project, this time using trigonometry. If you are interested in reading more about these projects, you will find more information at the following web site: http://mathforum.org/~sanders/creativegeometry/

I also asked them the following question: "We have found that there are certain postulates that we can use to prove two triangles congruent. More recently, we have studied ways to prove two triangles are similar. Let's compare these two methods. How do the methods for proving triangles similar differ from the methods use to prove triangles congruent? How are they the same?" In my writing-intensive geometry class, my students' explanations and reflections were an important part of these projects. Here are some of their answers and comments:

"The conditions that determine similar triangles differ from the conditions that determine congruent triangles, in some ways. One way is that for the similarity theorems. The corrresponding sides must be proportional instead of congruent. However, in both the similarity theorems and the congruence theorems, corresponding angles must me congruent. So it is the angles that give the triangle it's shape, and the side lengths that give the triangle it's size, to a certain extent. Another difference is that it always takes 3 pieces of information to prove triangles congruent, but you can prove two triangles similar with just two pieces of information if you know 2 angle pairs are congruent; you don't need to know anything about the sides at all." Adam

Janine had this to say about this project: "I am proud of the work I did on similar triangles because I feel that I did a good job of explaining it and seeing connnections. It was interesting to notice how the conditions for proving triangles similar are related in some ways to proving triangles congruent. It was fun to know that I can figure these things out for ourselves!"

"Numbers constitute the only universal language."

Nathanael West