The angles $A \widehat{B} C$ and $C \widehat{D} A$ of the quadrilateral $A B C D$ in the figure are right angles, and the four sides of the quadrilateral measure integers that are all distinct. If $A D=7$ and $B C=11$, what are the lengths of the sides $A B$ and $C D$?

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-026.jpg?height=268&width=368&top_left_y=2413&top_left_x=1461)
Since triangles $\triangle A B C$ and $\triangle A C D$ are right-angled and have the same hypotenuse $A C$, by the Pythagorean Theorem we have $x^{2}+11^{2}=y^{2}+7^{2}$, where $A B=x$ and $D C=y$. Then,

$$
(y-x)(y+x)=y^{2}-x^{2}=72=2^{3} \times 3^{2}
$$

and, therefore, $y-x$ and $y+x$ are divisors of 72. For each factorization of 72, we need to solve a system of two equations with two unknowns, as in the following table, and identify the cases where there are integer solutions.

| Factor of 72 |  | Measures of |  | Observations |
| :---: | :---: | :---: | :---: | :---: |
| $y+x$ | $y-x$ | $x$ | $y$ |  |
| 72 | 1 | - | - | No integer solution |
| 36 | 2 | 17 | 19 | Has integer solution |
| 24 | 3 | - | - | No integer solution |
| 28 | 4 | 12 | 16 | Has integer solution |
| 12 | 6 | 3 | 9 | Has integer solution |
| 9 | 8 | - | - | No integer solution |

Thus, there are three integer solutions for the lengths of the sides $x$ and $y$.