It is currently 21 Oct 2017, 11:26

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

A certain square is to be drawn on a coordinate plane

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

6 KUDOS received
Manager
Manager
avatar
Joined: 30 May 2010
Posts: 190

Kudos [?]: 244 [6], given: 32

A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 14 Sep 2010, 22:06
6
This post received
KUDOS
74
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

35% (00:48) correct 65% (00:55) wrong based on 601 sessions

HideShow timer Statistics

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


[Reveal] Spoiler:
I'll post the official explanation, but it doesn't make sense to me :)

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.

The correct answer is E.
[Reveal] Spoiler: OA

Last edited by Bunuel on 23 Mar 2016, 10:57, edited 2 times in total.
Edited the question

Kudos [?]: 244 [6], given: 32

Expert Post
29 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 41894

Kudos [?]: 129131 [29], given: 12194

Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 14 Sep 2010, 22:16
29
This post received
KUDOS
Expert's post
31
This post was
BOOKMARKED
jpr200012 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


This question becomes much easier if you visualize/draw it.

Let the origin be O and one of the vertices be A. Now, we are told that length of OA must be 10 (area to be 100). So if the coordinates of A is (x, y) then we would have \(x^2+y^2=100\) (distance from the origin to the point A(x, y) can be found by the formula \(d^2=x^2+y^2\))

Now, \(x^2+y^2=100\) has several integer solutions for \(x\) and \(y\), so several positions of vertex A, note that when vertex A has integer coordinates other vertices also have integer coordinates. For example imagine the case when square rests on X-axis to the right of Y-axis, then the vertices are: A(10,0), (10,10), (0,10) and (0,0).

Also you can notice that 100=6^2+8^2 and 100=0^2+10^2, so \(x\) can tale 7 values: -10, -8, -6, 0, 6, 8, 10. For \(x=-10\) and \(x=10\), \(y\) can take only 1 value 0, but for other values of \(x\), \(y\) can take two values positive or negative. For example: when \(x=6\) then \(y=8\) or \(y=-8\). This gives us 1+1+5*2=12 coordinates of point A:

\(x=10\) and \(y=0\), imagine this one to be the square which rests on X-axis and to get the other options rotate OA anticlockwise to get all possible cases;
\(x=8\) and \(y=6\);
\(x=6\) and \(y=8\);
\(x=0\) and \(y=10\);
\(x=-6\) and \(y=8\);
\(x=-8\) and \(y=6\);
\(x=-10\) and \(y=0\);
\(x=-8\) and \(y=-6\);
\(x=-6\) and \(y=-8\);
\(x=0\) and \(y=-10\);
\(x=6\) and \(y=-8\);
\(x=8\) and \(y=-6\).

Answer: E.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 129131 [29], given: 12194

1 KUDOS received
Senior Manager
Senior Manager
avatar
Status: GMAT Time...!!!
Joined: 03 Apr 2010
Posts: 292

Kudos [?]: 56 [1], given: 7

Schools: Chicago,Tuck,Oxford,cambridge
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 14 Sep 2010, 22:22
1
This post received
KUDOS
Bunuel wrote:
jpr200012 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


This question becomes much easier if you visualize/draw it.

Let the origin be O and one of the vertices be A. Now, we are told that length of OA must be 10 (area to be 100). So if the coordinates of A is (x, y) then we would have \(x^2+y^2=100\) (distance from the origin to the point A(x, y) can be found by the formula \(d^2=x^2+y^2\))

Now, \(x^2+y^2=100\) has several integer solutions for \(x\) and \(y\), so several positions of vertex A, note that when vertex A has integer coordinates other vertices also have integer coordinates. For example imagine the case when square rests on X-axis to the right of Y-axis, then the vertices are: A(10,0), (10,10), (0,10) and (0,0).

