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The Three Triangles

I recently added a guided inquiry worksheet on my website’s Pythagorean theorem page. It leads students along a proof of the theorem based on similarity. It is called The Three Triangles. This is a worksheet I used and honed over the years when I taught geometry. Upon rereading it, I thought it would make a good topic for a blog post, as it exemplifies several pedagogical concepts I’d like to discuss here. I will quote some of the worksheet as I comment on it, but before you read any further, I encourage you to print it out and look it over.

The Three Triangles

Before handing out the worksheet, I would start with a story of sorts, some version of this: “Once upon a time, there was a little girl who went for a walk in the woods. In a clearing, she saw a house and wondered who lived there. The door was ajar, so she peeked in, and then decided to go in. What do you think she saw in there?” (After students venture some ideas, usually including the basic “three bears” story:) “That’s what I thought! But no! She saw three triangles!” At this point, the students are disappointed, because obviously the story was super-short, it is about to end, and we’re about to start working. That’s fine — the story’s mission has been accomplished: the students are all alert and ready for the worksheet. I add: “There was a large triangle! A medium triangle! And a small baby triangle!” and hand out the worksheet.

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The introduction

Many students do not read unnumbered paragraphs such as the two that introduce this worksheet. They just want to get to the work they have to do. Nevertheless, those introductory paragraphs serve a purpose.

Here is one way to prove the Pythagorean Theorem using your newfound knowledge of scaling and similarity.

This is there largely for the teacher, to make clear what this activity is about. One risk in the guided inquiry approach is that students can be led by the nose, without a sense of what the goal is. This sentence clearly states our goal, and the strategy that we will use. I would make sure to say something to that effect before the students start working. 

The figure below shows three triangles. The points are indicated by capital letters, and the side lengths by lower case letters.

This is there as a reference. If later in the lesson I see a student writes “A” when they mean “a”, or vice-versa, I will point to this.

The figure

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1. Copy the original diagram and be sure that your diagram is clearly labeled.

The figure is clearly labeled on the worksheet, but alas some students do not see all the details of what’s right there in front of them. Having to copy it and label the copy helps to guarantee that they can follow the subsequent steps.

2. Write as many relationships as you can involving the figure’s acute angles (∠A, ∠B, ∠1, ∠2). Find at least six different equations and explain them (example: ∠1+∠2 = 90º) .

This is no longer a clerical task! Seeing and understanding these relationships is a prerequisite to establishing the similarity of the triangles. The reason I ask for six equations is to make sure students see and understand that ∠1=∠B and ∠2=∠A. Some hinting may be necessary!

3. There are three triangles in this diagram. Draw them separately and line up the corresponding parts. Be sure to label the sides and angles of all three triangles using the appropriate upper and lower case letters. Your triangles should look something like this:

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For students who can identify the corresponding parts of the triangles in the original figure, this step may not be necessary. Certainly, the teacher can do that, and could point them out at the board, but it is almost certain that many students could not follow such an explanation by watching passively. If the goal is for every student to understand the proof, drawing the triangles separately, and facing the same way is very helpful. And frankly, it’s also useful to reduce the risk of error even for the students who do not absolutely need this step.

Similar Triangles

4. Explain why the triangles are similar to each other.

If the students have learned about the AA criterion for similarity, this is not difficult. If they have not, they should not have been assigned this worksheet!

5. There are three pairs of similar triangles. For each pair, write that their sides are proportional by writing the ratios in the format:

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As in #3, some students may not need this much hand-holding, but it does not hurt. It also gives an approach that is difficult to forget and will come in handy any time students are dealing with similar right triangles.

A possible objection to this way of phrasing things is that it does not apply to isosceles right triangles, since there is no “short leg” or “long leg”. Fair enough. I failed to mention this to my students, and none of them ever brought it up. If a student or a colleague had pointed this out, we could have discussed that specific case separately.

Using the Proportions and the Pythagorean Theorem

6. If a = 4 and b = 5, find all of the other lengths (h, x, y, and c).

7. If x = 4 and y = 5, find all of the other lengths.

This is an important step, which helps to prepare students for doing the algebra in the next section. It is almost always a good idea to work with numbers before launching into all-variables manipulations, as it grounds the work in the concrete. Also, if students get different answers from their neighbors on these problems, it may indicate that they made one or more mistakes in #5. Such mistakes must be addressed before launching into the next section, as it depends on the answers to #5.

Finding Formulas

This, of course, was our destination all along. The example given before #8, and the work students did in #1-7 makes it straightforward to find the requested formulas.

#9 shows that we could have found the formula for #8c another way, and in fact provides a way to remember that particular one. However, in general, it’s not crucial for students to memorize these formulas: having worked through #1-7 means that they have a strategy to tackle problems of this type by using similar triangles.

Finally, #10 delivers the punch line: a proof of the Pythagorean theorem based on similar triangles and the distributive property.

Super Triples

It is not important that all students complete #11 or (especially) #12. They are there as “sponge” problems, to challenge students who finish #1-10 well in advance of their peers. (Though when they are working on this, they should still be available to help group members who are struggling with #7-10.)


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