GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

It is currently 28 May 2020, 00:56

Close

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
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.

Close

Request Expert Reply

Confirm Cancel

A rectangle ABCD is to be constructed on the XY plane

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Manager
Manager
User avatar
S
Joined: 28 Jan 2015
Posts: 86
Location: India
Concentration: Marketing, Entrepreneurship
GPA: 4
WE: Information Technology (Internet and New Media)
Premium Member
A rectangle ABCD is to be constructed on the XY plane  [#permalink]

Show Tags

New post Updated on: 16 Nov 2019, 21:58
15
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

54% (02:32) correct 46% (02:48) wrong based on 81 sessions

HideShow timer Statistics

A rectangle ABCD is to be constructed on the XY plane so that AB is || to the y axis ; if the X and Y co-ordinates of A,B,C & D are integers between -3 to 6 ; How many different rectangles can be constructed ?

A) 81
B) 100
C) 2025
D) 10000
E) 12100

_________________
Make Ur Skin like Rhino Never React unnecessary but Mind like Wolf make others' intuition vulnerable before U ~ Les Brown .

Official Mocks 1 : 710
Official Mocks 2 : 670
Official Mocks 3 : 590
Official Mocks 5 : 710

Originally posted by soumya170293 on 16 Nov 2019, 20:26.
Last edited by chetan2u on 16 Nov 2019, 21:58, edited 1 time in total.
Edited the Tags and formatted the question.
Math Expert
avatar
V
Joined: 02 Aug 2009
Posts: 8601
A rectangle ABCD is to be constructed on the XY plane  [#permalink]

Show Tags

New post 16 Nov 2019, 21:56
2
2
soumya170293 wrote:
A rectangle ABCD is to be constructed on the XY plane so that AB is || to the y axis ; if the X and Y co-ordinates of A,B,C & D are integers between -3 to 6 ; How many different rectangles can be constructed ?

A > 81
B > 100
C > 2025
D > 10000
E > 12100



So we have X and Y ranging from -3 to 6 that is 10 by 10 integer values.
You can choose 2 points in X coordinates in 10C2 ways, and for each of this way, we can choose 2 points in y coordinates in 10C2 ways.
Thus total ways = 10C2*10C2=45*45=2025

C

To understand the concept of 10C2*10C2, I have taken a 3 by 3 rectangle in which the answer will be 3C2*3C2=3*3=9

gvij2017
Attachments

Untitled5.png
Untitled5.png [ 32.25 KiB | Viewed 1210 times ]


_________________
Director
Director
User avatar
P
Joined: 08 Aug 2017
Posts: 726
Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

Show Tags

New post 16 Nov 2019, 23:37
1
Could you elaborate 10C2*10C2?
According to my understanding for your explanation, if I choose any two x coordinates from 10 x coordinates and two y coordinates from 10 y coordinates, then I could other shapes such as quadrilaterals, two intersecting lines as well.
Please help to understand this concept.

chetan2u wrote:
soumya170293 wrote:
A rectangle ABCD is to be constructed on the XY plane so that AB is || to the y axis ; if the X and Y co-ordinates of A,B,C & D are integers between -3 to 6 ; How many different rectangles can be constructed ?

A > 81
B > 100
C > 2025
D > 10000
E > 12100



So we have X and Y ranging from -3 to 6 that is 10 by 10 integer values.
You can choose 2 points in X coordinates in 10C2 ways, and for each of this way, we can choose 2 points in y coordinates in 10C2 ways.
Thus total ways = 10C2*10C2=45*45=2025

C
Manager
Manager
avatar
B
Joined: 20 Jul 2019
Posts: 63
Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

Show Tags

New post 17 Nov 2019, 00:14
chetan2u wrote:
soumya170293 wrote:
A rectangle ABCD is to be constructed on the XY plane so that AB is || to the y axis ; if the X and Y co-ordinates of A,B,C & D are integers between -3 to 6 ; How many different rectangles can be constructed ?

