square

akhp77 wrote: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

Is it possible to figure out in 2 minutes?

Yes, if you recognize that:

-- each side = 10 (since the area of each square is 100)
-- a line segment with a length of 10, with one endpoint at (0,0), and with coordinate values that are integers must be horizontal, vertical, or the hypotenuse of a 6-8-10 triangle

So we have the following options:
Draw a horizontal line from (-10,0) to (10,0) and a vertical line from (0,10) to (0,-10). This will enable us to draw 4 squares that satisfy all the requirements.
Draw a line from (-6,-8) to (6,8) and from (-8,6) to (8,-6). The distance from (0,0) to each of these points is 10, because each distance is the hypotenuse of a 6-8-10 triangle. Using these line segments, we can draw 4 more squares that satisfy all the requirements.
Draw a line from (-8,-6) to (8,6) and from (-6,8) to (6,-8). The distance from (0,0) to each of these points is 10, because each distance is the hypotenuse of a 6-8-10 triangle. Using these line segments, we can draw 4 more squares that satisfy all the requirements.

Total possible squares = 4+4+4 = 12.

See the attached file for a drawing of the 12 squares.