In a triangle $ABC$, the sides are $a=7, b=8,$ and $c=9$. Let $M$ be the midpoint of side $BC$ and $N$ be the midpoint of side $AC$. If the segment $MN$ is extended to a point $K$ outside the triangle such that the area of the quadrilateral $ABKM$ is equal to the area of triangle $ABC$, what is the ratio of the length $MK$ to the length $MN$?