The question given was to find the altitude to the side of length 14 in a triangle with sides of length 13,14,15. I keep trying proportions but I know I'm missing something or more likely, made a bad assumption somewhere. I did look up how to do it elsewhere which had the Pythagorean theorem twice and solving for a that way. What am I doing wrong here? I set up two similar triangles in my diagram to the right. Shouldn't the proportions match up and give the correct answer (a 5-12-13 and 9-12-15 triangle).
Your mistake is in trying proportions in the first place!
In particular, you don't necessarily know that the two triangles are similar - there's no reason to assume the angles you've marked as the same are the same. In fact, the triangles you're hoping for as the answer: 5-12-13 and 9-12-15 aren't similar.
To get the correct answer, you can either use the Pythagorean theorem twice as you mention or use something like Heron's formula for the area first.