A > 81
B > 100
C > 2025
D > 10000
E > 12100



So we have X and Y ranging from -3 to 6 that is 10 by 10 integer values.
You can choose 2 points in X coordinates in 10C2 ways, and for each of this way, we can choose 2 points in y coordinates in 10C2 ways.
Thus total ways = 10C2*10C2=45*45=2025

C
I didnt understand your approach , can you elaborate

Posted from my mobile device
SVP
SVP
avatar
V
Joined: 20 Jul 2017
Posts: 1506
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

Show Tags

New post 17 Nov 2019, 01:02
soumya170293 wrote:
A rectangle ABCD is to be constructed on the XY plane so that AB is || to the y axis ; if the X and Y co-ordinates of A,B,C & D are integers between -3 to 6 ; How many different rectangles can be constructed ?

A) 81
B) 100
C) 2025
D) 10000
E) 12100


A rectangle can be formed by selecting any 2 values of x and any 2 values of y

E.g: If we select \(x_1, x_2, y_1\) & \(y_2\), a rectangle can be formed with coordinates \((x_1, y_1), (x_1, y_2), (x_2, y_1)\) & \((x_2, y_2)\)

--> Number of different rectangles that can be formed = Number of ways of selecting 2 values of x & 2 values of y out of 10 values = \(10c_2*10c_2\) = \(45*45 = 2025\)

IMO Option C
Director
Director
User avatar
P
Joined: 08 Aug 2017
Posts: 726
Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

Show Tags

New post 17 Nov 2019, 04:57
I appreciate your elucidation. Actually I also missed the point that AB is parallel to Y axis.
In that case I would need 2 x coordinates and 2 y coordinates.
Arrangement doesn't matter here so I have to use combination concept.

There is one new learning from your every explanation.
Thanks!

chetan2u wrote:
soumya170293 wrote:
A rectangle ABCD is to be constructed on the XY plane so that AB is || to the y axis ; if the X and Y co-ordinates of A,B,C & D are integers between -3 to 6 ; How many different rectangles can be constructed ?

A > 81
B > 100
C > 2025
D > 10000
E > 12100



So we have X and Y ranging from -3 to 6 that is 10 by 10 integer values.
You can choose 2 points in X coordinates in 10C2 ways, and for each of this way, we can choose 2 points in y coordinates in 10C2 ways.
Thus total ways = 10C2*10C2=45*45=2025

C

To understand the concept of 10C2*10C2, I have taken a 3 by 3 rectangle in which the answer will be 3C2*3C2=3*3=9

gvij2017
VP
VP
User avatar
V
Joined: 07 Mar 2019
Posts: 1174
Location: India
GMAT 1: 580 Q43 V27
WE: Sales (Energy and Utilities)
Premium Member
A rectangle ABCD is to be constructed on the XY plane  [#permalink]

Show Tags

New post 18 Nov 2019, 01:57
1
Peddi wrote:
chetan2u wrote:
soumya170293 wrote:
A rectangle ABCD is to be constructed on the XY plane so that AB is || to the y axis ; if the X and Y co-ordinates of A,B,C & D are integers between -3 to 6 ; How many different rectangles can be constructed ?
A > 81
B > 100
C > 2025
D > 10000
E > 12100

So we have X and Y ranging from -3 to 6 that is 10 by 10 integer values.
You can choose 2 points in X coordinates in 10C2 ways, and for each of this way, we can choose 2 points in y coordinates in 10C2 ways.
Thus total ways = 10C2*10C2=45*45=2025
C
I didnt understand your approach , can you elaborate
Posted from my mobile device

Peddi
Tried as concise an explanation as it can be but this is best i can offer. You have to be a little patient. :)

The coordinates within which these rectangles can be formed are (-3,6) (6,6) (6,-3) (-3,-3).
From Point no. 1 to 9 = 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 285
From Point No. 10 to 17 = 2 * (72 + 63 + 54 + 45 + 36 + 27 + 18 + 9) = 648
From Point No. 18 to 24 = 2 * (56 + 48 + 40 + 32 + 24 + 16 + 8) = 448
From Point No. 25 to 30 = 2 * (42 + 35 + 28 + 21 + 14 + 7) = 294
From Point No. 31 to 35 = 2 * (30 + 24 + 18 + 12 + 6) = 180
From Point No. 36 to 39 = 2 * (20 + 15 + 10 + 5) = 100
From Point No. 40 to 42 = 2 * (12 + 8 + 4) = 48
From Point No. 43 to 44 = 2 * (6 + 3) = 18
From Point No. 45 = 2 * (2) = 4
Total formations are = 285 + 648 + 448 + 294 + 180 + 100 + 48 + 18 + 4 = 2025.

