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.
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:
In each webinar, we teach game-changing techniques and strategies for solving some of the most high-value GMAT quant questions. In addition, we will provide you with the opportunity to ask questions regarding how to best prepare for the GMAT.
You'll get a free, full-length GMAT practice test, our free Foundations of Math eBook and workshop, and access to free lessons on Sentence Correction and Data Sufficiency.
Reading Comprehension has been added to the Target Test Prep Verbal course. With our full Verbal course, including 1,000+ practice verbal questions and 400+ instructor-led videos, you now have access to everything you need to master GMAT Verbal.
46%
(02:20)
correct
54%
(02:33)
wrong
based on 371
sessions
HideShow
timer Statistics
Rectangle ABCD is constructed in the coordinate plane parallel to the x- and y-axes. If the x- and y-coordinates of each of the points are integers which satisfy 3 ≤ x ≤ 11 and -5 ≤ y ≤ 5, how many possible ways are there to construct rectangle ABCD?
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
12 Nov 2012, 00:59
3
Kudos
5
Bookmarks
Expert Reply
Archit143 wrote:
Rectangle ABCD is constructed in the coordinate plane parallel to the x- and y-axes. If the x- and y-coordinates of each of the points are integers which satisfy 3 ≤ x ≤ 11 and -5 ≤ y ≤ 5, how many possible ways are there to construct rectangle ABCD?
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
04 Jan 2018, 09:01
3
Kudos
3
Bookmarks
Expert Reply
Top Contributor
Archit143 wrote:
Rectangle ABCD is constructed in the coordinate plane parallel to the x- and y-axes. If the x- and y-coordinates of each of the points are integers which satisfy 3 ≤ x ≤ 11 and -5 ≤ y ≤ 5, how many possible ways are there to construct rectangle ABCD?
396 1260 1980 7920 15840
IMPORTANT:
First notice that, to construct this rectangle, the vertices will share several points. For example, if the 4 vertices are at (2, 5), (2, -3), (9, 5) and (9, -3), then we get a rectangle.
Notice that there are only 2 different x-coordinates (2 and 9) and only 2 different y-coordinates (-3 and 5)
So, to create the desired rectangle, we need only choose 2 different x-coordinates and 2 different y-coordinates
So, let's take the task of creating rectangles and break it into STAGES
STAGE 1: Select the 2 x-coordinates We can choose 2 values from the set {3, 4, 5, 6, 7, 8, 9, 10, and 11} In other words, we must choose 2 of the 9 values in the set Since the order in which we choose the numbers does not matter, we can use COMBINATIONS We can select 2 number from 9 numbers in 9C2 ways (= 36 ways)
STAGE 2: Select the 2 y-coordinates We can choose 2 values from the set {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5} In other words, we must choose 2 of the 11 values in the set We can select 2 number from 11 numbers in 11C2 ways (= 55 ways)
By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus create a rectangle) in (36)(55) ways (= 1980 ways)
Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
11 Nov 2012, 19:26
gnan wrote:
As the rectangle is parallel to coordinate axes, the coordinates of the points of the rectangle would be
(X1, Y1), (X2, Y1), (X2, Y2), (X1,Y2)
given that X1, X2 lie between 3 and 11..ie., 9 possible numbers
Possible combinations for X1,X2 would be 9C2 = 36
Similarly, Possible combinations for Y1, Y2 would be 11C2 = 55
Possible ways of constructing rectangle is by selecting any of the combination of X1,X2 and Y1,Y2
= 36 * 55 = 1980 Ans. C
Excellent explanation,
pls explain the similarity between between the triangle question and this, hope you would have solved it....I ll mention the question num for making it easy
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
11 Nov 2012, 21:36
1
Kudos
1
Bookmarks
We need to choose 2 numbers from the x domain [3,11], since they will form two lines parallel to the y axis. Similarly, we need two values from the y domain [-5,5] to form two values parallel to the x axis. There are 9 integers for x and there are 11 numbers for y.
Choose 2 from 9 for the sides parallel to y axis: 9C2 Choose 2 from 11 for the sides parallel to x axis: 11C2
Multiply to get the overall number which should give you C.
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
11 Nov 2012, 21:41
seriousmonkey wrote:
We need to choose 2 numbers from the x domain [3,11], since they will form two lines parallel to the y axis. Similarly, we need two values from the y domain [-5,5] to form two values parallel to the x axis. There are 9 integers for x and there are 11 numbers for y.
Choose 2 from 9 for the sides parallel to y axis: 9C2 Choose 2 from 11 for the sides parallel to x axis: 11C2
Multiply to get the overall number which should give you C.
I used the same method to slove just wanted to know the distinction between the question in my preceding post and the the question in this thread.
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
11 Nov 2012, 22:17
4
Kudos
7
Bookmarks
Just had a look- it is rather subtle. Here, we can choose 2 values of x and y as taking any 2 values of x and y will always yield a rectangle. For instance x=2 and x=3 are two lines parallel to y axis and y=1 and y=4 are two values parallel to x axis- plot these lines and you get a rectangle.
For the triangles query, we will need to constrain the values of the coordinates that x and y can take, say P is (x1,y1), Q(x1,y2) and R(x2,y1).
So we will need to pick one x to designate x1 and then we will have 9 more x values remaining from which we choose x2. Similarly we can do the same for y1 and y2.
