satyavegi 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

OA -E

Can Someone Explain this

One simple way to construct the square is to select the sides along the coordinate axes, which results in 4 possible squares.

But now the square can be slanted.
In which case the co-ordinates of the other vertices of the sides of the square originating from the origin should be at a distance of 10 from (0,0).

Say the those vertices are at (x,y).
Using the distance formula, x² + y² = 10²

Since we know both x and y are integers, the only possible values for (x, y) in 1st quadrant are (0,10), (10,0), (8,6), and (6,8). Out of these (0, 10) and (10, 0) are on the coordinate axes, and we have included that case.

Hence, two new solution in 1st quadrant. Similarly each of the other quadrants will also have two solutions.

Hence, total number of possible squares = 4 + 2*4 = 12

The correct answer is E.