If you draw a vertical line between the apexes of the two triangles you will exactly divide the shaded area in two. Each half will have a side of 1 and another of x, and also a fixed angle (opposite the side 1) of 150 degrees. In order to maximise the area of this triangle, we must maximise the distance between the vertex with angle 150, and the side of length 1. This will be achieved when the triangle is isosceles, such that the distance between the two apexes is also x.

Since cosine of 150 degrees is -sqrt(3)/2, it’s possible to calculate x to be sqrt(2-sqrt(3)), which is approximately 0.5176.