From the image below, the given and question is:
They are two identical squares and four identical pink triangles. $'A'$ is the midpoint, what fraction of the square on the right is shaded pink?
Now, I can very much understand that since $A$ is the midpoint, each of the $4$ identical triangles is split into a total of $8$ triangles with a {height:width}=${4:1}$ and that the sides of the square is split into two lengths with ratio $1:3$ on where the pink lines intersect with it.
However, with those I have no idea what to do next. Can anyone confirm the answer of $\frac{2}{5}$ and a solution?
Let $a$ be the edge of the square. Consider the following picture:
Note that $DE\parallel CK$, so $AI:IC= AE:EK= 1:4$. Thus $d(I,AB):d(C,AB)=AI:AC=1:5$, i.e. $d(I,AB)=\frac{a}{5}$. So $Area(AED)= \frac{a^2}{8}$, $Area(IEB)= \frac{EB\cdot d(I,AB)}{2}= \frac{3a^2}{40}$, so shaded area is $2\cdot(\frac{a^2}{8}+\frac{3a^2}{40})= 2\cdot \frac{a^2}{5}$. Therefore, the ratio of the shaded area and the area of the square is $2:5$.