In the quadrilateral $A B C D$, we have: $A B=5, B C=17, C D=5, D A=9$, and the measure of the segment $D B$ is an integer. Determine $D B$.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-09.jpg?height=176&width=344&top_left_y=752&top_left_x=1357)
Remember that, in a triangle, any side is greater than the difference and less than the sum of the other two. For triangle $A D B$, we have $A D - A B < B D < A D + A B$, and for triangle $C B D$, it follows that $B C - C D < B D < B C + C D$. Substituting the known values

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-35.jpg?height=215&width=419&top_left_y=470&top_left_x=1314)
we obtain:

$$
9 - 5 < B D < 5 + 9 \text{ and } 17 - 5 < B D < 17 + 5
$$

that is,

$$
4 < B D < 14 \text{ and } 12 < B D < 22
$$

From the two inequalities, we conclude that:

$$
12 < B D < 14
$$

Since $B D$ is an integer, we can only have $B D = 13$.