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A rectangle is plotted on the standard coordinate plane [#permalink]

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19 Jun 2011, 23:18

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A rectangle is plotted on the standard coordinate plane, with vertices at the origin and (0,6). If the coordinates of all vertices of the rectangle are non-negative integers, what are the coordinates of the other two vertices?

(1) The distance between the origin and one of the other vertices is 10 units.

(2) The distance between the origin and one of the other vertices is 8 units.

A rectangle is plotted on the standard coordinate plane, with vertices at the origin and (0,6). If the coordinates of all vertices of the rectangle are non-negative integers, what are the coordinates of the other two vertices?

Notice that we are told that the coordinates of all vertices of the rectangle are non-negative integers. That basically means that the rectangle is plotted entirely in the first quadrant: one vertex at the origin (0, 0), another on Y-axis (0, 6), third one somewhere in the first quadrant (x, 6) and the fourth vertex on the X-axis (x, 0):

Attachment:

Rectangle.png [ 5.05 KiB | Viewed 8991 times ]

(1) The distance between the origin and one of the other vertices is 10 units. 10 units is either the length of the other side or the length of the diagonal. The vertices could be: (0, 0), (0, 6), (10, 6) and (10, 0) OR (0, 0), (0, 6), (8, 6) and (8, 0), in this case the distance of 10 units is the distance between (0, 0) and (6, 8). Not sufficient.

(2) The distance between the origin and one of the other vertices is 8 units. Now, 8 units can not be the length of the diagonal since in this case the coordinates of the other two vertices won't be integers: if the diagonal=8, then the length of the other side of the rectangle is \(\sqrt{8^2-6^2}=2\sqrt{7}\), so the coordinates of the other two vertices are: \((2\sqrt{7}, \ 6)\) and \((2\sqrt{7}, \ 0)\). So, 8 units must be the length of the side, therefore the vertices are (0, 0), (0, 6), (8, 6) and (8, 0). Sufficient.

A rectangle is plotted on the standard coordinate plane, with vertices at the origin and (0,6). If the coordinates of all vertices of the rectangle are non-negative integers, what are the coordinates of the other two vertices?

(1) The distance between the origin and one of the other vertices is 10 units.

(2) The distance between the origin and one of the other vertices is 8 units.

stmnt 1- if the other coordinate is on y-axis we have coordinate as (0,10) and (6,10). Suff

but if the distance of 10 is b/w origin and diagonally opposite vertices then we get different coordinates. hence statement 1 becomes insufficient

stmnt2 - Same reasoning as above makes statement 2 as insufficient.

taking together we can conclude that distance of 10 is b/w origin and diagonally opposite vertices and distance of 8 lies on y axis. So we can find the values of the other vertices. hence together they are sufficient.

A rectangle is plotted on the standard coordinate plane, with vertices at the origin and (0,6). If the coordinates of all vertices of the rectangle are non-negative integers, what are the coordinates of the other two vertices?

(1) The distance between the origin and one of the other vertices is 10 units.

(2) The distance between the origin and one of the other vertices is 8 units.

What if one of the statements were: (1) The distance between the origin and one of the other vertices is 6 units.
_________________

What if one of the statements were: (1) The distance between the origin and one of the other vertices is 6 units.

Well in that case St 1 gives us no new information, since we know its a rectangle, we already know one side is of 6 units, the other side is either bigger or smaller than 6. So St 1 is insuff. We already know St 2 is insuff. But this time both together are also insuff since we don't know what 8 represents (the diagonal or one of the perpendicular sides).

What if one of the statements were: (1) The distance between the origin and one of the other vertices is 6 units.

Well in that case St 1 gives us no new information, since we know its a rectangle, we already know one side is of 6 units, the other side is either bigger or smaller than 6. So St 1 is insuff. We already know St 2 is insuff. But this time both together are also insuff since we don't know what 8 represents (the diagonal or one of the perpendicular sides).

Is my analysis valid?

Actually, st1 is sufficient in this case.

A square is a specialized rectangle. And it's only 6 units away from origin, thus it is directly above it.
_________________

A rectangle is plotted on the standard coordinate plane, with vertices at the origin and (0,6). If the coordinates of all vertices of the rectangle are non-negative integers, what are the coordinates of the other two vertices?

(1) The distance between the origin and one of the other vertices is 10 units.

(2) The distance between the origin and one of the other vertices is 8 units.

What if one of the statements were: (1) The distance between the origin and one of the other vertices is 6 units.

You could interpret that statement in two different ways, so I think the question becomes ambiguous. That statement is only sufficient if by 'one of the other vertices' you mean 'one of the vertices not mentioned in the question'. In that case, we must have a square, which is of course a type of rectangle. But, if by 'one of the other vertices' you mean 'one of the other vertices of the rectangle' (which is how I'd interpret it), then it's not sufficient; we already know that (0,0) and (0,6) are six apart.
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A rectangle is plotted on the standard coordinate plane, with vertices at the origin and (0,6). If the coordinates of all vertices of the rectangle are non-negative integers, what are the coordinates of the other two vertices?

(1) The distance between the origin and one of the other vertices is 10 units.

(2) The distance between the origin and one of the other vertices is 8 units.

What if one of the statements were: (1) The distance between the origin and one of the other vertices is 6 units.

