KöMaL 2006 K108

A square patch of grass, 100 m on a side is surrounded by a paved road along its boundary. Most people want to get from corner $A$ of the square to the midpoint $F$ of an opposite side. They walk along the path ABF , but some of them have started a path through the grassy area. They walk along the side $AB$ for a while, then they are heading towards $F$ along a straight line through the grass, as shown in the diagram. Thus they walk 25% less compared to the path along the border. How far is the turning point $K$ of the path from the point $A$?