I am completely new to this and have never worked with shapefiles. What I'm trying to find out is whether it is possible, using the TIGER data in conjunction with the ZCTA files, to determine the physical boundaries of a ZCTA and once that step is complete, to generate a list of random geographic points (LAT/LON pairs) that fall within that ZCTA boundary.

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Very possible. The TIGER Data website has data in shapefile or geodatabase format. Included in this data are ZCTA boundaries (ZIP Code Tabulation Areas). So you can download that data, and using a tool of your liking, generate points that fall within these boundaries

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