KöMaL 2009 A484

In the Cartesian plane, a convex polygonal disk $K$ is given. Apply the perpendicular affinity ($x$,$y$)$\mapsto$($x$,$py$) to the points of $K$, where 0<$p$<1. Denote the resulting polygonal disk by $K$'. Prove that $K$' can be covered by a polygonal disk which is congruent to $K$.