Explanation as below: (refer figure)
Attachment:
File comment: Rectangle ABCD
Rectangle .JPG
Rectangle .JPG [ 201.35 KiB | Viewed 1061 times ]

So, we can take rectangles(including squares) with dimensions as follows:
1. 1Hx1V (sides with two consecutive coordinates for example (-3,6) (-2,6) (-2,5) (-3,5) and so on). Total formations = 9 x 9 = 81
2. 2Hx2V (sides with two coordinates having 1 coordinate in between for example (-3,6) (-1,6) (-1,4) (-3,4) and so on) Total formations = 8*8 = 64
3. 3Hx3V (sides with two coordinates having 2 coordinates in between for example (-3,6) (0,6) (0,3) (-3,3) and so on) Total formations = 7*7 = 49
4. 4Hx4V (sides with two coordinates having 3 coordinates in between for example (-3,6) (1,6) (1,2) (-3,2) and so on) Total formations = 6*6 = 36
5. 5Hx5V (sides with two coordinates having 4 coordinates in between for example (-3,6) (2,6) (2,1) (-3,1) and so on) Total formations = 5*5 = 25
6. 6Hx6V (sides with two coordinates having 5 coordinates in between for example (-3,6) (3,6) (3,0) (-3,0) and so on) Total formations = 4*4 = 16
7. 7Hx7V (sides with two coordinates having 6 coordinates in between for example (-3,6) (4,6) (4,-1) (-3,-1) and so on) Total formations = 3*3 = 9
8. 8Hx8V (sides with two coordinates having 7 coordinates in between for example (-3,6) (5,6) (5,-2) (-3,-2) and so on) Total formations = 2*2 = 4
9. 9Hx9V (sides with two coordinates having 8 coordinates in between for example (-3,6) (6,6) (6,-3) (-3,-3) and so on) Total formations = 1*1 = 1
(where H and V are just for horizontal and vertical reference of side dimensions)

Up till now all formations were of square form.
Other rectangles are: (refer figure)
10. 1Vx2H (sides with two coordinates having 0 coordinates and having 1 coordinate along y-axis and x-axis respectively in between for example (-3,6) (-1,6) (-1,5) (-3,5) and so on) Total formations = 9 x 8 = 72
11. 1Vx3H (sides with two coordinates having 0 coordinates and having 2 coordinates along y-axis and x-axis respectively in between for example (-3,6) (0,6) (0,5) (-3,5) and so on) Total formations = 9 x 7 = 63
12. 1Vx4H (sides with two coordinates having 0 coordinates and having 3 coordinates along y-axis and x-axis respectively in between for example (-3,6) (1,6) (1,5) (-3,5) and so on) Total formations = 9 x 6 = 54
13. 1Vx5H (sides with two coordinates having 0 coordinates and having 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (2,6) (2,5) (-3,5) and so on) Total formations = 9 x 5 = 45
14. 1Vx6H (sides with two coordinates having 0 coordinates and having 5 coordinates along y-axis and x-axis respectively in between for example (-3,6) (3,6) (3,5) (-3,5) and so on) Total formations = 9 x 4 = 36
15. 1Vx7H (sides with two coordinates having 0 coordinates and having 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,5) (-3,5) and so on) Total formations = 9 x 3 = 27
16. 1Vx8H (sides with two coordinates having 0 coordinates and having 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,5) (-3,5) and so on) Total formations = 9 x 2 = 18
17. 1Vx9H (sides with two coordinates having 0 coordinates and having 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,5) (-3,5) and so on) Total formations = 9 x 1 = 9

