GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 17 Jan 2019, 01:17

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

## Events & Promotions

###### Events & Promotions in January
PrevNext
SuMoTuWeThFrSa
303112345
6789101112
13141516171819
20212223242526
272829303112
Open Detailed Calendar
• ### The winning strategy for a high GRE score

January 17, 2019

January 17, 2019

08:00 AM PST

09:00 AM PST

Learn the winning strategy for a high GRE score — what do people who reach a high score do differently? We're going to share insights, tips and strategies from data we've collected from over 50,000 students who used examPAL.
• ### Free GMAT Strategy Webinar

January 19, 2019

January 19, 2019

07:00 AM PST

09:00 AM PST

Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.

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

Author Message
Manager
Joined: 22 Feb 2007
Posts: 63
A certain square is to be drawn on a coordinate plane. One  [#permalink]

### Show Tags

27 Mar 2007, 18:54
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

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.
Senior Manager
Joined: 11 Feb 2007
Posts: 335

### Show Tags

27 Mar 2007, 21:01

I tried using 45-45-90 but the vertex coordinate would be (5sqrt(2), 5sqrt(2)) not integers..

I tried using 30-60-90 but the vertex coordinate would be (5,5sqrt(3)) not integers...

Unless two sides of the square are the x and y axis...hmm....
Senior Manager
Joined: 18 Jul 2006
Posts: 482

### Show Tags

Updated on: 28 Mar 2007, 13:31

Should be A.
Attachments

d1.doc [27.5 KiB]

Originally posted by Juaz on 27 Mar 2007, 22:05.
Last edited by Juaz on 28 Mar 2007, 13:31, edited 6 times in total.
Director
Joined: 14 Jan 2007
Posts: 734

### Show Tags

Updated on: 31 Mar 2007, 10:22
******deleted****see my post below...

Originally posted by vshaunak@gmail.com on 28 Mar 2007, 02:35.
Last edited by vshaunak@gmail.com on 31 Mar 2007, 10:22, edited 1 time in total.
Senior Manager
Joined: 20 Feb 2007
Posts: 253

### Show Tags

28 Mar 2007, 04:19

Fig is the master of coordinate geometry...Fig!!! please save us

Director
Joined: 14 Jan 2007
Posts: 734

### Show Tags

Updated on: 31 Mar 2007, 10:48
2
1
My approach is bit different.

Try to find the possible vertices,

1. Set1 - 4 squares, one in each quadrant. Vertices- (0,0) (0,10) (0,-10) (10,0) (-10,0)

2 Set2 - Rotate the set 1 of the squares in clockwise so that the the vertex of (10,0) becomes (8,6). There will be 4 squares with this orientation.

3 Set3 -Rotate the set1 squares further so that the the vertex of (10,0) becomes (6,8).There will be 4 squares with this orientation.

Hence there will be 4+4+4 = 12 squares.
It's hard to imagine that way without putting it diagrammatically.
Attached is the diagram. It;s not accurate but will certainly help you visualizing the squares orientation.
Attachments

square problem.doc [29 KiB]

Originally posted by vshaunak@gmail.com on 28 Mar 2007, 13:22.
Last edited by vshaunak@gmail.com on 31 Mar 2007, 10:48, edited 1 time in total.
Manager
Joined: 02 Jan 2007
Posts: 206

### Show Tags

28 Mar 2007, 14:40
E for me.
12 ways to draw the square.

4 like you all say plus another because
when the side of the square which willbe drawn diagonally such that co-ordinate x = 6 and y=8, forming a right triangle.

Plus another 4 when x & y interchange values
ie
when x=8 & y=10,
the side of the square will be 10.
SVP
Joined: 01 May 2006
Posts: 1775

### Show Tags

Updated on: 29 Mar 2007, 00:28
1
(E) for me ... Is it really a GMAT question

First of all, the square pocesses 4 sides of 10.

Let set:
o Point 1 : (x1,y1)
o Point 2 : (x2,y2) : diagonally opposed to the vertice on 0(0,0)
o Point 3 : (x3,y3)

Then, we have:
o x1^2+y1^2 = 100
o x3^2+y3^2 = 100
o x2^2+y2^2 = 200

Also, notice that y1/x1, y2/x2 and y3/x3 are the slopes of the equation passing by 0(0,0) and respectively the point 1, 2 and 3.

By the property of a square, we know that the sides 01 and 03 are perpendicular. Thus, we have:
o y1/x1 * y3/x3 = -1

Also, by properties of vectors, notice that:
o x2 = x1+x3
o y2 = y1+y3

But how could we obtain 100 with 2 integers?
o 8^2 + 6^2 = 100
o 10^2 + 0^2 = 100

Also, there are symetries that recreate similar squares by coordonates of integers in other quadrans.

To solve this issue, we should consider only 1 cadran and we should not exchange values of x1 and y1 : x1 to be equal to y1 and y1 to be equal x1 Indeed, it recreates only another point 3.

Arbitrarly, I choose the cadran 1.

How many square are done from x1 > 0 and y1 > 0 (Cadran 1)?
o If x1 = 8 and y1 = 6, then:
> x3=-6 and y3=8 and x2 = 2 and y2 = 14
or
> x3=6 and y3=-8 and x2 = 14 and y2 = -2

>>>> 2

Then, how many square possesses 2 sides : one on X and one on Y axes?
>>>> 4 : 1 by cadran.

