Altitude to the Hypotenuse

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Objective

SWBAT recognize similarity relationships that arise when the altitude is drawn to the hypotenuse of a right triangle.

Big Idea

There's more to right triangles than the Pythagorean Theorem.

Breaking Down the Geometric Mean

15 minutes

Knowledge of geometric means turns out to be pretty useful in a geometry course. For example, it's used in the proof of the Pythagorean Theorem and the proof of the slope criteria for perpendicular lines. In this section of the lesson, I spend some time developing the concept of the geometric mean.

I guide students through Understanding the Geometric Mean. Although it is nice if students have prior experience with arithmetic and geometric sequences, I teach the lesson as if students have no prior experience. 

I tried to design the resource for this section so that students would learn from it, work to make sense out of it, and also demonstrate their understanding and sense-making as they progress through it. As the teacher, I will basically set the pace for the class progressing through the activity, stopping at key points to make sure students are achieving the learning goals.

For example, after most of the fill-in-the-blanks, I stop for a pair share and then reveal/elaborate on the answer. At other key points, I stop for a more detailed analysis. See the following video for an example of this.

 

 

I also stop after the practice section to show the answers and how to get them. See Understanding the Geometric Mean[KEY] for answers to the handout.

Identifying Similarity Relationships

35 minutes

For this section, I give students Altitude to Hypotenuse. I designed this handout basically to capture a lot of what I would say to students if I were not using a handout. 

In an ideal world, I would like for students to be able to go through the handout on their own and get maximum learning out of it, but seeing as this is not guaranteed to happen for all students, I do guide students through it, stopping at key points for pair shares or to delve more deeply and elaborate on a particular part of the handout.

See the following video for one example of how I stop to share a helpful technique for creating proportions from similar triangles when it is not obvious which sides correspond with each other:

 

Generalizations

10 minutes

Making generalizations (MP8) is one of the most important practices in geometry. In this section, students will be extracting general truths from the examples they have seen in the previous section. For this section we work on the latter portion of Altitude to Hypotenuse.

I start by introducing key vocabulary:

  • hypotenuse
  • leg
  • hypotenuse segment
  • altitude

Then I basically lead my students through the reasoning process from the specific to the general. It goes something like this:

1. CD is the geometric mean of BD and AD becomes:

When the altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean of the hypotenuse segments (pretty straight forward)

2. CA is the geometric mean of AD and AB becomes

When the altitude is drawn to the hypotenuse of a right triangle, the leg is the geometric mean of the hypotenuse segment and the hypotenuse

3. CB is the geometric mean of DB and AB becomes

When the altitude is drawn to the hypotenuse of a right triangle, the leg is the geometric mean of the hypotenuse segment and the hypotenuse

Statements 2 and 3 leave us with some ambiguity (a word my students know is not a good thing in geometry). I ask them first to discuss with their A-B partner where the ambiguity is in the statements. Then I verify that the ambiguity is not in the term hypothesis (since that can only mean AB) but in the term hypotenuse segment (as that could mean AD or DB).

I then ask my students, "How can we be more precise in order to eliminate this ambiguity." "In other words, how could we generalize in order to capture what statements 2 and 3 are saying in one statement that is true and not ambiguous?" Upon closer inspection, students hopefully notice that the relevant hypotenuse segment is the one that is closest to the leg in question. From that I just help with the language, which is included in the Vocabulary section of the handout, so that we get to

4. When the altitude is drawn to the hypotenuse of a right triangle, a leg is the geometric mean of the hypotenuse and the hypotenuse segment adjacent to that leg.

Once we've made these generalizations, I have my students document these two important statements (1 and 4) in their notes. I also have two or three students volunteer to create a poster on chart paper that will serve as a classroom reference for these important truths.

Practice

15 minutes

Most textbooks have a section on similarity relationships that arise when the altitude is drawn to the hypotenuse of a right triangle. At this point in the lesson, I have students work on problems from our textbook using their new-found knowledge of these similarity relationships and the geometric mean as a useful tool in solving these types of problems.

Closure

5 minutes

As an Exit Ticket, I ask students to write about the day's lesson as if it were a movie.

  • How did it start?
  • What were the key events?
  • What was the exciting conclusion?
  • Was there a moral to the story? Etc.

In addition to giving students the opportunity to reflect on their learning and to see the lesson from a big picture perspective, it let's me know what they have taken away from the lesson. This will be important as our next lesson Proving the Pythagorean Theorem using Similar Triangles relies on what we learned in this lesson.