Cat Exam #4 Question 12:
"A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn?"
Answer is 12. Explanation discusses pythagorean triple and why a line with a vertice at the origin could have it's other vertice at (0,10), (6,8), (8,6),... I understand that part, and why there would then be 12 options. What I don't understand is how you can assume from that explanation that the other vertices of the square will also be integers. The explanation only says that "It is tedious and unnecessary to figure out all four coordinates for each square" but I don't see how it necessarily follows that the other vertices will also be integers just because one is. Please explain.