Notes
three aligned triangles solution

Three Aligned Triangles

Solution by Lengths in Equilateral Triangles and Area Scale Factor

Let $x$ be half the side length of the smallest triangle. Using the relationships between lengths in an equilateral triangle, the length of $E B$ is $\sqrt{3} x$. The area scale factors to the other triangles are $9$ and $4$, so the length scale factors are $3$ and $2$. Therefore $A B$ has length $4 x$ and $B C$ has length $3 x$, while $F A$ has length $3 \sqrt{3} x$ and $D C$ has length $2 \sqrt{3} x$.

The area of the shaded triangle can be found by taking the area of trapezium $F D C A$ and subtracting the areas of the trapezia $F E B A$ and $E D C B$. This is given by:

The area of the smallest equilateral triangle is given by $\sqrt{3} x^2$, so this is equal to $1$. Hence the area of the shaded region is $5$.