cuty 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


got it from MGMAT practice test !!

Night reader wrote:Hi cuty, to approach this problem decisively we need to look at the diagonal of one of our possible squares with one vertice in the origin at (x=0, y=0) and another in point (x=10, y=10). We are ready to derive the equation for line (diagonal) in our square --> y=x and we know that y*x=100 OR x^2=100. So x and y can be integers only when x=10, -10 {|10|} and y=10, -10 {|10|}, hence answer is A) 4 ways.

cuty 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


got it from MGMAT practice test !!

Anurag@Gurome wrote:

cuty 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

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.

Anurag@Gurome wrote:

cuty 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

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.

vineeshp wrote:
You have not taken into consideration the Pythagoras theorem and hence points 6,8 and 8,6 come into play.
So answer should be 12.