A client is building a new home and wants to position the home near a lake and a school. Which combination of analysis tools will create a polygon representing a potential area for building the home?

The combination of Buffer then Intersect is the most appropriate approach for this scenario because it effectively identifies an area of interest that meets the client's requirements.

First, the Buffer tool can be used to create a zone of influence around the lake and the school. This is done by defining a specific distance away from both the lake and school, allowing you to visualize areas that are sufficiently close to both features. For example, if the client wants to be within a certain distance from the lake and the school, the Buffer tool allows you to delineate these areas.

Once the buffer zones are established, the Intersect tool can then be applied to find the overlapping area between these buffers. The result will be a new polygon that represents potential locations for the new home that are both near the lake and the school, meeting the client's criteria for proximity.

Other methods such as Union with Spatial Join might involve combining layers and transferring attributes without reinforcing the proximity requirement, making them less suited for this specific analysis of creating a meaningful area based on distance from certain features. Therefore, the Buffer then Intersect approach distinctly serves the purpose of finding viable locations based on the spatial relationship with the identified key features.