On a secluded island, there lies a peculiar treasure map. The map shows an irregular quadrilateral plot of land with side lengths of 9 meters, 6 meters, 11 meters, and 8 meters.

A diagonal path cuts through the plot from one corner to the opposite corner, dividing the plot into two triangles. This diagonal path is 13 meters long.

One of these triangles has sides of lengths 9 meters, 8 meters, and 13 meters. The other triangle has sides of lengths 11 meters, 6 meters, and 13 meters.

There is a hidden treasure buried within the triangle that has the smallest side length of the quadrilateral. Find the area of the triangle that contains the treasure.