You are off track, but I can see you have the right idea. You have made an assumption which limits your choice of the sides of ABC. Here’s a clue. You are right to start with ABC, but look at it like this:
(AC-BC)(AC+BC)=25², then consider the factors of 25². If the factors are p and q:
AC-BC=p (you assumed p=1 and q=625 are the only factors).
AC+BC=q, so AC=(p+q)/2, BC=(q-p)/2. This will give you the dimensions of ABC. Note also that there are two similar triangles which will help you resolve ambiguities. There are in fact two solutions.