Tough one - Math experts pls help for smart solution?

[himu]

Saw a blog already...but wasn't of much help
hence re posting :

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?


4

6

8

10

12

[ceilidh.erickson]

A square with an area of 100 must have side lengths of 10, so the question really becomes: how many 10x10 squares have integer vertices, with one vertice at the origin? Or, in other words, how many points are there at a distance of 10 from the origin that can be written as (integer, integer)?

I think it's best to visualize the possibilities just in one quadrant. Of course, there's the square whose two sides are on the x and y axis.



Since there's one of these in every quadrant, that's 4 squares.

There are also squares with vertices at (6, 8) and (8, 6). Because a 6-8-10 triangle is a special right triangle, the points (6, 8) and (8, 6) will be at a distance of 10 from the origin:



Since there are two such squares in each quadrant, there are 2*4, or 8 other squares. 8 + 4 = 12 total squares.

The answer is E.

[himu]

Brilliant smart & best method!

Thanks a million...now I can visualize them

[Anurag@Gurome]

This problem has been solved quite a number of times in this forum.
Refer to the post here >> https://www.beatthegmat.com/coordinate-p ... tml#345142