 Solution

Regard the upper half-disc, with cyclic boundary vertices $-1,0,x,1$, as a conformal quadrilateral. Map it to a rectangle of width $w$ and height $h$, so the boundary arcs $[-1,0]$ and $[x,1]$ become its vertical sides. The rectangle formulas for the conformal modulus of a quadrilateral giveThe remaining two boundary arcs, $(0,x)$ and the semicircle, become the horizontal sides. Thus these are precisely the dual joining families in the same quadrilateral.