In a certain trapezoid, the longer of the two parallel sides is $3 \mathrm{~m}$ longer than the shorter one; the height is proportional to the arithmetic mean of the two parallel sides; what are the lengths of the two parallel sides and the height, if the area of the trapezoid is $7.75 \mathrm{~m}^{2}$ less than twice the area of the square formed above the shorter parallel side?
One side is $5 \mathrm{~m}$, the other is $8 \mathrm{~m}$, and the height is $6.5 \mathrm{~m}$.

Number of solutions: 20.