Problem 1341

1. Identify the given information and construct the necessary points:
- Trapezoid $ABCD$ with $AD \parallel BC$.
- Diagonals $AC$ and $BD$ with lengths $|AC| = 8$ and $|BD| = 6$.
- Midpoints $P$ and $S$ of $AD$ and $BC$ respectively, with $|PS| = 5$.

2. **Construct parallelograms $CADM$ and $BDMN$:**
- Extend $AD$ and $BC$ to form parallelograms $CADM$ and $BDMN$.
- Since $P$ and $S$ are midpoints, $PS$ is parallel to both $AD$ and $BC$.

3. Analyze the properties of the trapezoid and the midsegment:
- The midsegment $PS$ of a trapezoid is parallel to the bases and its length is the average of the lengths of the bases.
- Let $AD = a$ and $BC = b$. Then, $|PS| = \frac{a + b}{2}$.

4. **Use the given length of the midsegment to find the relationship between $a$ and $b$:**
- Given $|PS| = 5$, we have:
$$\frac{a + b}{2} = 5 \implies a + b = 10$$

5. Determine the height of the trapezoid using the diagonals:
- Since $AC \perp BD$, the diagonals intersect at right angles.
- The area of the trapezoid can be found using the formula for the area of a right-angled triangle formed by the diagonals:
$$\text{Area} = \frac{1}{2} \times AC \times BD = \frac{1}{2} \times 8 \times 6 = 24$$

6. Verify the area calculation:
- The area of the trapezoid is indeed given by the product of the diagonals divided by 2, confirming the calculation.

$\blacksquare$

The final answer is $\boxed{ 24 }$.