2003 AMC 10A Problem 22

Place $D = (0, 0),$ $A = (9, 0),$ $B = (9, 8),$ $C = (0, 8),$ $H = (3, 8),$ and $E = (4, 0).$ Line $AH$ has equation $y = -\dfrac{4}{3}x + 12,$ and line $EC$ has equation $y = -2x + 8.$ Setting them equal gives $x = -6$ and $y = 20,$ so $G = (-6, 20).$ Since $\overline{GF}$ is perpendicular to line $AD$ (the $x$-axis), its length is the height $20.$ Thus, the correct answer is B.