Let us assume the figure to be ACBDEF.
There are two equal trapeziums ACEF and CBDE.
If a is one of the sides of this figure,
base of trapezium = a units
height of trapezium = (a/2) units
side opposite 30 degrees : side opposite 90 degrees = 1:2 in the 30-60-90 triangle.
Area of the trapezium = base * height = (a^2) / 2 sq units.
Area of the whole figure = (a^2) sq units = 42 sq units.
a = root(42) units.
AB = 2 times height of each trapezium = 2 times (a/2) = a units = root(42) units.
Hope this helps. Is this correct ... Look forward to know a better method.
4GMAT - Mumbai