Hello, asyed94!
It's much easier thank you imagine . . .
In quadrilateral $\displaystyle ABCD\!:\;\;AB = 5,\;BC = 17,\;CD = 5,\;DA = 9.$
And the length of diagonal $\displaystyle BD$ is an integer.
Find the length of $\displaystyle BD.$ Code:
A
o
9 * *
* *
D * *
o * 5
* * *
5 * * *
* * *
C o * * * * * * o B
17
In $\displaystyle \Delta ABD$, the Triangle Inequality says: .$\displaystyle AB + AD \:> \:BD$
That is: .$\displaystyle 5 + 9 \:>\:BD \quad\Rightarrow\quad BD \:<\:14$
In $\displaystyle \Delta BCD$, the Triangle Inequality says: .$\displaystyle BD + CD \:>\:17$
That is: .$\displaystyle BD + 5 \:>\:17 \quad\Rightarrow\quad BD \:>\:12$
Therefore: $\displaystyle BD$ is an integer between 12 and 14 . . . $\displaystyle BD \,=\, 13.$