Also you can notice that 100=6^2+8^2 and 100=0^2+10^2, so \(x\) can tale 7 values: -10, -8, -6, 0, 6, 8, 10. For \(x=-10\) and \(x=10\), \(y\) can take only 1 value 0, but for other values of \(x\), \(y\) can take two values positive or negative. For example: when \(x=6\) then \(y=8\) or \(y=-8\). This gives us 1+1+5*2=12 coordinates of point A:

\(x=10\) and \(y=0\), imagine this one to be the square which rests on X-axis and to get the other options rotate OA anticlockwise to get all possible cases;
\(x=8\) and \(y=6\);
\(x=6\) and \(y=8\);
\(x=0\) and \(y=10\);
\(x=-6\) and \(y=8\);
\(x=-8\) and \(y=6\);
\(x=-10\) and \(y=0\);
\(x=-8\) and \(y=-6\);
\(x=-6\) and \(y=-8\);
\(x=0\) and \(y=-10\);
\(x=6\) and \(y=-8\);
\(x=8\) and \(y=-6\).

Answer: E.

hey Bunuel Thanx,but can u please come out with a Image,only 2 or 3 coordinate value drawn in it....i am sooooo confused....

Kudos [?]: 56 [1], given: 7

Expert Post
20 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 41894

Kudos [?]: 129131 [20], given: 12194

Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 14 Sep 2010, 23:13
20
This post received
KUDOS
Expert's post
3
This post was
BOOKMARKED
sandeep800 wrote:
hey Bunuel Thanx,but can u please come out with a Image,only 2 or 3 coordinate value drawn in it....i am sooooo confused....


All 12 squares.

Image posted on our forum by GMATGuruNY:
Attachment:
square.PNG
square.PNG [ 48.28 KiB | Viewed 37701 times ]

_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 129131 [20], given: 12194

Expert Post
1 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 41894

Kudos [?]: 129131 [1], given: 12194

Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 15 Sep 2010, 09:42
1
This post received
KUDOS
Expert's post
thirst4edu wrote:
Bunuel wrote:
sandeep800 wrote:
hey Bunuel Thanx,but can u please come out with a Image,only 2 or 3 coordinate value drawn in it....i am sooooo confused....


All 12 squares.

Image posted on our forum by GMATGuruNY:
Attachment:
square.PNG


Question says "One of the vertices must be on the origin", then why center of square is at origin of co-ordinate system? Shouldn't one of the vertices (the corner) of the square be at the origin?


Each diagram shows 4 squares not 1, so if you take first diagram you'll see 4 squares and each has one vertex at the origin.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 129131 [1], given: 12194

6 KUDOS received
CEO
CEO
User avatar
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2761

Kudos [?]: 1887 [6], given: 235

Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Reviews Badge
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 14 Oct 2010, 15:19
6
This post received
KUDOS
2
This post was
BOOKMARKED
Best way is to find for one quadrant and multiply by 4.

6,8 satisfy the point for the vertex of the square.

=> 8,6 will also satisfy => 2 squares per quadrant ---> if you are confused why this is true then draw the x-y axis and try to visualize what happens when x is replaced with y.
=> 4*2 = 8 squares

Now 10,0 also satisfy the point or the vertex.
but when we will replace x with y the same square is generated
=> 10,0 and 0,10 are part of same squares.
=> 1 per quadrant
=> 4*1 = 4 squares

total = 4+8 = 12 hence E
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned :)

Jo Bole So Nihaal , Sat Shri Akaal

:thanks Support GMAT Club by putting a GMAT Club badge on your blog/Facebook :thanks

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

Kudos [?]: 1887 [6], given: 235

1 KUDOS received
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 793

Kudos [?]: 1186 [1], given: 25

Location: London
GMAT ToolKit User Reviews Badge
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 14 Oct 2010, 15:27
1
This post received
KUDOS
I got this question on a MGMAT CAT as well, but I refuse to believe a question this hard can be on the real GMAT. It is not obvious at all how to solve this in a straight forward manner.
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

GMAT Club Premium Membership - big benefits and savings

Kudos [?]: 1186 [1], given: 25

1 KUDOS received
CEO
CEO
User avatar
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2761

Kudos [?]: 1887 [1], given: 235

Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Reviews Badge
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 14 Oct 2010, 16:13
1
This post received
KUDOS
shrouded1 wrote:
I got this question on a MGMAT CAT as well, but I refuse to believe a question this hard can be on the real GMAT. It is not obvious at all how to solve this in a straight forward manner.


I got this on my first Mgmat Cat as well. Most of the questions are time consuming in Mgmat cat's.
Have you seen similar level of mgmat cat in Gmat?
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned :)

Jo Bole So Nihaal , Sat Shri Akaal

:thanks Support GMAT Club by putting a GMAT Club badge on your blog/Facebook :thanks

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

Kudos [?]: 1887 [1], given: 235

Expert Post
1 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 41894

Kudos [?]: 129131 [1], given: 12194

Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 22 Dec 2010, 14:27
1
This post received
KUDOS
Expert's post
Merging similar topics.

