Each side of the square must have a length of 10

[star.estella.li]

This is from CAT Exam #1. I looked at the diagram shown in the answer explanation but still can't figure out how 12 squares can be drawn.

QUESTION: 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: Each side of the square must have a length of 10. If each side were to be 6, 7, 8, or most other numbers, there could only be four possible squares drawn, because each side, in order to have integer coordinates, would have to be drawn on the x- or y-axis. What makes a length of 10 different is that it could be the hypotenuse of a Pythagorean triple, meaning the vertices could have integer coordinates without lying on the x- or y-axis.

For example, a square could be drawn with the coordinates (0,0), (6,8), (-2, 14) and (-8, 6). (It is tedious and unnecessary to figure out all four coordinates for each square).

If we label the square abcd, with a at the origin and the letters representing points in a clockwise direction, we can get the number of possible squares by figuring out the number of unique ways ab can be drawn.

a has coordinates (0,0) and b could have the following coordinates, as shown in the picture:


(-10,0)
(-8,6)
(-6,8)
(0,10)
(6,8)
(8,6)
(10,0)
(8, -6)
(6, -8)
(0, 10)
(-6, -8)
(-8, -6)

There are 12 different ways to draw ab, and so there are 12 ways to draw abcd.

NOTE FROM STAR: I can't seem to post the diagram but basically, I understand that if you connect (0,0) to any of those 12 points, you'd get 12 lines. How you would then connect with them other points to form a square is where I get confused.

[StaceyKoprince]

It's really hard to visualize. The squares are all going to be drawn at an angle - they'll look more like (squared) diamonds. I would actually recommend that you grab a piece of graph paper, draw out the diagram, and actually sketch out the square that the explanation describes:

(0,0), (6,8), (-2, 14) and (-8, 6)

Just so you can visualize it. And then each new possibility for ab would represent one new square, just at a different angle than the others.

Here's the best news of all: unless you want a 780+ on this test, you really don't need to worry about this. Geometry in general isn't as frequently tested as other areas, and this particular problem is just about as hard as it gets. It might even be a little too hard for the real test. So don't worry about it. :)