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).
Using the pythagorean theorem and similar triangles
geometry
Best Answer
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.