• 1 Hour Free
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• Free Veritas GMAT Class
Experience Lesson 1 Live Free

Available with Beat the GMAT members only code

• Free Practice Test & Review
How would you score if you took the GMAT

Available with Beat the GMAT members only code

• Award-winning private GMAT tutoring
Register now and save up to \$200

Available with Beat the GMAT members only code

• Magoosh
Study with Magoosh GMAT prep

Available with Beat the GMAT members only code

• 5-Day Free Trial
5-day free, full-access trial TTP Quant

Available with Beat the GMAT members only code

• 5 Day FREE Trial
Study Smarter, Not Harder

Available with Beat the GMAT members only code

• Free Trial & Practice Exam
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• Get 300+ Practice Questions

Available with Beat the GMAT members only code

## Hard MGMAT 700-800 level question. Co-ordinate plane

This topic has 3 expert replies and 14 member replies
Goto page
• 1,
• 2
khurram Master | Next Rank: 500 Posts
Joined
07 Jan 2008
Posted:
231 messages
Followed by:
1 members
Thanked:
4 times

#### Hard MGMAT 700-800 level question. Co-ordinate plane

Mon Apr 14, 2008 6:37 pm
Elapsed Time: 00:00
• Lap #[LAPCOUNT] ([LAPTIME])
Thanks

Khurram

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
Each side of the square must have a length of 10. If each side were to be 6, 7, 8, or most other numbers, there could only be four possible squares drawn, because each side, in order to have integer coordinates, would have to be drawn on the x- or y-axis. What makes a length of 10 different is that it could be the hypotenuse of a Pythagorean triple, meaning the vertices could have integer coordinates without lying on the x- or y-axis.

For example, a square could be drawn with the coordinates (0,0), (6,8), (-2, 14) and (-8, 6). (It is tedious and unnecessary to figure out all four coordinates for each square).

If we label the square abcd, with a at the origin and the letters representing points in a clockwise direction, we can get the number of possible squares by figuring out the number of unique ways ab can be drawn.

a has coordinates (0,0) and b could have the following coordinates, as shown in the picture:

(-10,0)
(-8,6)
(-6,8)
(0,10)
(6,8)
(8,6)
(10,0)
(8, -6)
(6, -8)
(0, 10)
(-6, -8)
(-8, -6)

There are 12 different ways to draw ab, and so there are 12 ways to draw abcd.

Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!
aas550 Newbie | Next Rank: 10 Posts
Joined
26 Jul 2007
Posted:
9 messages
Thu Apr 17, 2008 11:11 am
Can someone explain this me in a different way?
What does smiley stand for?

Thanks

gabriel Legendary Member
Joined
20 Dec 2006
Posted:
986 messages
Followed by:
1 members
Thanked:
51 times
Thu Apr 17, 2008 11:25 am
aas550 wrote:
Can someone explain this me in a different way?
What does smiley stand for?

Thanks
The smiley stands for 8.

### GMAT/MBA Expert

Stuart Kovinsky GMAT Instructor
Joined
08 Jan 2008
Posted:
3225 messages
Followed by:
608 members
Thanked:
1710 times
GMAT Score:
800
Thu Apr 17, 2008 11:27 am
aas550 wrote:
Can someone explain this me in a different way?
What does smiley stand for?

Thanks
If you don't have smileys disabled (box you can click under the message that you type), if you ever type 8 followed by a closed bracket you'll get 8)

Free GMAT Practice Test under Proctored Conditions! - Find a practice test near you or live and online in Kaplan's Classroom Anywhere environment. Register today!
netigen Legendary Member
Joined
18 Feb 2008
Posted:
631 messages
Followed by:
3 members
Thanked:
29 times
Thu Apr 17, 2008 11:30 am

1. There are 4 quadrants we need to draw the squares in so if you find the number of square vertices possible in 1 quadrants , you can just multiple that with 4 to get the answer

2. The area of the sq = 100 so if a side is s then s^2 = 100 and for this to be possible s =10

3. Rephrasing the problem:

to get s = 10 the co-ordinates of each side of the square should be at a distance of 10 from (0,0)

lets say the vertices of sq are at (x,y), using the distance formula

x^2 + y^2 = 10^2

Since we know both x,y are integers the only possible values for x,y in quadrants I are

(0,10), (10,0) (8,6) (6,8)

