Welcome! Check out our free B-School Guides to learn how you compare with other applicants.
Login or Register
 

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

This topic has 3 expert replies and 14 member replies
Goto page
  • 1,
  • 2
Next
Bookmark
Previous Topic Next Topic
khurram Really wants to Beat The GMAT! Default Avatar
Joined
07 Jan 2008
Posted:
231 messages
Followed by:
1 members
Thanked:
3 times
Hard MGMAT 700-800 level question. Co-ordinate plane Post Mon Apr 14, 2008 7:37 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    Thanks
    I had 4 as answer. Can someone please elaborate on the answer explanation.

    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.

    The correct answer is E.

    Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!
    aas550 Just gettin' started! Default Avatar
    Joined
    26 Jul 2007
    Posted:
    9 messages
    Post Thu Apr 17, 2008 12:11 pm
    Can someone explain this me in a different way?
    What does smiley stand for?

    Thanks

    gabriel GMAT Destroyer!
    Joined
    20 Dec 2006
    Posted:
    986 messages
    Followed by:
    1 members
    Thanked:
    49 times
    Post Thu Apr 17, 2008 12:25 pm
    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:
    3156 messages
    Followed by:
    461 members
    Thanked:
    1433 times
    GMAT Score:
    800
    Post Thu Apr 17, 2008 12:27 pm
    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)

    _________________
    Stuart Kovinsky, B.A. LL.B.
    Kaplan Test Prep & Admissions
    Toronto Office
    1-800-KAP-TEST
    www.kaptest.com

    Learn more about me

    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 GMAT Destroyer! Default Avatar
    Joined
    19 Feb 2008
    Posted:
    631 messages
    Followed by:
    2 members
    Thanked:
    29 times
    Post Thu Apr 17, 2008 12:30 pm
    An alternative way to think about this:

    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 Rising GMAT Star Default Avatar
    Joined
    20 Jul 2008
    Posted:
    76 messages
    Thanked:
    1 times
    Post Tue Sep 09, 2008 2:44 pm
    very helpful thanks very much

    Montreal06 Rising GMAT Star Default Avatar
    Joined
    27 Jun 2008
    Posted:
    30 messages
    Test Date:
    October 2008
    Target GMAT Score:
    720
    Post Wed Oct 01, 2008 3:15 am
    Hi,

    I'm a bit confused about this part:

    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 GMAT Destroyer! Default Avatar
    Joined
    20 Jan 2007
    Posted:
    353 messages
    Thanked:
    7 times
    GMAT Score:
    720
    Post Sun Mar 01, 2009 12:39 am
    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 GMAT Destroyer! Default Avatar
    Joined
    11 Jun 2008
    Posted:
    418 messages
    Thanked:
    58 times
    Post Tue Mar 03, 2009 6:24 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

    This post contains an attachment. You must be logged in to download/view this file. Please login or register as a user.

    Thanked by: jzebra10
    gmatapril Rising GMAT Star Default Avatar
    Joined
    28 Jan 2011
    Posted:
    50 messages
    Post Mon Feb 28, 2011 5:53 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 Rising GMAT Star Default Avatar
    Joined
    24 Nov 2009
    Posted:
    58 messages
    Thanked:
    1 times
    Test Date:
    05-22-2010
    Target GMAT Score:
    740
    Post Mon May 02, 2011 9:30 am
    unbelievable.. this is a 2 minute question?? comon.. MGMAT

    GMAT/MBA Expert

    Whitney Garner GMAT Instructor
    Joined
    21 Sep 2010
    Posted:
    266 messages
    Followed by:
    65 members
    Thanked:
    132 times
    GMAT Score:
    770
    Most Thanked Expert
    Post Mon May 02, 2011 9:56 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.

    Smile
    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 Smile

    Thanked by: aseemdabur, rahulvsd, majidelzein, shawndx
    Free Manhattan GMAT online events - The first class of every online Manhattan GMAT course is free. Classes start every week.

    GMAT/MBA Expert

    GMATGuruNY GMAT Instructor
    Joined
    25 May 2010
    Posted:
    6110 messages
    Followed by:
    1043 members
    Thanked:
    5277 times
    GMAT Score:
    790
    Most Active Expert Most Responsive Expert Most Thanked Expert
    Post Mon May 02, 2011 3:12 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.

    The correct answer is E.

    _________________
    Mitch Hunt
    GMAT Private Tutor and Instructor
    GMATGuruNY@gmail.com
    If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
    Contact me about long distance tutoring!

    Thanked by: eaakbari
    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 Just gettin' started! Default Avatar
    Joined
    02 Jul 2011
    Posted:
    4 messages
    Post Mon Sep 26, 2011 9:35 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.

    The correct answer is E.
    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 Rising GMAT Star Default Avatar
    Joined
    28 Feb 2009
    Posted:
    87 messages
    Thanked:
    2 times
    Test Date:
    Not Decided
    Target GMAT Score:
    750
    Post Tue Nov 29, 2011 6:13 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.
    Please explain

    Goto page
    • 1,
    • 2
    Next

    Best Conversation Starters
    1 varun289 31 topics
    2 sana.noor 23 topics
    3 killerdrummer 21 topics
    4 Rudy414 19 topics
    5 sanaa.rizwan 14 topics
    See More Top Beat The GMAT Members...

    Most Active Experts
    1 image description Brent@GMATPrepNow

    GMAT Prep Now Teacher

    		 199 posts
    2 image description GMATGuruNY

    The Princeton Review Teacher

    		 134 posts
    3 image description Jim@StratusPrep

    Stratus Prep

    		 106 posts
    4 image description Anju@Gurome

    Gurome

    		 47 posts
    5 image description Jon@Admissionado

    Admissionado

    		 39 posts
    See More Top Beat The GMAT Experts