It is currently 18 Feb 2018, 18:10

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 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

Senior Manager
Joined: 28 Apr 2014
Posts: 269
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

11 May 2014, 20:56
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, 02:33, edited 1 time in total.
Senior Manager
Joined: 28 Apr 2014
Posts: 269
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

11 May 2014, 21:05
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: 43792
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

12 May 2014, 02:18
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.

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.
_________________
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 11025
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: 340 Q170 V170
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

29 May 2015, 14:29
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
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** Intern Joined: 06 Nov 2015 Posts: 27 GMAT 1: 690 Q49 V35 Re: A certain square is to be drawn on a coordinate plane [#permalink] Show Tags 07 Apr 2016, 16:09 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? Thanks for your response EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 11025 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: 340 Q170 V170 Re: A certain square is to be drawn on a coordinate plane [#permalink] Show Tags 07 Apr 2016, 18:41 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. GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Intern
Joined: 06 Nov 2015
Posts: 27
GMAT 1: 690 Q49 V35
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

08 Apr 2016, 05:13
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
Math Expert
Joined: 02 Aug 2009
Posts: 5649
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

08 Apr 2016, 05:42
1
KUDOS
Expert's post
thuyduong91vnu wrote:
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

Hi,

Yes,
and as you ask why?

we are not taking ONLY one set of integers as one vertice..
One is already existing as ORIGIN and the other we have taken as (8,6)..
the line joining origin and (8,6) and the ORIGIN and one diagonally opposite to (8,6) are at 90 degree or perpendicular..
their slope are in ratio -1..
1) let me show you with an example slope of line joining 0,0 and 8,6 ---$$m= \frac{(8-0)}{(6-0)}= \frac{8}{6}$$...
the line perpendicular to it will have -$$\frac{1}{m}$$..
so slope = $$-\frac{6}{8}=\frac{(x-0)}{(y-0)}$$..
thus x= -6 and y=8
so third point is also integer and similarly 4th vertice will also be integer
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

BANGALORE/-

Intern
Joined: 06 Nov 2015
Posts: 27
GMAT 1: 690 Q49 V35
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

08 Apr 2016, 14:58
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?

Thanks for your help
Math Expert
Joined: 02 Aug 2009
Posts: 5649
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

08 Apr 2016, 19:08
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?

Thanks for your help

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..
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

BANGALORE/-

Intern
Joined: 30 Apr 2016
Posts: 39
Location: India
Concentration: Entrepreneurship, General Management
Schools: Stern '19 (S)
GMAT 1: 640 Q44 V35
GMAT 2: 700 Q49 V37
GPA: 3.8
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

14 Jun 2016, 06:45
Where can I find questions on Co-Ordinate Geometry?
Bunuel bb walker
Math Expert
Joined: 02 Sep 2009
Posts: 43792
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

14 Jun 2016, 06:55
vedantkabra wrote:
Where can I find questions on Co-Ordinate Geometry?
Bunuel bb walker

First of all you might find the following post useful ALL YOU NEED FOR QUANT ! ! !.

Also, check our questions bank: viewforumtags.php

Theory on Coordinate Geometry: math-coordinate-geometry-87652.html

All DS Coordinate Geometry Problems to practice: search.php?search_id=tag&tag_id=41
All PS Coordinate Geometry Problems to practice: search.php?search_id=tag&tag_id=62

Hope it helps.
_________________
Manager
Joined: 01 Feb 2017
Posts: 79
A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

17 May 2017, 15:19
Area of square = 100. Therefore, Side = 10 and Diagnol = 10sqrt2.

First, calculate possible squares per quadrant:
Let end point of Diagnol be point A (x,y).
So, OA^2 (fixed length of diagnol) = x^2 + y^2 = 200.
Now, calculate [Integers]^2 satisfying above equation:
Only two matching pairs are found: (2,14) & (10,10).
So, we have three coordinates possible for point A: (2,14) (14,2) & (10,10).
Thus, possible squares: per quadrant : 3 / for all four quadrants : 4 × 3 = 12.

Ans. E
Manager
Joined: 23 Dec 2013
Posts: 235
Location: United States (CA)
GMAT 1: 710 Q45 V41
GMAT 2: 760 Q49 V44
GPA: 3.76
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

22 May 2017, 17:47
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.

This problem is easy if you simplify the possibilities in one quadrant and extend to the other quadrants. So for instance, in this case, the two obvious choices for a given quadrant are vertices that extend along the x-axis and y-axis. Those are two possible triangles, so we know that there are at least 8 possibilities and we can eliminate answer choice A.

Since the length of the side of the square is 10 and we know that it's a right triangle, two lengths of triangles would satisfy the pythagorean triple: 6^2 + 8^2 + 10^2. So (6,8) and (8,6) are two possible points for the vertex of the square. As a result, there are 4 possible squares that can be drawn in the first quadrant and 12 total squares. It's not equal to 16 because then you would double count the squares that extend along the x- and y-axes.
Intern
Joined: 30 Jan 2017
Posts: 6
GMAT 1: 680 Q60 V60
GPA: 4
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

09 Aug 2017, 07:49
Hello!

I had a different way to approach this...

Since we are looking for an area of 100 in a square, I factorised it and got 2,2,5,5

That's 4 factors and order matters.

Using permutations I get ---- 4 P 2 = 12.

Is this a correct way to look at this problem, or did I just get lucky?

Thanks for helping out !
Intern
Joined: 26 Sep 2016
Posts: 40
A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

27 Oct 2017, 03:12
Hi,
I am very confused about this question, I saw this on a Manhattan CAT test, the question might be hard, but other than that, I was thinking the vertices of a square are simply its corners, so in the question, I simply thought one of the corners should be in the origin. Just to clarify the concept, aren't the vertices of a square the corners?
Thanks.
Senior Manager
Joined: 03 Apr 2013
Posts: 290
Location: India
Concentration: Marketing, Finance
Schools: Simon '20
GMAT 1: 740 Q50 V41
GPA: 3
Re: A certain square is to be drawn on a coordinate plane [#permalink]

Show Tags

13 Nov 2017, 06:36
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 $$y=-6$$.

I did something exactly like you. But in the middle I got confused. Since we are only focusing on determining the coordinated of one vertex, how do we say for sure that the other two vertices will be necessarily integers?
_________________

Spread some love..Like = +1 Kudos

Re: A certain square is to be drawn on a coordinate plane   [#permalink] 13 Nov 2017, 06:36

Go to page   Previous    1   2   [ 37 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

 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®.