jmc

Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{CD},$ respectively. Then the hyperbola is the set of all points $P$ such that $$\left|PM-PN\right|=2a,$$and $2a$ is the distance between the two vertices of the hyperbola. To find the value of $2a,$ we set $P=A,$ so that $$2a=|AM-AN|=\left|\frac{1}{2}-\frac{\sqrt{5}}{2}\right|=\boxed{\frac{\sqrt{5}-1}{2}}.$$