Notes
three tilted rectangles solution

Solution to the Three Tilted Rectangles Puzzle

Solution by Symmetry and Pythagoras' Theorem

With the points labelled as above, the line $A E$ is a diagonal of the tilted rectangle and reflecting the tilted rectangle in this line results in the rectangle $A B E F$, showing that $A H$ is the same length as $A B$.

Let $x$ be the length of $A H$, so then $A F$ has length $2 x$. Let $y$ be the length of $G H$. Then $A G$ has length $2 x - y$ since $G H$ and $G F$ are the same length. Applying Pythagoras' theorem to triangle $G H A$ shows that:

which rearranges to $4 x y = 3 x^2$ so $y = \frac{3}{4} x$. The shaded region therefore has area $\frac{3}{4} x^2$.

The area of the total design is equivalent to three rectangles of area $2 x^2$ each with the red regions removed, so the total area is $\frac{21}{4} x^2$.

The shaded region is therefore $\frac{1}{7}$th of the total design.