Here is the question I am working on: For $s>2$ ,let $Q_s$ be the hyperbolic quadrilateral in $\mathbb{H}^2$ with vertices $−1+i$, $−1+2i$, $1+i$,and $1+si$.Determine the values of $s$ for which $Q_s$ is a hyperbolic parallelogram.
So after a quick sketch I can see that the left and right sides will always be parallel as they are vertical. The only time sides aren't parallel is when the upper bounding arc intersects the lower bounding arc. Now, my idea is to see the exact point at which the vertex $1+si$ becomes so high that both the upper and lower bounds start from the same point on the real axis, giving me a range for $s$ that tells me when the arcs are no longer parallel.
How can I determine where that point is? I think what I need to do is see what exact point it is where the fixed lower bounding arc meets the real axis, in order to determine how far the variable upper bounding arc can move its starting point.