a square on the coordinate plane

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

Redeem

This topic has expert replies

Post new topic Post Reply

Previous Topic Next Topic

sidon: Junior | Next Rank: 30 Posts; Posts: 16; Joined: Fri Apr 09, 2010 11:29 am

a square on the coordinate plane

by sidon » Sat Apr 24, 2010 5:19 am

Hi,

I have encountered the following question on one of my tests :

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?

I think there should be 16 , but apperntly the asnwer is 12 .

Can someone help ?

Thanks,
Sidon

Quote

Source: Beat The GMAT — Quantitative Reasoning | Sat Apr 24, 2010 5:19 am

iamseer: Master | Next Rank: 500 Posts; Posts: 170; Joined: Wed Jun 10, 2009 5:59 am; Thanked: 13 times

by iamseer » Sat Apr 24, 2010 7:16 am

Square with area 100, so side has to be 10

Look at it this way:
Square is placed initially at A(0,0),B(10,0),C(10,10),D(0,10).

Now you have to move this sqaure say anticlockwise with the vertex at origin fixed. But it has to be moved such that for the vertex B(x,y) x^2+y^2=100 and also x and y must be integers.
Now the only possibilities for B are (0,10), (6,8) and (8,6) and this can be repeated in all the remaining 3 quadrants.
so its 3*4 =12

perhaps you are getting 16 b'cos you are counting one of the positions for B in each quadrant twice.

HTH

"Choose to chance the rapids and dance the tides"

Quote

Fiver: Master | Next Rank: 500 Posts; Posts: 114; Joined: Mon Sep 22, 2008 3:51 am; Thanked: 8 times; GMAT Score:680

by Fiver » Sat Apr 24, 2010 7:31 am

Since the figure is a square and area of a square is (side)^2 = 100
we need to look for integer combinations, squares of which sum up to 100

Listing possibilities:
1] 10 & 0 Since 10^2 + 0 =100
2] 8 & 6 Since 64 + 36 = 100
3] 9 & 3 Since 81 + 9 = 100

Since it is given that one of the vertices at (0,0) and the fact that it is a sqaure that we need to plot,
we know the following:
1]The co-ordinates of 2 end-points of the sides extended from (0,0) will be either of the format (0,integer) or (integer,0).
2] Hence their lengths will be the square of the integer value.
3] If we plot 2 points then the 3rd point has no choice but to be fixed a value and thus we need not worry about the possibilities of this 3rd point.

e.g. If we choose the 1st option above one of our points will be (0,10) and the other will be (10,0) and hence the 3rd point has to be (10,10).
So my possibilities with this pair are (10&10), (-10&10), (10&-10), (-10&-10), fixing the square in each of the 4 quadrants.

Hence with the listed possibilities in mind, since each has 2 values and each value can take 2 signs (either + or -), each pair has 4 possibilites.
And since there are 3 such pairs total possibilities are 3*4 = 12.

Quote

sidon: Junior | Next Rank: 30 Posts; Posts: 16; Joined: Fri Apr 09, 2010 11:29 am

by sidon » Sat Apr 24, 2010 9:52 am

why can'y one side be 100 and the other side 1 ?

Quote

tpr-becky: GMAT Instructor; Posts: 509; Joined: Wed Apr 21, 2010 1:08 pm; Location: Irvine, CA; Thanked: 199 times; Followed by:85 members; GMAT Score:750