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# A rectangle ABCD is to be constructed on the XY plane

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A rectangle ABCD is to be constructed on the XY plane  [#permalink]

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Updated on: 16 Nov 2019, 21:58
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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

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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.
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A rectangle ABCD is to be constructed on the XY plane  [#permalink]

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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
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Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

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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.

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

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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
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Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

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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
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Joined: 08 Aug 2017
Posts: 723
Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

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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
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Joined: 07 Mar 2019
Posts: 1161
Location: India
GMAT 1: 580 Q43 V27
WE: Sales (Energy and Utilities)
A rectangle ABCD is to be constructed on the XY plane  [#permalink]

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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 [ 201.35 KiB | Viewed 1050 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.
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Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

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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.
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Re: A rectangle ABCD is to be constructed on the XY plane  [#permalink]

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