niketdoshi123 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

In each square, s=10.
For the coordinates of the vertices to be integers, each side must be either a horizontal or vertical line segment of length 10 or the hypotenuse of a 6-8-10 triangle.
Plot points at every combination of 0 and ±10 and every combination of ±6 and ±8.
Draw lines connecting these points to the origin.
Complete all the possible squares.
The result is the following:



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.

Total squares = 12.

The correct answer is E.

Alternative Approach:

Draw a circle with radius of 10 with center at the origin.
In quadrant 1 the points crosses the axis at y=10 and crosses the x axis at x=10.

At any point along the arc, y = sin (angle).
There are 5 angles that you should memorize:
Sin 0 = 0
Sin 30 = 0.5
Sin 45 = Sqrt 2 / 2
Sin 60 = Sqrt 3 / 2
Sin 90 = 1

If we multiply 10 times each of the angles above, we see that only 3 gives integers
The above approach applies to the 1st quadrant and there are 4 quadrants on the graph.

Therefore we have to multiple 3 x 4 = 12