In the city of Oropis, there is a triangular lake with each of its three sides being part of the perimeter of a square-shaped land with areas of $370 \mathrm{~m}^{2}$, $116 \mathrm{~m}^{2}$, and $74 \mathrm{~m}^{2}$, as shown in the first figure below. The mayor of Oropis, Arnaldo, wants to calculate the area of the lake but doesn't know how. The mayor's assistant, Bernaldo, has an idea. He says that it is enough to find integer values of $a, b, c$, and $d$ that satisfy the geometric conditions of the second figure, where the shaded area is congruent to the lake. He claims that after this, the task will become much simpler.
![](https://cdn.mathpix.com/cropped/2024_05_01_10107d2fba0ec26bc845g-33.jpg?height=892&width=1442&top_left_y=766&top_left_x=404)

(a) Determine integer values of $a, b, c$, and $d$ that satisfy the geometric conditions of Bernaldo's figure.

(b) Determine the area of the lake.
Solution

(a) See that the sides of the lake squared result in 74, 116, and 370. The conditions that must be met are the equations derived from three applications of the Pythagorean Theorem:

$$
\begin{aligned}
a^{2}+c^{2} & =74 \\
b^{2}+d^{2} & =116 \\
(a+b)^{2}+(c+d)^{2} & =370
\end{aligned}
$$

We can limit the tests of the first equation to numbers less than 9, since $9^{2}>74$. After a case analysis, we obtain $a=5$ and $c=7$ or $a=7$ and $c=5$ as the only possibilities. In the second equation, also after a case analysis, we have $b=4$ and $d=10$ or $b=10$ and $d=4$, and consequently, $a+b=5+4=9$ and $c+d=7+10=17$, since $(5+10)^{2}+(7+4)^{2} \neq 370$. Therefore, $a=5, b=4, c=7$, and $d=10$.
(b) Thus, the area of the lake is the area of the larger triangle minus two triangles and one rectangle. That is, the area of the lake, in $m^{2}$, is

$$
\begin{aligned}
\frac{17 \cdot 9}{2}-\frac{7 \cdot 5}{2}-\frac{10 \cdot 4}{2}-7 \cdot 4 & =\frac{118}{2}-20-28 \\
& =11
\end{aligned}
$$