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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

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

Show Tags

14 Sep 2010, 21:06

7

This post received KUDOS

55

This post was BOOKMARKED

00:00

A

B

C

D

E

Difficulty:

85% (hard)

Question Stats:

37% (01:58) correct
63% (00:54) wrong based on 262 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?

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:

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 \(y=-6\).

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

Show Tags

14 Oct 2010, 14:19

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

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.

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.
_________________

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

Show Tags

25 Jan 2013, 06:32

2

This post received KUDOS

2

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)... \(4+8 = 12\)

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.
_________________

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

Show Tags

14 Oct 2010, 14:27

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.
_________________

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

Show Tags

14 Oct 2010, 15:13

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?
_________________

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.
_________________

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

Show Tags

27 Mar 2013, 22:06

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)

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

Show Tags

28 Apr 2013, 09:39

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..!

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

Show Tags

10 Jan 2014, 06:40

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?

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:

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
_________________

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

Show Tags

14 Sep 2010, 21:22

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....

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

Show Tags

22 Dec 2010, 15:03

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.

Happy New Year everyone! Before I get started on this post, and well, restarted on this blog in general, I wanted to mention something. For the past several months...

It’s quickly approaching two years since I last wrote anything on this blog. A lot has happened since then. When I last posted, I had just gotten back from...

Post-MBA I became very intrigued by how senior leaders navigated their career progression. It was also at this time that I realized I learned nothing about this during my...