7. In a trapezoid, the diagonals are equal to 3 and 5, and the segment connecting the midpoints of the bases is equal to 2. Find the area of the trapezoid.
7. Let (Fig. 2) $ABCD$ be the given trapezoid ($BCAD$). Draw a line through $C$ parallel to $BD$, and denote by $K$ the point of intersection of this line with the line $AD$. The area of $\triangle ACK$ is equal to the area of the trapezoid, the sides $AC$ and $CK$ are equal to the diagonals of the trapezoid, and the median $CL$ is equal to the distance between the midpoints of $AD$ and $BC$. Extend $CL$ and take a point $M$ on the extension such that $|LLM| = |CL|$. We obtain $\triangle CKM$, which is equal in area to trapezoid $ABCD$, with sides equal to its diagonals and twice the distance between the midpoints of the bases. In our case, the sides of $\triangle CKM$ are 3.5 and 4.

Answer: $S=6$.