Problem Solving

Source: Manhattan GMAT

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: 4,6,8,10, or 12

OA in a few.

Answer is 8

Ans is E - 12

Explanation from Manhattan GMAT. I'm still confused by their 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 hyptoneuse 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 lable 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 coordinates:

(-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.

Visualization is the key to the answer. Let the origin be 0 and one of the vertices to be A. The square in question have an area of 100 which means the length of OA must be 10.
if the coordinates of A is (x,y), then we would have x^2 + y^2 = 100. (the distance from the origin 0 to the point A of (x,y) axis can be obtained with the Pythagoras theorem=
distance <d^2>= axis (x^2) + axis (y^2)
d^2=x^2 + y^2
x^2 + y^2 = 100 has several integers solutions for x and y.
you notice that
100= x^2 + y^2
= 6^2 + 8^2
and also
100= x^2 + y^2
= 10^2 + 10^2
which means that the coordinates can take these values - 10,8,6,0,-6,-8,-10.

1) Now if x=0 and y=0 through the axis, we have one square which rest on x-axis and to get the other option, we rotate length OA anticlockwise to get all possible cases which is ;
2) X=8 and y=6
3) X=6 and y=8
4) X=0 and y=10
5) X=-6 and y=8
6) X=-8 and y=6
7) X=-10 and y=0
8) X=-8 and y=-6
9) X=-6 and y=-8
10) X=0 and y=-10
11) X=6 and y=-8
12) X=8 and y=-6
if all the coordinate of the vertices must be integer, the square can definitely be drawn in 12 different ways.