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.
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.
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).
And then we can look for regularity in repeated reasoning to find the significance of the similar triangles. What do you notice?
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.
And then we used the geometric mean relationships to solve a few problems.
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?
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 …