BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Tough one - Math experts pls help for smart solution?

Expert replies
by himu » Wed Feb 27, 2013 8:33 pm
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
Join the discussion
Source: — Problem Solving |

by ceilidh.erickson » Wed Feb 27, 2013 8:56 pm
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.

Image

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:

Image

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.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
Join the discussion

by himu » Wed Feb 27, 2013 9:06 pm
Brilliant smart & best method!

Thanks a million...now I can visualize them :)
Join the discussion

by Anurag@Gurome » Wed Feb 27, 2013 9:35 pm
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
Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

Join Our Facebook Groups
GMAT with Gurome
https://www.facebook.com/groups/272466352793633/
Admissions with Gurome
https://www.facebook.com/groups/461459690536574/
Career Advising with Gurome
https://www.facebook.com/groups/360435787349781/
Join the discussion