That is why we get 11*10*10*9
If we choose 2 values directly , then we do not make a distinction in the order and if this happens we multiply the value by 4. Take x1=5, x2=8, y1=7 and y2=9 (so 8,5,7 and 9 can be arranged among one another and the combination does not take the changing values into account)
the four triangles you get are: (5,7), (8,7), (5,9) ; (8,7), (8,9), (5,7) ; (5,9), (8,9), (5,7) ; (8,9) , (5,9) , (5,7).
So we can also use : 4* 11C2*10C2 to get 9900
For the rectangle, choosing 2 values of x and y result in only 1 rectangle. This is the only difference
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
11 Nov 2012, 22:34
2
Kudos
1
Bookmarks
Sorry for the double post- but another easier way to think about this is as so: we can calculate the number of rectangles just as we have done for the original question you provided here.
Take any rectangle and you have two diagonals. Each diagonal divides the rectangle into two different right triangles. So taking the two diagonals into account, we can create 4 right triangles with each rectangle. So just multiply the number of rectangles by 4 to get the answer..
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
12 Nov 2012, 01:21
seriousmonkey wrote:
Sorry for the double post- but another easier way to think about this is as so: we can calculate the number of rectangles just as we have done for the original question you provided here.
Take any rectangle and you have two diagonals. Each diagonal divides the rectangle into two different right triangles. So taking the two diagonals into account, we can create 4 right triangles with each rectangle. So just multiply the number of rectangles by 4 to get the answer..
I was thinking on the same lines Now another had the question mentioned how many different squares instead of rectangle than what will be our answer???????????
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
27 Jun 2013, 13:36
consider rectangle to be ABCD
choosing A's X and Y co-ordinate >> 10c1*9c1 choosing B's X and Y co-ordinate >> 9c1.1 (because one coordinate is fixed) choosing C's X and Y co-ordinate >> 1.8c1 (because one coordinate is fixed) choosing D's X and Y co-ordinate >> 1.1 (Because both coordinate are fixed)
this comes equal to 7920. I think I am considering some of the cases twice or even four times. Please tell me what am I doing wrong ..
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
28 Jun 2013, 23:31
2
Kudos
1
Bookmarks
stunn3r wrote:
consider rectangle to be ABCD
choosing A's X and Y co-ordinate >> 10c1*9c1 choosing B's X and Y co-ordinate >> 9c1.1 (because one coordinate is fixed) choosing C's X and Y co-ordinate >> 1.8c1 (because one coordinate is fixed) choosing D's X and Y co-ordinate >> 1.1 (Because both coordinate are fixed)
this comes equal to 7920. I think I am considering some of the cases twice or even four times. Please tell me what am I doing wrong ..
Hi stunn3r
You have two errors:
(1) How did you come up with 7920, because 10C1*9C1*9C1*8C1 # 7920. Because there are 9 ways to choose X, and 11 ways to choose Y
(2) I assume your equations are correct. But the question here is "how many RECTANGLE?" not "how many combination of A,B,C and D ==> 4 points create only 1 rectangle ==> You should divide 7290/4 = 1980
Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
10 Sep 2014, 12:27
1
Kudos
The way I solved this problem was by thinking about the points and lines on the graph.
I chose two points on the horizontal axis between 3 & 11 . Because it doesn't specify that the points are integers I considered EVERY point, including the first, making the choices (9)(9) you then multiply by 11 because there are 11 points on the y axis. (9)(9)(11)= 891
You do the same thing for the y axis. 11 possible points and 2 must be chosen considering also that there are 9 horizontal points. (11)(11)(9) = 1089
Added together. (9)(9)(11) + (11)(11)(9) = 1980
An interesting thing is that the answer is a little smaller than 1980 because the axis points cannot be the same on the graph... but since there are an infinite number of points between 3 & 11, the difference is negligible.
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
29 Nov 2017, 16:50
seriousmonkey wrote:
Just had a look- it is rather subtle. Here, we can choose 2 values of x and y as taking any 2 values of x and y will always yield a rectangle. For instance x=2 and x=3 are two lines parallel to y axis and y=1 and y=4 are two values parallel to x axis- plot these lines and you get a rectangle.
For the triangles query, we will need to constrain the values of the coordinates that x and y can take, say P is (x1,y1), Q(x1,y2) and R(x2,y1).
So we will need to pick one x to designate x1 and then we will have 9 more x values remaining from which we choose x2. Similarly we can do the same for y1 and y2.
That is why we get 11*10*10*9
If we choose 2 values directly , then we do not make a distinction in the order and if this happens we multiply the value by 4. Take x1=5, x2=8, y1=7 and y2=9 (so 8,5,7 and 9 can be arranged among one another and the combination does not take the changing values into account)
the four triangles you get are: (5,7), (8,7), (5,9) ; (8,7), (8,9), (5,7) ; (5,9), (8,9), (5,7) ; (8,9) , (5,9) , (5,7).
So we can also use : 4* 11C2*10C2 to get 9900
For the rectangle, choosing 2 values of x and y result in only 1 rectangle. This is the only difference
Can you explain this part in detail-That is why we get 11*10*10*9
Re: Rectangle ABCD is constructed in the coordinate plane parall
[#permalink]
21 Oct 2021, 09:28
Top Contributor
Hello from the GMAT Club BumpBot!
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________
One of the fastest-growing graduate business schools in Southern California, shaping the future by developing leading thinkers who will stand at the forefront of business growth. MBA Landing | School of Business (ucr.edu)