You're supposed to find the general answer, although at least one of you gave a pretty good experimental answer!
Focus on the objective: we're not focusing on the golden rectangle here, we're trying to show whether the ratio of sides of two rectangles are the same (have a gold-like property).
Don't assume an isosceles triangle! When they ask you whether something works for right triangles, draw the wickedest right triangle you know and ask! "Test at the extremes."
"Is cheese good?"
Answer: if you like limburger cheese, then probably the answer for you is yes!