Finally,

we have : 4*2 + 4 = 12

Originally posted by Fig on 28 Mar 2007, 14:50.
Last edited by Fig on 29 Mar 2007, 00:28, edited 1 time in total.
SVP
Joined: 01 May 2006
Posts: 1775

### Show Tags

Updated on: 28 Mar 2007, 14:55
Summer3 wrote:

Fig is the master of coordinate geometry...Fig!!! please save us

.... A bit overstated :p

Originally posted by Fig on 28 Mar 2007, 14:54.
Last edited by Fig on 28 Mar 2007, 14:55, edited 1 time in total.
GMAT Instructor
Joined: 04 Jul 2006
Posts: 1251
Re: how many squares in a plane  [#permalink]

### Show Tags

28 Mar 2007, 14:55
amd08 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

Such a square must have a point P(x,y) that is sqrt(200) from the origin
i.e. x^2+y^2=200 where x and y are integers

|x|=sqrt(200-y^2)

|y| could be 2,10,14

So, I guess there will be 12 such squares

Let's confirm:

Let's look at the squares in an organized way. Let P(x,y) be the vertex of the side OP we would encounter if we started at the y axis and looked clockwise. How many such pairs (x,y) are there? |x|=sqrt(100-y^2), where x and y are both integers, so we can make a chart:

|y| ..... |x|
0..........10
6...........8
8...........6
10.........0

So P(x,y) must be one of {(0,10),(6,8),(8,6),(10,0), (8,-6), (6,-8),(0,-10), (-6,-8),(-8,-6),(-10,0),(-8,6),(-6,-8)} Thus there are 12 squares, confirming the simplier method above
Senior Manager
Joined: 20 Feb 2007
Posts: 253

### Show Tags

28 Mar 2007, 15:29

No Fig, I did not overstate

Could you please give the figures from your favorite webpage showing these squares?

Thanks
Manager
Joined: 22 Feb 2007
Posts: 63

### Show Tags

28 Mar 2007, 16:47
indeed its E. Thanks!
SVP
Joined: 01 May 2006
Posts: 1775

### Show Tags

Updated on: 29 Mar 2007, 03:17
At least, I can try

I have represented the point that I epressed from 1 (x1,y1) in the cadran 1 and I have added the special square with 1 and 3 on X & Y axes to give a point 2 in cadran 1.

As u can see, by mirroring with X and Y axes of the figure, we obtain the other squares in cadran II, III and IV.

And also, to avoid confusion, we should not flip X1 > Y1 and Y1 > X1 because we recreate a point 3 and so a square already considered.
Attachments

Fig1_Sqaure for (x1,y1) in cardan 1.gif [ 9.29 KiB | Viewed 1796 times ]

Originally posted by Fig on 29 Mar 2007, 01:02.
Last edited by Fig on 29 Mar 2007, 03:17, edited 1 time in total.
Senior Manager
Joined: 20 Feb 2007
Posts: 253

### Show Tags

29 Mar 2007, 01:49
wow!! nice Fig!!!

Thanks!
Manager
Joined: 09 Feb 2007
Posts: 77

### Show Tags

29 Mar 2007, 06:11
Fig's explanation is good. There is also a visual way to do this:

First, if one vertex is on integer coordinates, then all of them will be. You can draw a picture and use geometric similarities to see this. Thus, the problem reduces to rotating a vertex adjacent to the origin in a circle, and counting the times it passes through an integer coordinate. This then, will give (going counterclockwise) (10,0), (8,6), (6,8), (0,10), (-6,8), ... (8,-6) for a total of 12.
CEO
Joined: 20 Nov 2005
Posts: 2774
Schools: Completed at SAID BUSINESS SCHOOL, OXFORD - Class of 2008

### Show Tags

29 Mar 2007, 08:59
I think GMAT doesn't ask for the whole solution but want you to find the trick.

Trick is to find points on circle (origin 0,0 and radius 10).
Equation of circle
x^2 + y^2 = 10

how many times both x and y will be integers?
only when |x| and |y| (6,8) and (0,10)
so there are 12 points
(0,10), (10,0) (0,-10) (-10,0)
(6,8) (8,6) (8,-6) (6,-8) (-6,-8) (-8,-6) (-8,6) (-6,8)

Ans : 12
Director
Joined: 17 Sep 2005
Posts: 822

### Show Tags

15 Sep 2007, 05:28
ps_dahiya wrote:
I think GMAT doesn't ask for the whole solution but want you to find the trick.

Trick is to find points on circle (origin 0,0 and radius 10).
Equation of circle
x^2 + y^2 = 10

how many times both x and y will be integers?
only when |x| and |y| (6,8) and (0,10)
so there are 12 points
(0,10), (10,0) (0,-10) (-10,0)
(6,8) (8,6) (8,-6) (6,-8) (-6,-8) (-8,-6) (-8,6) (-6,8)

Ans : 12

Small correction,

Equation of Circle:

x^2 + y^2 = r^2 = 10^2 = 100 (If center of the circle is at origin)

However, indeed a good approach.

- Brajesh
Non-Human User
Joined: 09 Sep 2013
Posts: 9420
Re: A certain square is to be drawn on a coordinate plane. One  [#permalink]

### Show Tags

20 Jan 2018, 00:05
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.

_________________
Re: A certain square is to be drawn on a coordinate plane. One &nbs [#permalink] 20 Jan 2018, 00:05
Display posts from previous: Sort by