Let's analyze the problem. We are given a coordinate grid where each unit represents a mile. The library is located at (-2, 2) and the grocery store is at (3, 2). The distance between the library and the grocery store is the same as between the grocery store and the post office. Additionally, the post office is located south of the grocery store.

Step 1: Calculate the distance between the library and the grocery store.

The library and the grocery store lie on the same horizontal line (both at $y = 2$). The distance between them is just the difference in their $x$-coordinates: $\text{Distance} = 3 - (-2) = 3 + 2 = 5 \text{ miles}.$ Thus, the post office must be 5 miles south of the grocery store.

Step 2: Determine possible locations of the post office.

Since the post office is 5 miles south, we need to subtract 5 from the $y$-coordinate of the grocery store, which is $y = 2$. So the $y$-coordinate of the post office will be: $2 - 5 = -3.$ The $x$-coordinate will remain the same as that of the grocery store, which is $x = 3$.

Thus, the only possible location for the post office is $(3, -3)$.

Final Answer:

The post office can only be located at $(3, -3)$.

Would you like further clarification or details on any part of this? Here are five related questions to extend this concept:

1. What is the general formula for finding the distance between two points on a coordinate plane?
2. How would the answer change if the post office was to the east or west of the grocery store instead of south?
3. Can you explain how translations on the coordinate plane work when shifting points vertically or horizontally?
4. What would happen if the post office was a different fixed distance from the grocery store? How would you handle that?
5. How could we extend this problem to three-dimensional coordinates?

Tip: When working with distances on a coordinate grid, it's often helpful to first consider whether the points lie on the same horizontal or vertical line to simplify calculations.