Altitude to the Hypotenuse

27 Jan

This is another one of those theorems dealing with similarity that I am not sure we should cover. We did, though, at the recommendation of my former students who say that they have had to use the information on high stakes standardized tests for college entrance and scholarships.

CCSS-M G-SRT.4 says Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

My students explored what the altitude to the hypotenuse buys us mathematically.

Screen Shot 2014-01-26 at 6.08.09 PM

What happens when we construct the altitude to the hypotenuse?

Some students were not ready to reason abstractly, but they were okay with reasoning quantitatively. One acute angle measure will let us deduce all of them.

Screen Shot 2014-01-26 at 6.11.52 PM

And make it more evident that the three right triangles are similar by AA~.

Our technology makes it more evident that the three right triangles are similar because of a dilation and a rotation (from Altitude to the Hypotenuse on Geometry Nspired).

01-26-2014 Image017 01-26-2014 Image018 01-26-2014 Image019 01-26-2014 Image020 01-26-2014 Image021

And then we can look for regularity in repeated reasoning to find the significance of the similar triangles. What do you notice?

Screen Shot 2014-01-26 at 6.12.15 PM

We had been working some problems with the geometric mean throughout this unit. And so after several minutes (several long minutes, because I wanted them to see it instead of me telling them and so I waited), more than one student noticed the geometric mean relationships.

Screen Shot 2014-01-26 at 6.15.39 PM

Screen Shot 2014-01-26 at 6.31.49 PM

And then we used the geometric mean relationships to solve a few problems.

Screen Shot 2014-01-26 at 6.15.48 PM

So we haven’t actually used the altitude to the hypotenuse to prove the Pythagorean Theorem yet. I will admit that I had not taken the time to think through that proof until now – and it only happened because I started reading Steven Strogatz’ The Joy of X earlier today.

After showing an “elegant” visual proof of the Pythagorean Theorem that “illuminates” (one that shows, literally, how to decompose the squares on the legs of a right triangle to show that the sum of their areas is equal to the square on the hypotenuse of a right triangle), Steven Strogatz suggests that a proof of the Pythagorean Theorem using the triangle similarity that occurs when the altitude to the hypotenuse of the triangle is drawn is “ugly” and “murky”. (From chapter 12, “Square Dancing”)

How can you use the proportional relationships in the similar right triangles to prove the Pythagorean Theorem?

Screen Shot 2014-01-26 at 6.48.56 PM

Like any good author of a mathematics book, Strogatz leaves the actual proof up to the reader (with missing steps in the Notes section at the end of the book).

So I get why Strogatz calls the proof “ugly” and “murky” compared to the visual we have all seen to show the sum of the areas of the squares on the legs is equal to the area of the square on the hypotenuse. But I wonder if using the technology to make sense of the proportions can change the similarity proof to “elegant”? And I wonder if the beauty of a proof is in the eye of the one who makes sense of it? Because I got pretty excited about using the proportional relationships to show that the Pythagorean Theorem is true. And I have a feeling that at least a few of my students will, too, as the journey continues …


Posted by on January 27, 2014 in Dilations, Geometry, Right Triangles


Tags: , , , , ,

6 responses to “Altitude to the Hypotenuse

  1. Travis

    February 18, 2014 at 2:20 am

    I continue to enjoy your blog….[typo: Theorems include: a line parallel to one side of a triangle—>perpendicular

    • jwilson828

      February 24, 2014 at 1:53 am

      The theorems were listed like that in the CCSS Standard, the first one of which doesn’t actually go with this post, as you noted! I’ve been trying to include the standard we are using in each post, but maybe it would be helpful if I bold or highlight the part we are actually addressing. Thank you!

  2. Mr. Deyo

    August 21, 2014 at 12:01 am

    Click to access geometric_mean.pdf

    Here is a graphic organizer I made a few years ago to illustrate the 3 geometric means in a diagram with 3 different sized right triangles. Enjoy.

    • jwilson828

      August 21, 2014 at 5:33 am

      Thank you for sharing your graphic organizer!

  3. Karyn Harrelson

    February 27, 2019 at 11:09 pm

    I use this proof every year with my students.


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