Out of these possibilities (0,10) and (10,0) lie on the same sq so it means we can have 3 unique possibilities

so total = 4*3 = 12

Thanked by: shawndx
mberkowitz Senior | Next Rank: 100 Posts
Joined
20 Jul 2008
Posted:
76 messages
Thanked:
1 times
Tue Sep 09, 2008 1:44 pm

Montreal06 Senior | Next Rank: 100 Posts
Joined
27 Jun 2008
Posted:
30 messages
Test Date:
October 2008
Target GMAT Score:
720
Wed Oct 01, 2008 2:15 am
Hi,

to get s = 10 the co-ordinates of each side of the square should be at a distance of 10 from (0,0)

lets say the vertices of sq are at (x,y), using the distance formula

x^2 + y^2 = 10^2

Since we know both x,y are integers the only possible values for x,y in quadrants I are

(0,10), (10,0) (8,6) (6,8 )

If the coordinate needs to be 10 units away from (0,0), how does the vertice (8,6) or (6,8 ) come into play? i.e. 8^2*6^2 = 10^2

but how does the pythagorean principle come into play if we're just talking sides of the square?

thanks,
r

mjjking Master | Next Rank: 500 Posts
Joined
20 Jan 2007
Posted:
353 messages
Thanked:
7 times
GMAT Score:
720
Sat Feb 28, 2009 11:39 pm
can somebody explain how and why we find 8,6 and 6,8?

_________________
Beat The GMAT - 1st priority
Enter a top MBA program - 2nd priority
Loving my wife: MOST IMPORTANT OF ALL!

REAL THING 1 (AUG 2007): 680 (Q43, V40)
REAL THING 2 (APR 2009): 720 (Q47, V41)

bluementor Master | Next Rank: 500 Posts
Joined
11 Jun 2008
Posted:
418 messages
Thanked:
65 times
Tue Mar 03, 2009 5:24 am
mjjking wrote:
can somebody explain how and why we find 8,6 and 6,8?
I think its best to use a diagram to illustrate this.

We know that each of the sides of the square is 10. If you drew a line from the origin to point A(10, 0), you have a line of length 10. You can form a (red) square using this line as the basis.

Now, move point A anti-clock wise (with the other end of the line fixed to the origin).Notice the right-angled triangle that is formed with OA' and the x-axis. With the hypoteneus equal to 10, the only possible integer combinations for the height and base of this triangle is 6 and 8 (remember the 6-8-10 triangle). With OA', you have height = 6, base =8, and you will be able to form a (blue) square.

You can also move point A' to A'' and construct a triangle with OA'' and the x-axis. The other combination for the height and base is 8 and 6, respectively. And with this you have another (green) square.

If you repeat this for the other quadrants, you will have a total of 3x4 squares = 12 squares.

-BM-
Attachments

gmatapril Senior | Next Rank: 100 Posts
Joined
28 Jan 2011
Posted:
50 messages
Mon Feb 28, 2011 4:53 pm
why we have to take points 6, 8 .( we know that side of square is 10 )please correct me

bluementor wrote:
mjjking wrote:
can somebody explain how and why we find 8,6 and 6,8?
I think its best to use a diagram to illustrate this.

We know that each of the sides of the square is 10. If you drew a line from the origin to point A(10, 0), you have a line of length 10. You can form a (red) square using this line as the basis.

Now, move point A anti-clock wise (with the other end of the line fixed to the origin).Notice the right-angled triangle that is formed with OA' and the x-axis. With the hypoteneus equal to 10, the only possible integer combinations for the height and base of this triangle is 6 and 8 (remember the 6-8-10 triangle). With OA', you have height = 6, base =8, and you will be able to form a (blue) square.

You can also move point A' to A'' and construct a triangle with OA'' and the x-axis. The other combination for the height and base is 8 and 6, respectively. And with this you have another (green) square.

If you repeat this for the other quadrants, you will have a total of 3x4 squares = 12 squares.

-BM-

jerryragland Senior | Next Rank: 100 Posts
Joined
24 Nov 2009
Posted:
58 messages
Thanked:
1 times
Test Date:
05-22-2010
Target GMAT Score:
740
Mon May 02, 2011 8:30 am
unbelievable.. this is a 2 minute question?? comon.. MGMAT

### GMAT/MBA Expert

Whitney Garner GMAT Instructor
Joined
21 Sep 2010
Posted:
273 messages
Followed by:
73 members
Thanked:
154 times
GMAT Score:
770
Mon May 02, 2011 8:56 am
khurram 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
A couple of notes on this one. I think everyone is comfortable with the fact that the square must have sides of length 10, so I will ignore that.

