Let $ABCD$ be a unit square. A hyperbola has its foci at the midpoints of $\overline{AB}$ and $\overline{CD},$ and passes through all the vertices of the square. Compute the distance between the two vertices of the hyperbola.
Question:
Let $ABCD$ be a unit square. A hyperbola has its foci at the midpoints of $\overline{AB}$ and $\overline{CD},$ and passes through all the vertices of the square. Compute the distance between the two vertices of the hyperbola.
Answer:
Let's think step by step.

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\frac{\sqrt5-1}{2}
Let $ABCD$ be a unit square. A hyperbola has its foci at the midpoints of $\overline{AB}$ and $\overline{CD},$ and passes through all the vertices of the square. Compute the distance between the two vertices of the hyperbola.
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