Diagonal touch

                  

Image

 

$\frac{16}{81}$ is shaded.

Let $x$ and $y$ be the distances shown. Then the shaded area is $8y + x$. But there are a number of similar triangles and from one pair $$\frac{x}{8}=\frac{y}{1}$$ i.e. $$x=8y$$ So, $$\frac{\text{shaded area}}{\text{total area}}=\frac{8y+x}{9(x+y)}=\frac{8y+8y}{9\times 9y}=\frac{16}{81}$$