When we accidentally derived the law of cosines

We’ve been working through some of the challenging proofs in chapter 9 of Art of Problem Solving’s Introduction to Geometry book. As I’ve written, my son’s been finding these proofs to be pretty difficult:

 

BUT – he’s staying with them and seeming to get a little more out of each one that we go through.

Today’s challenge was prove that in an obtuse triangle, if we call the sides A, B, and C, with C opposite the obtuse angle, then

We spent about 30 minutes working through this proof and then I wanted to go through it again on camera. To my surprise my son took the proof in a completely different direction in the video and we ended up essentially proving the law of cosines by accident. Ha!

The first 5 minutes was introducing the problem, drawing a little picture, and starting down the path toward the proof:

 

In the second half, we looked at how our picture helped is sort through some complicated looking equations. It doesn’t seem to be progressing too quickly, then at 2:23 – “oooooooooohhh”

I wrap up at the end talking very briefly about the interesting expression that we stumbled on – the law of cosines!

 

Looking at some slightly more difficult proofs

We started a new section in our Geometry book this week and it starts with some properties of triangles that require slightly more difficult proofs that we’ve seen so far. My son is struggling with the difficulty and I have a lot of sympathy – learning techniques of proof was absolutely one of the most difficult things for me.

Today we were exploring non-right triangles and, in particular, when a triangle’s sides might satisfy .

It took about 30 minutes to explore the idea this morning. Even with that long exploration, reviewing the idea again in our movie took a long time. Initially we just talked through about the problem and the initial ideas in the proof.

I broke this video after we write down three algebraic relationships in the triangle.

 

Once we had those relationships, the idea was to see how we could apply them to our problem. I think there are two parts of learning math that are particularly hard here. The first is understanding the connection between the geometry and the algebra. The second is understanding the relationships between the numbers / sides implied by the algebra.

 

Again, I have a lot of sympathy for my son as learning more difficult proofs really is hard. A line from an old Indigo Girls song comes to mind – “the hardest to learn was the least complicated.”