Hmm.. maybe this will clear things up slightly. So the thing is, while we can't assume that diagrams are drawn to scale, we can definitely "assume" that intersections are held. ie, we know that KL and ML intersect at point L (as obvious as this may sound). Also, we know that there are intersections on the boundaries and one in the interior of the diagram.
There is honestly one way to draw the diagram (unless you're just a smart alec or something) such that the intersections are all made correctly and the dimensions are all correct.
If I were you, I would never just substitute numbers in for geometry problems (to solve the ENTIRE problem... if you're trying to find a counterexample, go for it!). Instead, I would set the shorter side to length a and the longer side to length b.
In this case, we see that length ML is of size 2a and length KN is of size b. So, b = 2 * a. Now, we are trying to find KN/MN = b / (a + b). Let's try and solve an easier problem..
If we knew MN/KN, we'd be okay.. just take the reciprocal! So, can we do this? Well,
MN/KN = (a+b)/b = (a/b) + 1
But b = 2a, so MN/KN = 3/2, so KN/MN = 2/3. Does this make sense? If this seems really contrived, let me know and I'll try to make it a bit less contrived.
). It's true that you can't assume that things aren't to scale in the GMAT, but in this case, you can see it as valid.
