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Coordinates for the first side must be (0,0) to (m,n) where m and n are integers, then the other vertices will also have integer coordinates. You want m and n such that the line will have a length of 10, so m^2+n^2=10^2.
Make a table for m:
m | m^2 | n^2
0 | 0 | 100
1 | 1 | 99
2 | 4 | 96
3 | 9 | 91
4 | 16 | 84
5 | 25 | 75
6 | 36 | 64
7 | 49 | 51
8 | 64 | 36
9 | 81 | 19
10 | 100 | 0
When m is 0 or 100, the sides will be parallel/perpendicular to the axis and there will be 4 such rotations of the square.
When m is 6, n is 8 and that square will have 4 rotations.
When m is 8, n is 6 and that square will also have 4 rotations. Don't forget, vertex (8,6) looks similar to (6,8) but the squares are not the same, so each has 3 unique rotations.
12, E
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