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The figure above shows a rectangle ABCD in the xy- coordinate plane

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The figure above shows a rectangle ABCD in the xy- coordinate plane  [#permalink]

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New post Updated on: 23 Feb 2018, 23:45
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The figure below shows a rectangle ABCD in the xy- coordinate plane. The sides AB and AD are parallel to the x- and the y- axis respectively. How many squares of side 1 unit that lie on or inside the rectangle ABCD can be drawn?

(1) The length of side AB is 6 units
(2) The coordinates of points A and C are (3,2) and (9, 5)

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Originally posted by saswata4s on 23 Feb 2018, 09:49.
Last edited by Bunuel on 23 Feb 2018, 23:45, edited 1 time in total.
Edited the question.
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Re: The figure above shows a rectangle ABCD in the xy- coordinate plane  [#permalink]

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New post 23 Feb 2018, 09:59
saswata4s wrote:
The figure below shows a rectangle ABCD in the xy- coordinate plane. The sides AB and AD are parallel to the x- and the y- axis respectively. How many squares of side 1 unit that lie on or inside the rectangle ABCD can be drawn?

(1) The length of side AB is 6 units
(2) The coordinates of points A and C are (3,2) and (9, 5)


To know the number of square on or inside rectangle ABCD, we need to know the area of rectangle ABCD. Hence we need to know the length and breadth of the rectangle ABCD

Statement 1: provides the value of length, but neither the width or any co-ordinates are mentioned. Insufficient

Statement 2: \(A=(3,2)\) & \(C=(9,5)\), then \(B=(9,2)\) because B will have the same x-coordinate as that of point C & same y-coordinate as that of point A

so length \(AB=9-3=6\) and width \(BC=5-2=3\). Sufficient

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Re: The figure above shows a rectangle ABCD in the xy- coordinate plane  [#permalink]

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New post 23 Feb 2018, 10:30
saswata4s wrote:
The figure below shows a rectangle ABCD in the xy- coordinate plane. The sides AB and AD are parallel to the x- and the y- axis respectively. How many squares of side 1 unit that lie on or inside the rectangle ABCD can be drawn?

(1) The length of side AB is 6 units
(2) The coordinates of points A and C are (3,2) and (9, 5)


I personally think that the question should mention 'How many squares of side 1 unit and with all its vertices having integer coordinates' that lie on or inside the rectangle ABCD can be drawn.

That is because if we do not mention that the vertices of square have to be integers, then actually infinite squares can be made.

Eg, we start with the first square of side 1 unit which is at the leftmost bottom position in this rectangle, i.e., the square whose bottom left vertex is A. Now if we assume the coordinates of A to be (3,2) then the other three vertices will be at (4,2); (4,3) and (3,3).
Now if we just shift this square by 0.1 units on the right, then we will have another different square with side 1 unit having the four vertices as: (3.1,2); (4.1,2); (4.1,3) and (3.1,3).

Similarly we can shift that first square 0.1 units up (instead of right) and we will have another square. Similarly, instead of shifting 0.1 units we could also shift by 0.01 units or 0.02 units. So we can just make the first square anywhere in this rectangle and then keep shifting it very slightly to right/left and/or top/bottom - thereby resulting in infinite squares.
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Re: The figure above shows a rectangle ABCD in the xy- coordinate plane  [#permalink]

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New post 23 Feb 2018, 14:53
Hi , what is meant by "How many squares of side 1 unit that lie on or inside the rectangle ABCD can be drawn?"
If it just said inside the rectangle it would be plain , we need the area of the rectangle and then divide it to the square. This wording on or confused me.
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Re: The figure above shows a rectangle ABCD in the xy- coordinate plane  [#permalink]

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New post 23 Feb 2018, 21:16
saswata4s wrote:
The figure below shows a rectangle ABCD in the xy- coordinate plane. The sides AB and AD are parallel to the x- and the y- axis respectively. How many squares of side 1 unit that lie on or inside the rectangle ABCD can be drawn?

(1) The length of side AB is 6 units
(2) The coordinates of points A and C are (3,2) and (9, 5)


I think if at all there is any deficiency in the wording of the question then that is that it should specify "maximum number of square (without any overlap) of side 1 unit that lie on or inside". By using Maximum word, it will imply that small squares need to be drawn starting from any of the four vertices and this is what I have assumed in my solution. By starting from any other point inside the big square will not yield "maximum" number of small squares.
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Re: The figure above shows a rectangle ABCD in the xy- coordinate plane  [#permalink]

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New post 24 Feb 2018, 07:26
niks18 wrote:
saswata4s wrote:
The figure below shows a rectangle ABCD in the xy- coordinate plane. The sides AB and AD are parallel to the x- and the y- axis respectively. How many squares of side 1 unit that lie on or inside the rectangle ABCD can be drawn?

(1) The length of side AB is 6 units
(2) The coordinates of points A and C are (3,2) and (9, 5)


I think if at all there is any deficiency in the wording of the question then that is that it should specify "maximum number of square (without any overlap) of side 1 unit that lie on or inside". By using Maximum word, it will imply that small squares need to be drawn starting from any of the four vertices and this is what I have assumed in my solution. By starting from any other point inside the big square will not yield "maximum" number of small squares.


I agree. But this is how the question was framed in the source.

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Re: The figure above shows a rectangle ABCD in the xy- coordinate plane &nbs [#permalink] 24 Feb 2018, 07:26
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The figure above shows a rectangle ABCD in the xy- coordinate plane

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