we know that the square, 4 equal sides, has a perimeter of 24, meaning each sides is simply 24 ÷ 4 = 6, since the area of a square is simply the side², that means the area of the square is 6² or just 36. We also know that the the trapezium has double the area of the square, namely 2*36 = 72, Check the picture below.
[tex]\textit{area of a trapezoid}\\\\ A=\cfrac{h(a+b)}{2}~~ \begin{cases} h=height\\ a,b=\stackrel{parallel~sides}{bases}\\[-0.5em] \hrulefill\\ h=8\\ a=AB\\ b=3AB\\ A=72 \end{cases} \implies \begin{array}{llll} 72=\cfrac{8(AB+3AB)}{2}\\\\ 72=4(4AB)\implies 72=16AB\\\\ \cfrac{72}{16}=AB\implies \cfrac{9}{2}=AB \end{array}[/tex]
