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# Square ABCD is to be drawn in the xy-plane such that the origin

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Joined: 31 Jul 2017
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Square ABCD is to be drawn in the xy-plane such that the origin  [#permalink]

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09 Sep 2018, 22:06
1
4
00:00

Difficulty:

85% (hard)

Question Stats:

36% (02:00) correct 64% (02:07) wrong based on 52 sessions

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Square ABCD is to be drawn in the xy-plane such that the origin is one of vertices and the perimeter is 20. If all the rest three vertices have integer coordinates (x, y), how many different squares can be drawn?

1. 12
2. 10
3. 8
4. 6
5. 4

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Joined: 02 Aug 2009
Posts: 7684
Re: Square ABCD is to be drawn in the xy-plane such that the origin  [#permalink]

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10 Sep 2018, 01:45
rahul16singh28 wrote:
Square ABCD is to be drawn in the xy-plane such that the origin is one of vertices and the perimeter is 20. If all the rest three vertices have integer coordinates (x, y), how many different squares can be drawn?

1. 12
2. 10
3. 8
4. 6
5. 4

so one vertex is fixed and we rotate this square. In a way we are rotating the square around its diagonal.
thus it is a circle with radius as diagonal of the square..
perimeter = 20...side = 20/4=5
diagonal = $$5\sqrt{2}$$
thus $$x^2+y^2=r^2=(5\sqrt{2})^2=50$$
now 50 = 1+49=1^2+7^2 , so x and y can take values 1 or -1 and 7 or -7
In each quadrant - 2 values

next 50=5^2+5^2 so 5 or -5
in each quadrant = 1 value

total (2+1)*4=12

A
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Re: Square ABCD is to be drawn in the xy-plane such that the origin  [#permalink]

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11 Sep 2018, 22:02
Bunuel could you please explain this
Re: Square ABCD is to be drawn in the xy-plane such that the origin   [#permalink] 11 Sep 2018, 22:02
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