Construction

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.

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Click here for a GSP file showing the constructed segment.