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

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

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01 Nov 2012, 08:17
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sanjoo 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

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

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

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27 Mar 2013, 22:40
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
Need help
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27 Mar 2013, 23:06
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)

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

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28 Mar 2013, 03:04
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mun23 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
Need help

Responding to a pm:
Discussed here: a-certain-square-is-to-be-drawn-on-a-coordinate-plane-127018.html?hilit=square%20vertices%20origin%20integer
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Status: struggling with GMAT Joined: 06 Dec 2012 Posts: 228 Location: Bangladesh Concentration: Accounting GMAT Date: 04-06-2013 GPA: 3.65 Followers: 14 Kudos [?]: 264 [0], given: 46 Re: A certain square is to be drawn on a coordinate plane [#permalink] ### Show Tags 28 Mar 2013, 04:30 I am not understanding how the answer is E .I thought the the answer is A. how 8,6 occurs.........need help.Finding this math difficult for me Manager Joined: 23 Jan 2013 Posts: 173 Concentration: Technology, Other Schools: Haas GMAT Date: 01-14-2015 WE: Information Technology (Computer Software) Followers: 3 Kudos [?]: 38 [0], given: 41 Re: A certain square is to be drawn on a coordinate plane [#permalink] ### Show Tags 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 !! Intern Joined: 28 Apr 2013 Posts: 1 Followers: 0 Kudos [?]: 0 [0], given: 40 Re: A certain square is to be drawn on a coordinate plane [#permalink] ### Show Tags 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 [ 48.28 KiB | Viewed 1774 times ] Manager Joined: 24 Mar 2013 Posts: 63 Followers: 0 Kudos [?]: 1 [0], given: 10 Re: A certain square is to be drawn on a coordinate plane [#permalink] ### Show Tags 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? Senior Manager Joined: 15 Aug 2013 Posts: 328 Followers: 0 Kudos [?]: 38 [0], given: 23 Re: A certain square is to be drawn on a coordinate plane [#permalink] ### Show Tags 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? Senior Manager Joined: 28 Apr 2014 Posts: 291 Followers: 1 Kudos [?]: 31 [0], given: 46 Re: A certain square is to be drawn on a coordinate plane [#permalink] ### Show Tags 11 May 2014, 21:56 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 Bunuel I have a doubt in the figure. The question says that one of the vertices must be origin but in the figure it shows the centre of the square at origin. Isn't this a fallacy ? In other words one of the vertex of the circle will always have to be (0,0) . Now rotating along this point and considering any one quadrant at a time , we can say distance of any adjacent vertex ( x,y) must be 10 units. So x^2 + y^2 = 100. Given the constraint of co-ordinates being integers , we see that 8,6 and 6,8 satisfy this . So considering quadrant one only two vertex are possible i.e. (6,8) and (8,6) . Thus 2 squares are possible in quad 1. For four quadrants the possibilities are 4* 2 = 8. Now squares can be also be formed along the x-y axis . They would be 4 in number i.e. 1 in each quadrant with two adjacents sides as x/y axis. This makes the total as 8+4 = 12. So although the same answer is coming but the figure in question is confusing. Is this the correct approach ? Last edited by himanshujovi on 12 May 2014, 03:33, edited 1 time in total. Senior Manager Joined: 28 Apr 2014 Posts: 291 Followers: 1 Kudos [?]: 31 [0], given: 46 Re: A certain square is to be drawn on a coordinate plane [#permalink] ### Show Tags 11 May 2014, 22:05 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 The figure seems to indicate that the area is 20*20 = 400 sq units. Math Expert Joined: 02 Sep 2009 Posts: 33562 Followers: 5948 Kudos [?]: 73840 [0], given: 9903 Re: A certain square is to be drawn on a coordinate plane [#permalink] ### Show Tags 12 May 2014, 03:18 Expert's post himanshujovi 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 The figure seems to indicate that the area is 20*20 = 400 sq units. Please read the whole thread: a-certain-square-is-to-be-drawn-on-a-coordinate-plane-127018.html#p782255 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. _________________ Intern Joined: 09 Mar 2015 Posts: 16 Concentration: General Management, Entrepreneurship GMAT 1: 660 Q47 V34 GPA: 1.2 WE: Project Management (Energy and Utilities) Followers: 0 Kudos [?]: 4 [0], given: 82 Re: A certain square is to be drawn on a coordinate plane [#permalink] ### Show Tags 29 May 2015, 12:34 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 I think more than 12 ways are possible. EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 6701 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: 340 Q170 V170 Followers: 289 Kudos [?]: 1973 [0], given: 161 Re: A certain square is to be drawn on a coordinate plane [#permalink] ### Show Tags 29 May 2015, 15:29 Expert's post Hi pawanCEO, This question includes a number of "restrictions" that you must follow: 1) You have to draw a SQUARE 2) One of the vertices MUST be at the ORIGIN (0, 0) 3) EVERY vertices MUST be an INTEGER 4) Since the area is 100, each side length MUST be 10 Given these restrictions, there are only 12 possible squares that can be drawn. You mentioned that you think that there are MORE than 12 possibilities.....if so, then why do you think that? Do you have any examples? GMAT assassins aren't born, they're made, Rich _________________ # Rich Cohen Co-Founder & GMAT Assassin # Special Offer: Save$75 + GMAT Club Tests

