To determine the area of the rectangle after dilation, we first need to understand how dilation affects the area of shapes.

Steps:

1. Understanding the Dilation Rule: The given dilation rule is $(x, y) \to (13x, 13y)$. This means that the rectangle's dimensions (length and width) will be scaled by a factor of 13.

2. Effect on Area: When a two-dimensional shape is dilated by a factor $k$, its area is scaled by a factor of $k^2$. In this case, the dilation factor is 13, so the area will be scaled by $13^2 = 169$.

3. Original Area: If you know the original area of the rectangle, the new area can be found by multiplying it by 169.

For example, if the original area of the rectangle is $A_{\text{orig}}$, then the new area $A_{\text{new}}$ is: $A_{\text{new}} = 169 \times A_{\text{orig}}$

Conclusion:

To get the exact area after dilation, I need to know the original area of the rectangle. Could you provide that, or should I assume a sample rectangle to calculate the new area for you?

Let me know how you’d like to proceed!