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A certain square is to be drawn on a coordinate plane MGMAT

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by doclkk » Fri Sep 04, 2009 7:51 am
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

I've read the MGMAT explanation and I don't understand how they got the coordinates 6,8. I mean. I think figuring out 8 is easy. Figuring out 12? No idea how they did it.

Any insight would be great

OA: (E)
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by bharathh » Fri Sep 04, 2009 10:12 am
How do you get 8? If you get 8, 12 should be easy.

1 square can be made by keeping two sides of the square along the axis.

Now another way to get the square with one vertex on the origin is to slant the square. If you slant the square it will form a right angled triangle with the axis where one side of the square is the hypotenuse of the triangle.

A right angled triangle with h=10 can have sides 6 and 8

So you can get a right angle triangle with (6,8) and another with (8,6).

Thus there are 3 possible squares that can be formed in one quadrant of the x-y plane. There are 4 planes... So 12 squares
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