Proportions in a Trapezoid

Assign the points $A$ , $B$ , $C$ , $D$ and $E$ as shown in the figure. Let the area of the blue triangle be $x$ , $AB=a$ and $BD=b$ .

We note that the blue $\triangle ABC$ is similar to $\triangle ADE$ . Since areas of similar figures are directly proportional to the square of their respective linear dimensions, we have:

$\begin{aligned} \frac {[ABC]}{[ADE]} & = \frac x{x+9} = \left(\frac a{a+b}\right)^2 & \small \color{#3D99F6} [\triangle] \text{ denotes the area of }\triangle. \end{aligned}$

Since rectangles with areas 56 and 14 share the same base length, their area is proportional to their heights $a$ and $b$ . Therefore,

$\begin{aligned} \frac x{x+9} & = \left(\frac a{a+b}\right)^2 = \left(\frac {56}{56+14}\right)^2 = \frac {16}{25} \\ \implies 25 x & = 16x + 144 \\ 9 x & = 144 \\ \implies x & = \boxed{16} \end{aligned}$