CAT -1 : How many squares?

[arvindshastry]

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?
4
6
8
10
12

Manhattan SC Explanation:

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.

The correct answer is E.

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However, the points (6,8) and (-8,6) appear on the same square and hence will not constitute different squares -->

For eg, If b = (6,8) and c = (-8,6) in one square, then in the same square, we can have c = (-8,6) and b = (6,8). So there is a double counting of squares. According to me the answer should be 8 and not 12.

Any comments?

[rfernandez]

If b = (6, 8), as defined by the solution, the other three vertices are: (-2, 14), (-8, 6), and (0, 0)

If b = (-8, 6), the other three vertices are: (-14, -2), (-6, -8), and (0, 0)

These are unique squares, and so there is not double-counting. Both of these squares share a side that extends from (0,0) to (-8, 6), but then the two squares lie on either side of this side.