Good Guessing Strategy
- I can quickly sketch the 4 "easy" squares that sit on the axes, so 4 feels too easy - eliminate A.
- I can also see that anything I can draw in one coordinate will be mirrored in the other 3, so the answer must be a multiple of 4 - eliminate B and D.
- This just leaves C and E - not a bad guess if stuck (and that would only have taken a minute or so to do).

Actually Solving
The problem gives us a square in a coordinate plane so step 1 should be to sketch something. I would draw at least one of the "easy" squares (the ones that sit on the axes), but then start to think about rotating it. The side length 10 becomes a distance from the origin = This should be sending off a beacon for you to use Pythagorean Theorem rules. Add to that the fact that the vertices have to be integers = which right triangles have integer side lengths?? AH thats right, the Pythagorean triples.

We should have committed the first few iterations of the 3-4-5 triangle to memory, so hypotenuse 10 would be the 6-8-10 triangle. We just need to use the coordinates (6,8) or (8,6) to see if our hunch is correct. Draw one (see image attached).

Now, I chose to make the x-axis length 8 (using the coordinate (8,6) as the vertices), but I could have just as easily picked (6,8). So that means there are 2 of these "weird" squares in each quadrant - so 8 of those when I mirror them around. And then the 4 "easy" squares. Total = 12.

Whit

_________________
Whitney Garner
GMAT Instructor & Instructor Developer
Manhattan Prep

Contributor to Beat The GMAT!

Math is a lot like love - a simple idea that can easily get complicated

Free Manhattan Prep online events - The first class of every online Manhattan Prep course is free. Classes start every week.

### GMAT/MBA Expert

GMATGuruNY GMAT Instructor
Joined
25 May 2010
Posted:
13356 messages
Followed by:
1779 members
Thanked:
12880 times
GMAT Score:
790
Mon May 02, 2011 2:12 pm
Quote:
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
Step 1: If area = 100, side = 10.
Step 2: Recognize that the hypotenuse of a 6-8-10 triangle is 10.
Step 3: Plot coordinate pairs using every possible combination of (±6,±8), (±8,±6),(0,±10) and (±10,0).
Step 4: Using the plotted points, draw sets of squares centered about the origin. The following sets are possible:

Number of possible squares = 12.

_________________
Mitch Hunt
GMAT Private Tutor
GMATGuruNY@gmail.com
If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
Available for tutoring in NYC and long-distance.

Thanked by: eaakbari, mevicks
Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.
sarang_gmt11 Newbie | Next Rank: 10 Posts
Joined
02 Jul 2011
Posted:
4 messages
Mon Sep 26, 2011 8:35 pm
GMATGuruNY wrote:
Quote:
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
Step 1: If area = 100, side = 10.
Step 2: Recognize that the hypotenuse of a 6-8-10 triangle is 10.
Step 3: Plot coordinate pairs using every possible combination of (±6,±8), (±8,±6),(0,±10) and (±10,0).
Step 4: Using the plotted points, draw sets of squares centered about the origin. The following sets are possible:

Number of possible squares = 12.

Hi ,
In the above diagram you have shown the centre of the square on the origin but in the problem it is mentioned that one of the vertices must be on the origin ?

imhimanshu Senior | Next Rank: 100 Posts
Joined
28 Feb 2009
Posted:
87 messages
Thanked:
2 times
Test Date:
Not Decided
Target GMAT Score:
750
Tue Nov 29, 2011 5:13 am
Sorry to bring up the old post..but somehow I am not able to get the solution.. request you to please clarify my below doubts -

1- I understand that we are employing Pythagorean triplets, but I am not been able to follow how that triplets will help me in solving/understanding that all other points will be integer.

### Best Conversation Starters

1 Vincen 180 topics
2 lheiannie07 61 topics
3 Roland2rule 54 topics
4 ardz24 44 topics
5 VJesus12 14 topics
See More Top Beat The GMAT Members...

### Most Active Experts

1 Brent@GMATPrepNow

GMAT Prep Now Teacher

155 posts
2 Rich.C@EMPOWERgma...

EMPOWERgmat

105 posts
3 GMATGuruNY

The Princeton Review Teacher

101 posts
4 Jay@ManhattanReview

Manhattan Review

82 posts
5 Matt@VeritasPrep

Veritas Prep

80 posts
See More Top Beat The GMAT Experts