Question 100
Let $[AB]$ be a segment and $\sigma$ be one of the halfplanes determined by the straight line $AB$. The segments $[AP]$ and $[BQ]$ with integer lengths are situated in $\sigma$ and are perpendicular to the straight line $AB$. The intersection point $M$ of the straight lines $AQ$ and $BP$ is distanced at 8 units to the straight line $AB$. Find the lengths of the segments $[AP]$ and $[BQ]$, if it is known that the triangle $BQM$ has a greatest area.