CAT 3- How may squares?

[dixitsandeep]

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?


A 4
B 6
C 8
D 10
E 12


MGMAT Explaination:
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:


My question:
Using line ab (refer to explaination) as one of the sides, two square can be drawn. each square will be mirror-image of the other { think line ab as a mirror }.
However,the explaination does not count mirror-image-squares... nor does it discuss the possibility that mirror-image-squares will overlap counted squares.

[dixitsandeep]

I found other explaination.

number of possible squares is number of possible diagonals with one end at origin.

Length of diagonal will be 10*sqrt(2).

i.e. sum of squares of x and y for diagonal points must be 200.

possible values for diagonal point is (10,10) (14,2) (2,14) in first quardent.

Such set will exist in all 4 quardants.

Number of squares = 4x3 =12

[mschwrtz]

Very clever.

About mirror-image squares. They're all already counted, though not discussed in those terms. Draw the 12 squares described, and you'll see that they form three groups of four squares per group, such that each of the squares has two reflections in it's group. Confusing? Not bad if you draw it.

About using the hypotenuse. I like that an awful lot, but I don't think that it'll be accessible to may on a time test. the 6, 8, 10 triplet, on the other hand, is one that we should recognize right away. You realize that it's yield the same 12 squares, right?