Question 11
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A = (20, 100)$ and $D = (21, 107).$ The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$