To find the difference in the lengths of the diagonals of the two tiles, we can subtract the length of the second diagonal from the length of the first diagonal.
The diagonals are:
- First tile: \(9\sqrt{2}\) cm
- Second tile: \(8\sqrt{2}\) cm
Now, we calculate the difference:
\[ 9\sqrt{2} - 8\sqrt{2} = (9 - 8)\sqrt{2} = 1\sqrt{2} = \sqrt{2} \text{ cm} \]
Thus, the difference in the lengths of the diagonals is \(\sqrt{2}\) centimeters.