Similar Triangles Problem: Two inscribed squares and a 3-4-5 right triangle! (2017 AMC10A P21)

This difficult-looking problem will be solved purely with similar triangles. This is a very fun one since it involves two inscribed squares sitting inside of a 3-4-5 right triangle. They are orientated differently but the method of attack will be the same for both: use the fact that we have many pairs of similar triangles with smaller triangles sitting inside of bigger triangles, set up our similarity relationships and ratios and solve. But what exactly will we be solving for? x? y? Not necessarily. All we care about is the ratio x/y… Important problem-solving principle Sometimes if you are asked to find the ratio x/y or even the product xy or even something funky like xy^2, you DO NOT NECESSARILY NEED TO KNOW WHAT THE INDIVIDUAL VALUES FOR X AND Y ACTUALLY ARE. If we are clever enough, we can get the answer DIRECTLY without going through unnecessary work in the middle!!!