Coordinate plane

[Nigogo]

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 :roll:

[papgust]

Is the answer 10?

[NikolayZ]

I ve seen this problem on this forum already afaik.

1) lets solve this for just one (first quadrant)
in order to make the square with integer sides - we have to know what pairs of (x;y) give us the desired square.
Assume we need to know just one pair of (x;y) coordinates of a square side - let it be OB ( 0 - origin)
- O (0:0),B (10:0)
- O (0:0),B (8:6)
- O (0:0),B (6:8)
SO, in each quadrant there will be 3 different square positions.
==> 12 IMO

[Nigogo]

the answer is 12, vertices with 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.