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# A Square is to be drawn on the xy-plane. If one vertex of the square

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Re: A Square is to be drawn on the xy-plane. If one vertex of the square [#permalink]
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pushpitkc wrote:
A Square is to be drawn on the xy plane. If one vertex of the square needs to be at
the origin, sides of the square need to be 5 units each, and all coordinates of all
vertices need to be integers, how many different squares are possible?

A. 4
B. 6
C. 8
D. 12
E. 16

Source: Experts Global

Similar question to practice: https://gmatclub.com/forum/a-certain-sq ... 27018.html
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Re: A Square is to be drawn on the xy-plane. If one vertex of the square [#permalink]
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pushpitkc wrote:
A Square is to be drawn on the xy plane. If one vertex of the square needs to be at
the origin, sides of the square need to be 5 units each, and all coordinates of all
vertices need to be integers, how many different squares are possible?

A. 4
B. 6
C. 8
D. 12
E. 16

Source: Experts Global

The square can be formed with points when Y = 5,3,4 and X correspondingly will be X = 0,4,3. So, for each Quadrant there will be 3 Different Squares, hence for all the 4 Quadrants, it will be 12.
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Re: A Square is to be drawn on the xy-plane. If one vertex of the square [#permalink]
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The square could rotate around the origin, making a cycle.
(x^2)+(y^2)=Diagonal of the square
(x^2)+(y^2)=50
Solving above Equation with Integer solutions=12
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A Square is to be drawn on the xy-plane. If one vertex of the square [#permalink]
KarishmaB chetan2u  KrishnakumarKA1
Will you be kind enough to show me how ,  starting with the point (-3,4), we can form 4 different squares ?  Finding it a bit confusing to visualize..Any simpler approach..
KrishnakumarKA1 wrote:
Hi,

This is a very interesting question.

Easily someone could fall for the trap answer as 4 here.

If one visualize the squares only on the x and y axis, then they end up falling for the trap.

So the vertices need not be only on the x and y axis, they have mentioned all the vertices should be integers,

So, Let’s try to visualize it,

We can see that,

Co-ordinates (-4,3) forms a side length of the square as 5 from the origin (0,0), Because 3-4-5 are pythogorean triplets.

As shown in the figure(In the attachment), we can form four squares starting with point (-4,3).

So similarly starting with the point (-3,4), we can form 4 different squares.

Already on the x and y axis there are 4 different squares.

So totally, we can form 12 different squares with side length as 5 and one vertex at the origin. So the answer is D.

Most important factor of solving a co-ordinate geometry question is visualizing.

Hope this helps.

­
A Square is to be drawn on the xy-plane. If one vertex of the square [#permalink]
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