You could interpret that statement in two different ways, so I think the question becomes ambiguous. That statement is only sufficient if by 'one of the other vertices' you mean 'one of the vertices not mentioned in the question'. In that case, we must have a square, which is of course a type of rectangle. But, if by 'one of the other vertices' you mean 'one of the other vertices of the rectangle' (which is how I'd interpret it), then it's not sufficient; we already know that (0,0) and (0,6) are six apart.

Good that pointed out the ambiguity.

I was just trying to create a situation where knowing one of the vertices of the rectangle that doesn't lie on the x-axis can still suffice to answer the question.

So, I rephrase my statement. (1) The distance between the origin and one of the rectangle's other vertices that don't lie on the x-axis is 6 units.
_________________

[ I was just trying to create a situation where knowing one of the vertices of the rectangle that doesn't lie on the x-axis can still suffice to answer the question.

I think it's a more interesting question that way, and a more GMAT-like one as well - if one of the statements turns out to be sufficient alone. If, in your rephrased statement, you make the distance any number less than 6, then there would be no ambiguity and the statement would be sufficient alone, since then the distance cannot possibly be the length of the diagonal.
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If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

Hhmm I think C, statement 1 and 2 could both refer to either the diagonal distance between the vertices or just the sides.. Combined, 10 is bigger so it must be the diagonal leaving 8 for the sides, from there you can figure out the coordinates..

Re: A rectangle is plotted on the standard coordinate plane, [#permalink]

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23 Jun 2012, 21:58

Am I right in stating that when the question means that the distance between one of the vertices and origin, it may be construed as that distance between the origin and diagonal/ length of the side...??

Because, then C makes sense and diagonally opposite would be 10, 6 and the other would be 8,0.

Re: A rectangle is plotted on the standard coordinate plane [#permalink]

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17 Aug 2013, 15:05

Bunuel wrote:

Answer to this question is B, not C.

A rectangle is plotted on the standard coordinate plane, with vertices at the origin and (0,6). If the coordinates of all vertices of the rectangle are non-negative integers, what are the coordinates of the other two vertices?

Notice that we are told that the coordinates of all vertices of the rectangle are non-negative integers. That basically means that the rectangle is plotted entirely in the first quadrant: one vertex at the origin (0, 0), another on Y-axis (0, 6), third one somewhere in the first quadrant (x, 6) and the fourth vertex on the X-axis (x, 0).

(1) The distance between the origin and one of the other vertices is 10 units. 10 units is either the length of the other side or the length of the diagonal. The vertices could be: (0, 0), (0, 6), (10, 6) and (10, 0) OR (0, 0), (0, 6), (8, 6) and (8, 0), in this case the distance of 10 units is the distance between (0, 0) and (6, 8). Not sufficient.

(2) The distance between the origin and one of the other vertices is 8 units. Now, 8 units can not be the length of the diagonal since in this case the coordinates of the other two vertices won't be integers: if the diagonal=8, then the length of the other side of the rectangle is \(\sqrt{8^2-6^2}=2\sqrt{7}\), so the coordinates of the other two vertices are: \((2\sqrt{7}, \ 6)\) and \((2\sqrt{7}, \ 0)\). So, 8 units must be the length of the side, therefore the vertices are (0, 0), (0, 6), (8, 6) and (8, 0). Sufficient.

Answer: B.

Hope it's clear.

What if you rotate the rectangle by few degrees upwards in the first quadrant but let the length of the two sides remain 6 and 8 respectively? The coordinates will remain positive but will be different from what you have listed above.

A rectangle is plotted on the standard coordinate plane, with vertices at the origin and (0,6). If the coordinates of all vertices of the rectangle are non-negative integers, what are the coordinates of the other two vertices?

Notice that we are told that the coordinates of all vertices of the rectangle are non-negative integers. That basically means that the rectangle is plotted entirely in the first quadrant: one vertex at the origin (0, 0), another on Y-axis (0, 6), third one somewhere in the first quadrant (x, 6) and the fourth vertex on the X-axis (x, 0).

(1) The distance between the origin and one of the other vertices is 10 units. 10 units is either the length of the other side or the length of the diagonal. The vertices could be: (0, 0), (0, 6), (10, 6) and (10, 0) OR (0, 0), (0, 6), (8, 6) and (8, 0), in this case the distance of 10 units is the distance between (0, 0) and (6, 8). Not sufficient.

(2) The distance between the origin and one of the other vertices is 8 units. Now, 8 units can not be the length of the diagonal since in this case the coordinates of the other two vertices won't be integers: if the diagonal=8, then the length of the other side of the rectangle is \(\sqrt{8^2-6^2}=2\sqrt{7}\), so the coordinates of the other two vertices are: \((2\sqrt{7}, \ 6)\) and \((2\sqrt{7}, \ 0)\). So, 8 units must be the length of the side, therefore the vertices are (0, 0), (0, 6), (8, 6) and (8, 0). Sufficient.

Answer: B.

Hope it's clear.

What if you rotate the rectangle by few degrees upwards in the first quadrant but let the length of the two sides remain 6 and 8 respectively? The coordinates will remain positive but will be different from what you have listed above.

Am I right?

Will the figure still be a rectangle then?
_________________

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