[ Square](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) $ABCD$ has sides of length 2.  [Set](https://artofproblemsolving.com/wiki/index.php/Set) $S$ is the set of all [line segments](https://artofproblemsolving.com/wiki/index.php/Line_segment) that have length 2 and whose [endpoints](https://artofproblemsolving.com/wiki/index.php/Endpoint) are on adjacent sides of the square. The [midpoints](https://artofproblemsolving.com/wiki/index.php/Midpoint) of the line segments in set $S$ enclose a region whose [area](https://artofproblemsolving.com/wiki/index.php/Area) to the nearest hundredth is $k$.  Find $100k$.