A gardener is cultivating a garden plot shaped like a right triangle. The legs of the triangle measure $8$ meters and $15$ meters. At the right-angle corner, a small square region $S$ is paved and left unplanted. The rest of the garden is planted. The shortest distance from the vertex of $S$ furthest from the right angle to the hypotenuse of the triangle is $3$ meters. What fraction of the garden plot is planted?