Working through a tough geometry problem with my younger son

My younger son is working through Art of Problem Solving’s Introduction to Geometry book this school year. He works for about 30 min ever day and seems to really enjoy the challenge problems. Today the last challenge problem from the chapter on similar triangles gave him some difficulty.

I could tell he was having trouble understanding the problem and I asked him what was wrong. His answer was interesting -> they didn’t give any side lengths.

The problem is a pretty good challenge problem for kids learning geometry, but I thought talking through it tonight would make a good project. So, here’s the problem and a short introduction to what was giving him trouble:

In the next part of the project we began to solve the problem. There are (I think) two critical ideas -> (i) finding all of the similar triangles, and (ii) finding the parts in the diagram which have the same length.

It takes a few minutes for my son to find all of the relationships, but he does get there. Despite being a little confused, his thought process is really nice to hear.

Now we moved to the last side to see if we could find another relationship that would simplify the equation that we are hoping will be equal to 1.

After we had that last relationship he was able to see how the expressions in the equations corresponded to various side lengths in the picture. From there he was able to see why the sum was indeed 1.

I like this problem a lot and am happy that my son wanted to struggle with it.