Struggling with a challenging AMC 10 geometry problem

This problem from the 2016 AMC 10 a gave my son some trouble yesterday:

We talked through it last night and it was interesting to see where his intuition was off:

After finding a solution in the last video that he knew was wrong (because it wasn’t one of the choices on the test) we looked back to see if there was alternate approach to the problem. This approach led us to find the actual solution and also the mistake in the first approach:

It is always fascinating to the though process on a challenging problem. Sometimes the thought process is so close to the right approach that the mistake is really tricky to spot.