What is the area of this Trapezium

Question

enter image description hereThere is a triangle that contains 3 triangles and a trapezium. The area of these triangles are respectively 4, 7, and 14. What is the area of the Trapezium?

Answer 1

In the figure below (not drawn to scale), break the trapezium into two triangles with areas $x$ and $y$. Notice that segment $FC$ is has twice the length of segment $DF$, since triangle $\triangle FCA$ has twice the area of triangle $\triangle DFA$, and these triangles have the same 'height' relative to their bases $FC$ and $DF$. Observing that segments $FC$ and $DF$ are also bases for triangles $\triangle ECF$ and $\triangle DEF$, we deduce $x=2$.

To compute $y$, notice that the area of triangle $\triangle BED$ bears the same relationship to the area of triangle $\triangle DEA$ as does triangle $\triangle BCD$ to triangle $\triangle DCA$. (We deduce this by considering the segments $BD$ and $DA$ as bases for these triangles.) That is: $$\frac y{x+7}=\frac{BD}{DA}=\frac {y+x+4}{7+14}.$$ Plug in $x=2$ and solve for $y=\frac92$.