A rectangle is plotted on the standard coordinate plane

Question

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.

Bunuel · Accepted Answer

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):

Attachment:

Rectangle.png [ 5.05 KiB | Viewed 20617 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.

Answer: B.

Hope it's clear.

kp1811 · Answer

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.

Ans C

fluke · Answer

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

microair · Answer

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?

fluke · Answer

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.

IanStewart · Answer

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.

fluke · Answer

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.

IanStewart · Answer

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.

keenys · Answer

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?

Bunuel · Answer

Will the figure still be a  rectangle  then?

abhisheksharma · Answer

Bunuel, if the co-ordinates are 2root7 thn wht is the harm in it ??

Posted from my mobile device

Bunuel · Answer

Stem says that the coordinates of all vertices of the rectangle are non-negative    integers   .

nariskat · Answer

I think it should be c as 'non-negative integers' means it can be either negative non integer or positive integers. Please correct me if I'm wrong. Thanks!

Bunuel · Answer

Non-negative integers mean  integers  more than or equal to zero: 0, 1, 2, ...

Hope it's clear.

vad3tha · Answer

B is sufficient. The distance from (0,0) to other vertices is 8. So can be point (0,8) or the distance can be the the diagonal length of 8. Then the other length of the rectangle will be \sqrt{( 8^2 -6^2)} = 2\sqrt{7}. And 2\sqrt{7} is not an integer, there is left with one choice. (0,8)

abullet · Answer

I think in either statement you have a distance that can be the x coordinate you need or the diagonal. With 10 you can have (10, 0) or a diagonal 10 in which case with the Pythagorean theorem the coordinates would be (8,0). This is possible so you don't now the solution.
If the distance is 8 you can have (8,0) or a diagonal that gives coordinates (square root of 28, 0) which is not an integer. So in this case the only possibility is for (8,0). Therefore Stat 2 is suf.

bankerboy30 · Answer

Bunuel can you explain why the rectangle can't be rotated a few degrees?

Bunuel · Answer

Because we re told that the coordinates of all vertices of the rectangle are non-negative integers. If you rotate it this condition would be violated. I think the image here a-rectangle-is-plotted-on-the-standard-coordinate-plane-115697.html#p1098927 should help to visualize this. Hope it helps.

shallow9323 · Answer

The question states that it is a rectangle, so if u look at statement 2. 
 8 units has to be the length along the x plain because for it to be a square a min length of 6*1.4 =8.4 is needed.  
If the length is lower that 8.4, then rectangle would have lower part move into the 4th quadrant, hence 8 has to be along the x axis and not
the diagonal length.

and statement 1 is not sufficient as its is more than 8.4 and hence the rectangle can be formed if it is a the diagonal or the horizontal length.

A rectangle is plotted on the standard coordinate plane

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

A rectangle is plotted on the standard coordinate plane

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards