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# A certain square is to be drawn on a coordinate plane. One

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Director
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14 Sep 2007, 19:57
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
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14 Sep 2007, 20:20
I get 8 as well. By drawing 1 big square from 4 little squares resting on the axis. Then I get another big square that has the diagonals of each square resting on an axis.

I have no idea if this is correct though.
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14 Sep 2007, 20:32
beckee529 wrote:
ashkrs 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

I get 8 (C). Two different ways in each of the four regions.

oh shoot.. I think this is a trick question. I change my answer to A (4) because if the set of four squares drawn diagonally cannot generate non-integer coordinates (hence (0, 10sqrt2), etc) we cannot consider those!
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14 Sep 2007, 20:53
Should be 8.
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14 Sep 2007, 21:05
Nice drawing exactly what I was saying. However, I I agree w/ Kumar now. We can't have a vertice of (0, 10sqrt2)

Nice catch.
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14 Sep 2007, 21:07
GMATBLACKBELT wrote:
Nice drawing exactly what I was saying. However, I I agree w/ Kumar now. We can't have a vertice of (0, 10sqrt2)

Nice catch.

yes, that's exactly what i was thinking too.. but do you mean you agree with my revised answer of 4 or kumar's answer of 8??
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15 Sep 2007, 03:37
should be 'E'

There will be 12 squares. See the link below:
http://www.gmatclub.com/forum/t43813
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15 Sep 2007, 09:54
vshaunak@gmail.com wrote:
should be 'E'

There will be 12 squares. See the link below:
http://www.gmatclub.com/forum/t43813

Yes OA is E .
And I am to get in terms with this question.
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15 Sep 2007, 09:57
vshaunak@gmail.com wrote:
should be 'E'

There will be 12 squares. See the link below:
http://www.gmatclub.com/forum/t43813

thanks vshaunak.. completely forgot about the 6-8-10 right triangle.. tricky!!
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19 Aug 2008, 11:44
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
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19 Aug 2008, 11:50
I think its A, just because it is a square with area 100, so each side must be 10, it cannot be any other shape than a 10x10 square. so if this is the case, then it can only be drawn four ways, one in each quadrant bc the origin (0,0) has to be one of the vertices. unless I am missing something...
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19 Aug 2008, 11:57
lexis 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?
4
6
8
10
12

12

one vertice must be on (0,0) each quadrant find how many points (integer points which makes the line with side 10 )
(0,10) , (8,6), (6,8).. So you can draw 3 squares in each quadrant .

Figure is on the way
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22 Aug 2008, 10:39
yes that makes a lot of sense, I guess you could just kind of figure out which numbers squared equals 10 squared, which are 8 and 6. I assume this is like an 800 level question?
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22 Aug 2008, 20:58
When the top left vertex moved to (6,8), did you check if the diagonal vertex coordinates are integer or not? i still feel there are only 4 solutions.
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14 Sep 2008, 14:44
pallavi79us 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?
4
6
8
10
12
Please explain how you arrive at a solution. OA will follow.

thanks,

We need to find out coordinates such that the distance between (0,0) and (x,y) is 10 (since area is 100, side =10)

(0,0) (10,0) 4 such squares in 4 Quadrants

(0,0) (8,6) 4 such squares (8,-6) ( -8,6) (-8,-6)

(0,0) (6,8) 4 such squares (6,-8) (-6,8) and (-6,-8)

Total 12
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22 Sep 2008, 01:20
This is previously posted.

For co-ordinates to be integers, possible co-ordinates for one of the vertices are (6,8), (8,6), (10,0) in one quadrant. Multiply this number of three different co-ordinates with four quadrants....and hence the answer should be 12.
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22 Sep 2008, 02:07
Hey dude I am not getting how you come with (6,8) and (8,6) as valid vertices point.

As per my knowledge... if we have one vertices to be (0,0) then one of vertices will lie on X axis (10,0) and other one must lie on Y axis (0,10) ..

So we are left with only one choice for fourth vertices... (10,10)

This is for 1 quadrants , as we have total 4 quadrants answer should be 4 .

Hope i have not done anything wrong here and i had understand question.

Please correct me if i am wrong

Thanks
Vishal Shah
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22 Sep 2008, 10:46
vr4indian wrote:
Hey dude I am not getting how you come with (6,8) and (8,6) as valid vertices point.

As per my knowledge... if we have one vertices to be (0,0) then one of vertices will lie on X axis (10,0) and other one must lie on Y axis (0,10) ..

So we are left with only one choice for fourth vertices... (10,10)

This is for 1 quadrants , as we have total 4 quadrants answer should be 4 .

Hope i have not done anything wrong here and i had understand question.

Please correct me if i am wrong

Thanks
Vishal Shah

Distance between (0,0) and (6,8) or (8,6) will be 10.
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Geometry Question - need help [#permalink]

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04 Feb 2009, 07:00
I've been having a hard time understanding the solution to this problem. Can anyone offer a simple explanation? Thanks in advance.

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?

***The Answer is 12 but I don't understand why.
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Re: Geometry Question - need help [#permalink]

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04 Feb 2009, 08:11
I've been having a hard time understanding the solution to this problem. Can anyone offer a simple explanation? Thanks in advance.

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?

***The Answer is 12 but I don't understand why.

Look for the clue: If all coordinates of the vertices must be integers.

How is a squre with area 100 possible? If its sides are each 10. How is a side with 10 is possible subject to the coordinates that make the the side 10? Its possible only with 6 abd 8 coordinates. So possibilities are:

first qd:
i: 10 and 0
ii: 8 and 6
ii: 6 and 8

Second qd:
i: 0 and 10
ii: -6 and 8
ii: -8 and 6

similarly there are 3 more in each of 3rd and 4th quadrant.

so altogather 12 possibilities.
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Re: Geometry Question - need help   [#permalink] 04 Feb 2009, 08:11

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