18. 2Vx3H (sides with two coordinates having 1 coordinate and 2 coordinates along y-axis and x-axis respectively in between for example (-3,6) (0,6) (0,4) (-3,4) and so on) Total formations = 8 x 7 = 56
19. 2Vx4H (sides with two coordinates having 1 coordinate and 3 coordinates along y-axis and x-axis respectively in between for example (-3,6) (1,6) (1,4) (-3,4) and so on) Total formations = 8 x 6 = 48
20. 2Vx5H (sides with two coordinates having 1 coordinate and 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (2,6) (2,4) (-3,4) and so on) Total formations = 8 x 5 = 40
21. 2Vx6H (sides with two coordinates having 1 coordinate and 5 coordinates along y-axis and x-axis respectively in between for example (-3,6) (3,6) (3,4) (-3,4) and so on) Total formations = 8 x 4 = 32
22. 2Vx7H (sides with two coordinates having 1 coordinate and 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,4) (-3,4) and so on) Total formations = 8 x 3 = 24
23. 2Vx8H (sides with two coordinates having 1 coordinate and 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,4) (-3,4) and so on) Total formations = 8 x 2 = 16
24. 2Vx9H (sides with two coordinates having 1 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,4) (-3,4) and so on) Total formations = 8 x 1 = 8
Again
25. 3Vx4H (sides with two coordinates having 2 coordinate and 3 coordinates along y-axis and x-axis respectively in between for example (-3,6) (1,6) (1,3) (-3,3) and so on) Total formations = 7 x 6 = 42
26. 3Vx5H (sides with two coordinates having 2 coordinate and 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (2,6) (2,3) (-3,3) and so on) Total formations = 7 x 5 = 35
27. 3Vx6H (sides with two coordinates having 2 coordinate and 5 coordinates along y-axis and x-axis respectively in between for example (-3,6) (3,6) (3,3) (-3,3) and so on) Total formations = 7 x 4 = 28
28. 3Vx7H (sides with two coordinates having 2 coordinate and 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,3) (-3,3) and so on) Total formations = 7 x 3 = 21
29. 3Vx8H (sides with two coordinates having 2 coordinate and 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,3) (-3,3) and so on) Total formations = 7 x 2 = 14
30. 3Vx9H (sides with two coordinates having 2 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (2,6) (2,3) (-3,3) and so on) Total formations = 7 x 1 = 7

31. 4Vx5H (sides with two coordinates having 3 coordinate and 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (2,6) (2,2) (-3,2) and so on) Total formations = 6 x 5 = 30
32. 4Vx6H (sides with two coordinates having 3 coordinate and 5 coordinates along y-axis and x-axis respectively in between for example (-3,6) (3,6) (3,2) (-3,2) and so on) Total formations = 6 x 4 = 24
33. 4Vx7H (sides with two coordinates having 3 coordinate and 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,2) (-3,2) and so on) Total formations = 6 x 3 = 18
34. 4Vx8H (sides with two coordinates having 3 coordinate and 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,2) (-3,2) and so on) Total formations = 6 x 2 = 12
35. 4Vx9H (sides with two coordinates having 3 coordinate and 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,2) (-3,2) and so on) Total formations = 6 x 1 = 6

36. 5Vx6H (sides with two coordinates having 4 coordinate and 5 coordinates along y-axis and x-axis respectively in between for example (-3,6) (3,6) (3,1) (-3,1) and so on) Total formations = 5 x 4 = 20
37. 5Vx7H (sides with two coordinates having 4 coordinate and 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,1) (-3,1) and so on) Total formations = 5 x 3 = 15
38. 5Vx8H (sides with two coordinates having 4 coordinate and 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,1) (-3,1) and so on) Total formations = 5 x 2 = 10
39. 5Vx9H (sides with two coordinates having 4 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,1) (-3,1) and so on) Total formations = 5 x 1 = 5

40. 6Vx7H (sides with two coordinates having 5 coordinate and 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,0) (-3,0) and so on) Total formations = 4 x 3 = 12
41. 6Vx8H (sides with two coordinates having 5 coordinate and 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,0) (-3,0) and so on) Total formations = 4 x 2 = 8
42. 6Vx9H (sides with two coordinates having 5 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,0) (-3,0) and so on) Total formations = 4 x 1 = 4

43. 7Vx8H (sides with two coordinates having 6 coordinate and 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,-1) (-3,-1) and so on) Total formations = 3 x 2 = 6
44. 7Vx9H (sides with two coordinates having 6 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,-1) (-3,-1) and so on) Total formations = 3 x 1 = 3

45. 8Vx9H (sides with two coordinates having 7 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (6,0) (-3,0) and so on) Total formations = 2 x 1 = 2
Since point no. 10 to 45 will have a mirror image along the diagonal (-3,6) – (6,-3), the number of each formations calculated above for point no. 10 to 45 would be doubled.
Example: For Point No. 10, 1Vx2H (sides with two coordinates having 0 coordinates and having 1 coordinate along y-axis and x-axis respectively in between for example (-3,6) (-1,6) (-1,5) (-3,5)) would have an exact replica as 2Vx1H (sides with two coordinates having 0 coordinates and having 1 coordinate along y-axis and x-axis respectively in between for example (-3,6) (-2,6) (-2,4) (-3,4)) and so on). Then Total formations would be = 2 x 9 x 8 = 144 and so on…

gvij2017 Have a look. Since side of rectangle is always || to an axis, only these are possible formations.
Hope this is helpful.
_________________
Ephemeral Epiphany..!

