## 39. Square Plot

Mathieu has a square plot of land. Surrounding this plot is an alley of constant width with an area of $464 \mathrm{~m}^{2}$. Walking around his plot, Mathieu noticed that the difference in length between the outer and inner edges of the alley is 32 m.

What is the total area of Mathieu's plot, including the alley?
39. Let $x$ be the length of the outer edge of the alley, and $y$ be the length of the inner edge of the alley. We have the following system of equations:

$$
\begin{gathered}
x^{2}-y^{2}=464 \\
4 x-4 y=32
\end{gathered}
$$

From the second equation, it follows that $x=y+8$. Then the first equation gives $x=33$ and $y=25$. (Note that, by dividing the first equation by the second, we get $x+y=$ $=464: 8=58$.)

Thus, the total area of Mathieu's plot is 1089 m.