What fraction of the rectangle is shaded? (You may assume that each line, other than the diagonal of the rectangle, is parallel to some side of the rectangle.)
Is there a way to solve this without doing algebraic manipulations to get the other side length of the rectangle? How can the two rectangles being formed inside help in the calculation of the diagonal?
You do not need the length of the diagonal to find the proportion.
If $H$ is the height of the whole rectangle, then the small rectangle in the bottom-left corner is of height $\frac H9$ because of similarity.
Hence, the proportion of the shaded region is
$$\frac{1\cdot \frac 89H + 8\cdot \frac 19 H}{9H} = \frac{16}{81}$$.
Another way is to see that the whole rectangle is tiled by $9\times 9$ rectangles congruent to the small bottom-left rectangle. Now, counting gives as well $\frac{16}{81}$.