jmc

Triangle $AFG$ is similar to triangle $AHI$, and $$\frac{AF}{AH}=\frac{3\cdot AB}{4\cdot AB}=\frac{3}{4}.$$ It follows that the ratio of the area of $△AFG$ to the area of $△AHI$ is ${\left(\frac{3}{4}\right)}^2=\frac{9}{16}$. Since $△AFG$ takes up $\frac{9}{16}$ of the area of $△AHI$, trapezoid $FGIH$ takes up the other $\frac{7}{16}$ of the area. More formally, $$\begin{array}{rl}[AFG]+[FGIH]& =[AHI]⟹\\ \frac{[AFG]}{[AHI]}+\frac{[FGIH]}{[AHI]}& =1⟹\\ \frac{[FGIH]}{[AHI]}& =1-\frac{[AFG]}{[AHI]}\\ & =1-\frac{9}{16}\\ & =\boxed{\frac{7}{16}}.\end{array}$$