# Find Fraction of the Area of Square Shaded With Pink

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? ## 1 Answer

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 $$2:5$$.