TheBirla wrote:
Great approach nookway, but in the 2nd part of your your solution, you are assuming the 4rth vertex is also an integer. And though your assumption is correct, i.e. the 4rth vertex is (14,2) , (2,14) and so on and so forth, I am not sure if this is a 2 minute problem and i got this in one of the mock CAT's that i was doing. Is this the level of problems one has to expect if you are aiming for a 750 + ? Thanks.


As for your question I doubt that this is a realistic GMAT question. Though if you find that # of squares should be multiple of 4 you'll be left with A, C and E choices right away. Next, you can also rule out A as at least 2 squares per quadrant can be easily found and then make an educated guess for E thus "solving" in less than 2 minutes. Refer for complete solution to the posts above.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 129131 [1], given: 12194

Expert Post
3 KUDOS received
Veritas Prep GMAT Instructor
User avatar
G
Joined: 16 Oct 2010
Posts: 7676

Kudos [?]: 17380 [3], given: 232

Location: Pune, India
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 22 Dec 2010, 14:40
3
This post received
KUDOS
Expert's post
1
This post was
BOOKMARKED
TheBirla wrote:
Great approach nookway, but in the 2nd part of your your solution, you are assuming the 4rth vertex is also an integer. And though your assumption is correct, i.e. the 4rth vertex is (14,2) , (2,14) and so on and so forth, I am not sure if this is a 2 minute problem and i got this in one of the mock CAT's that i was doing. Is this the level of problems one has to expect if you are aiming for a 750 + ? Thanks.


The reason the other vertex will be integral is that square is a symmetrical figure. I have explained this in the following post:
http://gmatclub.com/forum/coordinate-plane-90772.html#p807400

So you don't need to find the 4th vertex and hence don't need to spend that time. You just need to figure out the integral values of x and y such that x^2 + y^2 = 100 which is quite straight forward.
Take x = 0. y = 10 satisfies.
Now check for x = 1/2/3 etc which will take just a few secs each. You will see that x = 6 and y = 8 satisfies.

x can be 0/10/6/8, y will be 10/0/8/6 or -10/-8/-6.
Taking negative sign of x, you will get: x = -10/-6/-8 and y will be 0/8/6 or -8/-6.

Total 12 such squares.

And yes, it is one of the tougher questions, definitely above 700.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199

Veritas Prep Reviews

Kudos [?]: 17380 [3], given: 232

Intern
Intern
avatar
Joined: 23 Oct 2010
Posts: 31

Kudos [?]: [0], given: 1

Location: London
WE 1: Consulting - 1.5 Yrs
WE 2: IB Finance - 5 Yrs
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 22 Dec 2010, 16:03
Thanks a lot Bunuel and Karishma.

Karishma, took me a while to get my head around the solution (symmetry of squares), but once i did its just given me a different perspective for these sort of problems. Great approach ! And thanks once again.

Kudos [?]: [0], given: 1

Senior Manager
Senior Manager
avatar
Joined: 06 Aug 2011
Posts: 390

Kudos [?]: 235 [0], given: 82

Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 01 Nov 2012, 11:51
how can 8 and 6 be X and Y.. when we multiply both we r geting 48 ..bt ans should be 100..

m not geting how can 6 8 and 0 be the x and y value :(
_________________

Bole So Nehal.. Sat Siri Akal.. Waheguru ji help me to get 700+ score !

Kudos [?]: 235 [0], given: 82

3 KUDOS received
Senior Manager
Senior Manager
User avatar
Joined: 13 Aug 2012
Posts: 458

Kudos [?]: 541 [3], given: 11

Concentration: Marketing, Finance
GPA: 3.23
GMAT ToolKit User
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 25 Jan 2013, 07:32
3
This post received
KUDOS
3
This post was
BOOKMARKED
jpr200012 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



To construct a square, we think of one of its side which is a line from (0,0) to some (x,y).
Since the area of the square is 100, its side will have a side = 10.
We can use the distance formula: \(d^2 = x^2 + y^2\) Thus, \(100 = x^2 + y^2\)

Let's think of combinations of perfectly squared x and y that adds up to 100.
{0,10} and {6,8} The best way to think of these combinations is to list the perfect squares and experiment on the combinations that adds up to 100.

