MANHATTAN GMAT QUESTION COORDINATE GEOMETRY

[karthick451]

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

ANY HELP IS APPRECIATED.

[niks_01.27]

If this is supposed to be a GMAT question and then I assume that the vertices talked about here are of the square, where one of the vertices must be on origin. Then only 4 squares can be drawn.

What is the answer? Is it A

Will explain if its correct.

[karthick451]

A is not the answer niks and it is a GMAT question.

I will give the answer after a few more responses.

[mayonnai5e]

My guess is E. You can rotate the square along the vertex at the origin and 360 degrees. The 4 obvious positions are where two sides align with the x and y axis. However, there should be two other positions in each quadrant where the vertex will just so happen to fall on the integer coordinates. What's the OA?

[kajcha]

I will go with 12.

One vertex can lie on origin and one on any of these points

(10,0), (-10,0), (0,10), (0,10), (6,8), (6,-8), (-6,8), (-6,-8), (8,6), (8,-6), (-8,6), (-8, -6)

From any of these points and origin you can calculate other 2 points.

e.g. Suppose 3rd coordinate is at (x,y) - diagonally opposite to origin

so, distance between (x,y) and (6,8) will be 10

and distance between (x,y) and origing will be 10root2

For this example other 2 points will be (2,14) and (-3,4)

I don't think you would really need to calculate for each of these points individually. If you are able to prove one point fits the formala, I think you can safely assume others will satisfy too.

I would definitely like to see some other answers.

[karthick451]

E is the answer.

Kajcha, I think you explanation is spot on. Kudos.

[samirpandeyit62]

I agree with Kajcha the ans should be E 12, I solved it like
Kajcha only.

[niks_01.27]

Agree with 12. Thanks, kajcha for a good explanation.