Altitude to the Hypotenuse

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 …

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Similarity Theorems

 

Similarity Theorems

Prove theorems involving similarity

G-SRT 4. 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.

G-SRT 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

I have struggled some to know which additional theorems we should include when teaching G-SRT 4. So, at least for now, my best source for whether to teach something that we have traditionally included has been to ask my juniors and seniors who have recently taken a high stakes standardized test for college entry and scholarships. I get that my only reason to teach something shouldn’t be whether or not it is “on the test”, but I don’t want my students to be at a disadvantage when they go to take the test.

We started the Dilations unit by showing that two triangles are similar when there is a dilation and if needed, set of rigid motions, to map one figure onto another. In our lesson on Similarity Theorems, we discuss similarity criteria for triangles in terms of the traditional similarity theorems: AA~, SAS~, SSS~. We formatively assess that students can use the similarity theorems.

      

Are the figures similar? If so, why?

      

If the figures are similar, then what is the value of x?

Students always do well with the first triangle but not with the second. Asking the second uncovers student misconceptions.

We briefly explored 3 other theorems dealing with similar figures.

 

1. A midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. What is the relationship between the midsegment and the third side of the triangle? We used part of the Geometry Nspired activity called Triangle Midsegments for this exploration.

This is an example of where technology makes me slow down. It would be so much faster for me to tell students that the midsegment is parallel to the third side of the triangle and that it is half the length of the third side of the triangle. But I have found that students remember the information longer when they make sense of it: from constructing the midsegment to observing what happens as the triangle changes.

We formatively assess to be sure that students can use what they have observed.

And then we prove the theorem.

Where would you start?

What Standard for Mathematical Practice would be helpful?

 

2. The Side-Splitter Theorem.

Observe what happens.

Formatively assess to be sure students can use what they observed.

And then prove the theorem.

Where would you start?

What Standard for Mathematical Practice would be helpful?

We used look for and make use of structure. The auxiliary line gives us something two cases of “a line parallel to one side of a triangle divides the other two proportionally”, and then we can use the parallel lines and transversal to show that the two triangles are similar to each other by AA~.

 

3. The Angle Bisector Theorem.

Observe what happens.

Formatively assess to be sure students can use what they observed.

And then prove the theorem.

Where would you start?

What Standard for Mathematical Practice would be helpful?

What auxiliary lines would be helpful?

We actually didn’t get to completely prove The Angle Bisector Theorem…our time ran out. But I think that’s okay. We covered the Angle Bisector Theorem because my former students suggested it is helpful. And because we have taken care all year to observe what happens and prove most of what happens, my students realize that the result isn’t magic.

I started reading The Joy of X this morning. I love Steven Strogatz’ quote at the bottom of page 5: “Logic leaves us no choice. In that sense, math always involves both invention and discovery: we invent the concepts but discover their consequences. … in mathematics our freedom lies in the questions we ask – and in how we pursue them – but not in the answers awaiting us.”

And so the journey to ask more questions continues …