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

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07 Apr 2016, 17:09
Bunuel wrote:
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).

Hi Bunuel,

Could you help to explain why we can assure that whenever vertex A has integer coordinates other vertices also have integer coordinates? I mean do we have some theorem about this one or do we have some way to justify it?

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

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07 Apr 2016, 19:41
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Hi thuyduong91vnu,

If we only knew that we were drawing a square with one vertice at the Origin, then the other 3 vertices COULD be on non-integer co-ordinates. However, the original prompt STATES that all 3 vertices are on integer co-ordinates, so we have to use the 'restrictions' that the question places on us.

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

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08 Apr 2016, 06:13
Hi Rich,

Thanks for your response. I understand your explanation, but that is not my point though

My question is, let's say we have to calculate the number of squares OABC, whose each side has to equal 10 and the coordinates of all 4 vertices have to be integers. By determining possible combinations of x and y-coordinates of A vertice, we could find the questioned number, right? But, assume that we have found these combinations of x and y-coordinators of A vertice (like (8,6) or (-8,-6)..), then how can we assure that the remaining vertices B and C will also have integer coordinates?

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

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08 Apr 2016, 15:58
Hi chetan2u,

Thanks for your clarification. I think I got it now But, one more question, as I found a new concept here: "diagonally opposite". As you explained, after figuring out the slope of perpendicular line (which is $$\frac{-6}{8}$$), we could use such slope to get the coordinates of diagonally opposite point (which are x= -6 and y=8) by substituing $$\frac{x-0}{y-0}$$ for $$\frac{-6}{8}$$, right? But to do this, we should not reduce the fractional value of the slope, I mean we should not reduce $$\frac{−6}{8}$$ to $$\frac{-3}{4}$$? It is the way to find out coordinates of any diagonally opposite point?

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

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08 Apr 2016, 20:08
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thuyduong91vnu wrote:
Hi chetan2u,

Thanks for your clarification. I think I got it now But, one more question, as I found a new concept here: "diagonally opposite". As you explained, after figuring out the slope of perpendicular line (which is $$\frac{-6}{8}$$), we could use such slope to get the coordinates of diagonally opposite point (which are x= -6 and y=8) by substituing $$\frac{x-0}{y-0}$$ for $$\frac{-6}{8}$$, right? But to do this, we should not reduce the fractional value of the slope, I mean we should not reduce $$\frac{−6}{8}$$ to $$\frac{-3}{4}$$? It is the way to find out coordinates of any diagonally opposite point?

hi,
we did not reduce the ratio because the length of the line is the same..
Had it been a rectangle, the ratio could have changed..
Even if we reduce the ratio, we will still get the same answer

even for this example

x/y = -6/8=-3/4..
let the common ratio be a..
so x= -3a and y =4a...

the length of each side is 10..
so $$\sqrt{(-3a)^2+(4a)^2}$$ = 10..
$$9a^2+16a^2 = 100$$..
$$a^2 = \frac{100}{25}=4$$..
a= 2, -2..
depending on which Quad the point is we can calculate the coord..
x=-3*2; y=4*2..
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Re: A certain square is to be drawn on a coordinate plane   [#permalink] 08 Apr 2016, 20:08

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