GMATPREP1 590(Q48,V23) March 6, 2019
GMATPREP2 610(Q44,V29) June 10, 2019
GMATPREPSoft1 680(Q48,V35) June 26, 2019
Manager
Manager
avatar
S
Joined: 24 Jan 2013
Posts: 90
Location: India
GMAT 1: 660 Q49 V31
GMAT ToolKit User CAT Tests
Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

Show Tags

New post 21 Mar 2020, 11:11
1
HI

Can anyone tell the difference between this question and Triangle question.

Why cant solution to this question be --- 10C1 * 9C1 * 10C1*9C1 like in below question ---

Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and Y coordinates of P,Q and R are to be integers that satisfy the inequalitites -4≤ X≤ 5 and 6≤ y≤ 16. How many different triangles with these properties could be constructed?

A)110
B)1100
C)9900
D)10000
E)12100

thanks..


We have the rectangle with dimensions 10*11 (10 horizontal dots and 11 vertical). PQ is parallel to y-axis and PR is parallel to x-axis.

Choose the (x,y) coordinates for vertex P (right angle): 10C1*11C1;
Choose the x coordinate for vertex R (as y coordinate is fixed by A): 9C1, (10-1=9 as 1 horizontal dot is already occupied by A);
Choose the y coordinate for vertex Q (as x coordinate is fixed by A): 10C1, (11-1=10 as 1 vertical dot is already occupied by A).

10C1*11C1*9C1*10C1=9900.
_________________
Please highlight if you found this helpful, Thanks in advance
Intern
Intern
User avatar
B
Joined: 17 Nov 2014
Posts: 11
Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

Show Tags

New post 19 Apr 2020, 05:10
1
zs2 wrote:
HI

Can anyone tell the difference between this question and Triangle question.

Why cant solution to this question be --- 10C1 * 9C1 * 10C1*9C1 like in below question ---

Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and Y coordinates of P,Q and R are to be integers that satisfy the inequalitites -4≤ X≤ 5 and 6≤ y≤ 16. How many different triangles with these properties could be constructed?

A)110
B)1100
C)9900
D)10000
E)12100

thanks..


We have the rectangle with dimensions 10*11 (10 horizontal dots and 11 vertical). PQ is parallel to y-axis and PR is parallel to x-axis.

Choose the (x,y) coordinates for vertex P (right angle): 10C1*11C1;
Choose the x coordinate for vertex R (as y coordinate is fixed by A): 9C1, (10-1=9 as 1 horizontal dot is already occupied by A);
Choose the y coordinate for vertex Q (as x coordinate is fixed by A): 10C1, (11-1=10 as 1 vertical dot is already occupied by A).

10C1*11C1*9C1*10C1=9900.


Thanks for bringing up this triangle question zs2. So the basic difference between the rectangle question and triangle question is the usage of the permutations concept, thereby reflecting in the formula to solve these individual questions.

In the rectangle question, we are choosing the two points on each axis (x1, x2 or y1, y2) at the same time. Thus we are opting for the combination formula nCr. The shape of the rectangle doesn't change when we swap x1 with x2 or y1 with y2. Hence the whole number of cases should be divided by 2 in order to eliminate the repeating cases. Hence (10x9)/2 which is nothing but 10C2.

In case of triangle question, we are choosing the points in similar fashion (x1, x2 or y1, y2) except that if we swap the points the shape of the triangle changes. Hence there is no need to divide the whole cases by 2. Since the order of the chosen points matter here, we use permutation formula which is nPr. Hence (11x10) which is nothing 11P10.

Hope this helped.
GMAT Club Bot
Re: A rectangle ABCD is to be constructed on the XY plane   [#permalink] 19 Apr 2020, 05:10

A rectangle ABCD is to be constructed on the XY plane

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne