Stephen Morris has provided a very detailed response to this question, which I have copied to the end of the original post.
It turns out that my intuition was correct (the tiling is possible if and only if r is a rational number lying in the half open interval (0,1]), but the proof is not obvious. The maths involved seems to be rather lovely though, so Stephen’s comments are well worth a look.
Obviously, the tiles must not overlap one another and must not overlap the boundary of the blue square.
[UPDATE: See the bottom of this post for the answer to this question, provided by Stephen Morris.]