In a trapezoid, the midsegment has length 17 and the distance between the midpoints of the diagonals is 7 . Find the lengths of the bases.
Let $a$ and $b$ be the bases, with $a>b$. The length of the midsegment is the average of the bases, so $(a+b) / 2=17$, and the distance between the midpoints of the diagonals is half their difference, so $(a-b) / 2=7$. Adding the two equations gives $a=24$, and subtracting gives $b=10$.