Now {0,10}, Both numbers could be x,y or reversed and 10 can be negative or positive. Thus, we already have \(2*2 = 4\) points.
Now {6,8}, Both numbers could be x,y or reversed and 6 and 8 could be negative or positive. Thus, we have \(2*2*2 = 8\)points

And your possible points that form a distance of 10 from the (0,0)... \(4+8 = 12\)

Answer: 12
_________________

Impossible is nothing to God.

Kudos [?]: 541 [3], given: 11

1 KUDOS received
Intern
Intern
avatar
Joined: 27 Mar 2013
Posts: 1

Kudos [?]: 1 [1], given: 1

Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 27 Mar 2013, 23:06
1
This post received
KUDOS
Perimeter of a circle of radius 10 centered at the origin will coincide with 12 points where where both the x and y values are integers. (10, 0); (8, 6); (6, 8); (0, 10); (-8, 6), (-6, 8); (-10, 0); (-8, -6); (-6, -8); (0, -10); (8, -6); (6, -8)

E. 12

Kudos [?]: 1 [1], given: 1

Manager
Manager
avatar
Joined: 23 Jan 2013
Posts: 172

Kudos [?]: 56 [0], given: 41

Concentration: Technology, Other
Schools: Haas
GMAT Date: 01-14-2015
WE: Information Technology (Computer Software)
GMAT ToolKit User
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 28 Apr 2013, 09:55
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?

Doesn't one of the vertices must be on the origin mean one vertices of the square has always to be (0,0) hence only 4 possibilities ???
Please clarify !!

Kudos [?]: 56 [0], given: 41

1 KUDOS received
Intern
Intern
avatar
Joined: 14 Feb 2013
Posts: 31

Kudos [?]: 60 [1], given: 14

Schools: Duke '16
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 28 Apr 2013, 10:39
1
This post received
KUDOS
Since area of the square is 100, each side = 10
One of the vertices of the square = (0,0)
Let the co-ordinates of another vertex of the square be (x,y)
Using the formula \(d^2 = x^2 + y^2\) (d = Distance from the origin to any point in the co-ordinate)
So, \(100 = x^2 + y^2\)
As the vertices, must be integers, solve for different values for x and y
When x = 0, y = 10
Also, x = 0, y = -10
x=10, y = 0
x = -10, y = 0
Also x = 6 , y = 8 (as \(100 = 6^2+8^2\))
x = 6, y = -8
x = -6, y= 8
x = -6, y = -8
Similarly, x = 8, y = 6
x = 8, y = -6
x = -8, y = 6
x = -8, y = -6

that is 12 possible values
_________________

Consider giving +1 Kudo :) when my post helps you.
Also, Good Questions deserve Kudos..!

Kudos [?]: 60 [1], given: 14

Intern
Intern
avatar
Joined: 28 Apr 2013
Posts: 1

Kudos [?]: [0], given: 40

Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 28 Apr 2013, 20:50
karishmatandon wrote:
Since area of the square is 100, each side = 10
One of the vertices of the square = (0,0)
Let the co-ordinates of another vertex of the square be (x,y)
Using the formula \(d^2 = x^2 + y^2\) (d = Distance from the origin to any point in the co-ordinate)
So, \(100 = x^2 + y^2\)
As the vertices, must be integers, solve for different values for x and y
When x = 0, y = 10
Also, x = 0, y = -10
x=10, y = 0
x = -10, y = 0
Also x = 6 , y = 8 (as \(100 = 6^2+8^2\))
x = 6, y = -8
x = -6, y= 8
x = -6, y = -8
Similarly, x = 8, y = 6
x = 8, y = -6
x = -8, y = 6
x = -8, y = -6

that is 12 possible values


This diagram posted earlier in the forum explains everything
Attachments

square.png
square.png [ 48.28 KiB | Viewed 4273 times ]

Kudos [?]: [0], given: 40

Manager
Manager
avatar
Joined: 24 Mar 2013
Posts: 59

Kudos [?]: 4 [0], given: 10

Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 29 Dec 2013, 02:16
Got this question wrong on the mgmat cat also.

