Given a trapezoid ABCD with parallel sides of length a and b.
Construct a segment CE of lenght a perpendicular to CD at C. Draw CD.
The length of DE is
Construct the midpoint M of DE and construct a perpendicular segment MN from the midpoint that has length half of DE. Construct DN.
DN has length
Now, locate P on DC such that DP = DN. Construct a parallel line to AD and locate its intersection with BC at R. Construct RS parallel to CD. RS is the desired segment.
Click here for a GSP file showing the constructed segment.