A square with an area of 100 must have side lengths of 10, so the question really becomes: how many 10x10 squares have integer vertices, with one vertice at the origin? Or, in other words, how many points are there at a distance of 10 from the origin that can be written as (integer, integer)?

I think it's best to visualize the possibilities just in one quadrant. Of course, there's the square whose two sides are on the x and y axis.



Since there's one of these in every quadrant, that's 4 squares.

There are also squares with vertices at (6, 8) and (8, 6). Because a 6-8-10 triangle is a special right triangle, the points (6, 8) and (8, 6) will be at a distance of 10 from the origin:



Since there are two such squares in each quadrant, there are 2*4, or 8 other squares. 8 + 4 = 12 total squares.

The answer is E.

This problem has been solved quite a number of times in this forum.
Refer to the post here >> https://www.beatthegmat.com/coordinate-p ... tml#345142