2. The bases of a trapezoid are $a$ and $b$, $(a>b)$, and the sum of the angles at the larger base is $90^{\circ}$. What is the length of the segment whose endpoints are the midpoints of the bases?
Solution. Let $\overline{A B}=a$ and $\overline{C D}=b$ be the bases of the trapezoid $A B C D$, and let $\alpha+\beta=90^{\circ}$.

By extending the sides $A D$ and $B C$ of the trapezoid $A B C D$ to their intersection point $S$, we obtain the right triangles $A B S$ and $D C S$. If $M$

![](https://cdn.mathpix.com/cropped/2024_06_05_d14a67e360422a3dcaa3g-1.jpg?height=332&width=569&top_left_y=1031&top_left_x=893)
and $N$ are the midpoints of the bases $A B$ and $C D$ respectively, then the segments $S M$ and $S N$ are medians of these right triangles, so it follows that $\overline{S M}=\frac{a}{2}, \overline{S N}=\frac{b}{2}$.

From this, it follows that $\overline{M N}=\overline{S M}-\overline{S N}=\frac{a-b}{2}$.