What I still don't understand how you confirm that a square that has a hypotenuse as an edge running from 0,0 to 6,8, would also have vertices at integer co-ordinates 8,6, and 14,2 and not fractional co-ordinates. How do you reach that conclusion?

Kudos [?]: 4 [0], given: 10

1 KUDOS received
Current Student
User avatar
Joined: 06 Sep 2013
Posts: 1978

Kudos [?]: 719 [1], given: 355

Concentration: Finance
GMAT ToolKit User
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 10 Jan 2014, 07:40
1
This post received
KUDOS
jpr200012 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


[Reveal] Spoiler:
I'll post the official explanation, but it doesn't make sense to me :)

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.

The correct answer is E.


Think of this as a circle with radius 10 ok?

10 will be the hypothenuse of the triangle.

Now x^2 + y^2 = 100 since it is based in the origin as per the question.

Now since x and y must be integers we only have 2 sets of numbers that satisfy this

0 and 10 obviously and 6 and 8 (Think of pythagorean triples 3-4-5 only doubled)

Now, since we have 4 quadrants we are going to have a bunch of different combinations between (x,y) order and signs but shouldn't be too hard

10 and 0, since 0 does not have a sign then we have 0,10, 10,0, -10,0 and 0,-10 easy to count them out total of 4

Then, since 6 and 8 will have positive and negative signs then its better to use a combinatorics approach instead of counting

Two slots _ _

We have 2 options for the first (6 and 8), 1 option for the first (either 6 or 8). And then for each of them we have 2 possible signs (+ or -)

Then (2)(2)*(2)(1) = 2^3 = 8

Now add em up 8 + 4 = 12

E is the correct answer

Hope it helps
Gimme Kudos

Cheers
J :)

Kudos [?]: 719 [1], given: 355

Senior Manager
Senior Manager
avatar
Joined: 15 Aug 2013
Posts: 302

Kudos [?]: 82 [0], given: 23

Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

New post 10 May 2014, 13:11
Bunuel wrote:
jpr200012 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


This question becomes much easier if you visualize/draw it.

Let the origin be O and one of the vertices be A. Now, we are told that length of OA must be 10 (area to be 100). So if the coordinates of A is (x, y) then we would have \(x^2+y^2=100\) (distance from the origin to the point A(x, y) can be found by the formula \(d^2=x^2+y^2\))

Now, \(x^2+y^2=100\) has several integer solutions for \(x\) and \(y\), so several positions of vertex A, note that when vertex A has integer coordinates other vertices also have integer coordinates. For example imagine the case when square rests on X-axis to the right of Y-axis, then the vertices are: A(10,0), (10,10), (0,10) and (0,0).

Also you can notice that 100=6^2+8^2 and 100=0^2+10^2, so \(x\) can tale 7 values: -10, -8, -6, 0, 6, 8, 10. For \(x=-10\) and \(x=10\), \(y\) can take only 1 value 0, but for other values of \(x\), \(y\) can take two values positive or negative. For example: when \(x=6\) then \(y=8\) or \(y=-8\). This gives us 1+1+5*2=12 coordinates of point A:

\(x=10\) and \(y=0\), imagine this one to be the square which rests on X-axis and to get the other options rotate OA anticlockwise to get all possible cases;
\(x=8\) and \(y=6\);
\(x=6\) and \(y=8\);
\(x=0\) and \(y=10\);
\(x=-6\) and \(y=8\);
\(x=-8\) and \(y=6\);
\(x=-10\) and \(y=0\);
\(x=-8\) and \(y=-6\);
\(x=-6\) and \(y=-8\);
\(x=0\) and \(y=-10\);
\(x=6\) and \(y=-8\);
\(x=8\) and \(y=-6\).

Answer: E.


Hi Bunuel,

How did you come up with 8 & 6 being the viable options? I can obviously see it once you point it out but how did you come up with that in the first place?

Additionally, if the square had an area of 50 and we still had to maintain integer lengths, then our answer would be 4, correct?

Kudos [?]: 82 [0], given: 23

Re: A certain square is to be drawn on a coordinate plane   [#permalink] 10 May 2014, 13:11

Go to page    1   2    Next  [ 36 posts ] 

Display posts from previous: Sort by

A certain square is to be drawn on a coordinate plane

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


cron

GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.