Problem Solving

Hi,

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?

Ans: 12


In my opinion if all co-ordinates must be integers and if i try to draw a diagonal square yes i will have 8 points but wont two points co-ordinate to the same square? As it is necessary to make every coordinate integer. With this logic i got 8 as my ans. I am a little weak at coordinate geometry can someone please explain. Thanks.

Here is a drawing of the 12 squares that can be formed:



In the top two figures:
There are two sets of 4 squares -- for a total of 8 -- all centered about the origin.
Each side is the hypotenuse of a 6-8-10 triangle.

In the bottom figure:
There are 4 squares centered about the origin.
Each side is a horizontal or vertical line segment of length 10.