[ทิดมี สึกใหม่]

We may assume that the sides of lengths 1 and 8 are adjacent sides of the quadrilateral,
as otherwise we can flip over the shaded triangle in the first diagram.
Now the quadrilateral may be divided into two triangles as shown in the second diagram.
In each triangle, two sides have fixed length.


Hence its area is maximum if these two sides are perpendicular to each other.
Since $1^2$+$8^2$=$4^2$+$7^2$,
both maxima can be achieved simultaneously.
In that case, the area of the unshaded triangle is 4
and the area of the shaded triangle is 14.
Hence the maximum area of the quadrilateral is 18.