In the xy-plane, the points (c, d), (c, -d), and (-c, -d) are three ve

Hi, there. I'm happy to help with this.

The question: In the xy-plane, the points (c, d), (c, -d), and (-c, -d) are three vertices of a certain square. If c < 0 and d > 0, which of the following points is in the same quadrant as the fourth vertex of the square?

I marked the tricky part in red. It seems c is a negative number and d is a positive number. This means

Vertex #1 = (c, d) is in QII (that is, negative x and positive y)
Vertex #2 = (c, -d) is in QIII (that is, both x & y negative)
Vertex #3 = (-c, -d) is in QIV (that is y is negative, but x is positive)

That means the last vertex should be in the first quadrant --- the only first quadrant point is (5, 3), answer = E.

Does that make sense? Please let me know if you have any questions on this.

Mike
_________________

As the three vertices are (c, d), (c, -d), and (-c, -d) then the fourth will be (-c, d), the only combination which is left. Now, as c < 0 and d > 0, then \(-c=-negative=positive\) and \(d=positive\). Thus the fourth vertex \((-c, d)=(positive, \ positive)\) is in the I quadrant, only answer choice E offers I quadrant point: (3, 5).

Answer: E.
_________________

Your logic was fine.. You figured that after (c, d), (c, -d), and (-c, -d), the fourth vertex would be (-c, d). Perfect!
The only problem was the additional info that said that 'c' is negative and 'd' is positive. This means that in (-c, d), both (-c) and d will be positive! Hence the OA is (E) i.e. the one in which both, the x and the y co-ordinate, are positive.

From the question since this is given If c < 0 and d > 0 .... you can do c = -1 and d = 1

Then map the coordinates for all three points (c,d) = (-1, 1) , (c, -d) = (-1, -1) and (-c, -d) = (1,-1) .....

Well if you draw all these three points in the plane you will realise that the missing point has positive x, y so E

Since c < 0 and d > 0:

(c, d) = (neg, pos) = quadrant II

(c, -d) = (neg, neg) = quadrant III

(-c, -d) = (pos, neg) = quadrant IV

Thus, the 4th vertex is in quadrant I and has a point that is (pos, pos). Thus, choice E is correct, since it’s the only point that is (pos, pos).

Answer: E
_________________

Check the diagram below:

The fourth vertex is at (-c, d), which is (positive, positive), so in the first quadrant.

Hope it helps.
_________________

Although I understood the problem but is it possible for someone who can draw the vertices in a picture and present it. Visually still it is difficult for me to understand, how the -c,d simply lying in the quadrant II.
_________________

Hi All,

You might find it helpful to TEST VALUES.

We're told that C < 0 and D > 0.

Let's TEST
C = -2
D = +2

We're given 3 of 4 co-ordinates for a square....
(C,D) = (-2, +2) which is in the 2nd Quadrant
(C, -D) = (-2, -2) which is in the 3rd Quadrant
(-C, -D) = (+2, -2) which is in the 4th Quadrant

Since we're dealing with a square, the 4th co-ordinate will be in 1st Quadrant...There's only one answer that's also in the 1st Quadrant...

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Simply test values for c = -1 and d = 1 (per constraints of the question)
Since all the sides are equal, the point must lie in the first quadrant and EVERY point in this quadrant is in the form (x,y) where x>0 and y > 0.

E is the only answer choice that aligns.

Hi Mihiri,

We're told that C < 0 and D > 0... so before you do anything to those values (such as multiply them by "-1"), you have to make sure that you make sure that your C and D fit those facts first.

To help you with how all of this applies to this question, you might find it helpful to TEST VALUES.

Let's TEST
C = -2
D = +2

We're given 3 of 4 co-ordinates for a square....
(C,D) = (-2, +2) which is in the 2nd Quadrant
(C, -D) = (-2, -2) which is in the 3rd Quadrant
(-C, -D) = (+2, -2) which is in the 4th Quadrant

Since we're dealing with a square, the 4th co-ordinate will be in 1st Quadrant...There's only one answer that's also in the 1st Quadrant...

GMAT assassins aren't born, they're made,
Rich

Hi EMPOWERgmatRichC

A quick question.
When the question stem says, "which of the following points is in the same quadrant as the fourth vertex of the square?".

If one were to write this in the full from does it mean "which of the following points is in the same quadrant as the fourth vertex of the square is in?
In simple words, a point that lies in the same quadrant as the fourth vertex of the square lies in. However, that point doesn't necessarily have to be the vertex of the square. It just have to lie on that specific quadrant.
I ask because I often find the wording of the GMAT to be very tricky.

Thank you.

Hi gurudabl,

This prompt gives us 3 of the 4 vertices of a SQUARE, so it makes sense that we would asked something about the 4th vertex of that square. From the answer choices, we can see 5 different co-ordinates that appear in various quadrants of the xy-plane; we're asked which of those answers is in the SAME quadrant as the 4th vertex of the square.

To answer your immediate question: NO, the points that are referenced in the 5 answers are not necessarily a vertex of the square. Considering how the answer choices are written though, you can actually prove that they can't be on the square. If you plug in the respective values for C and D from any of the Answers into the 3 vertices that we're given, you'll end up with 3 points of a RECTANGLE - not a SQUARE).

GMAT assassins aren't born, they're made,
Rich

This makes complete sense but I thought the answer choices also follow the C<0 and D>0 conditions. Hence I went with B wherein (-*-5,3)= (5,3).

\(Given\) \(c<0, \ d>0\)

So,
\((c,d)= (-c,d)\) will go to the quadrant \(II\)

\((c,d)= (-c,-d)\) will to quadrant \(III\)

\((-c, -d)= (c,-d)\) will go to the quadrant \(IV\)

Thus to be a square the last point will go to the quadrant \(I\)

Only \((3,5)\) will go to the quadrant \(I\)

The answer is \(E.\)

Take an example in such cases. You know that the fourth vertex would be (-c, d).
Now since c is negative, c can be say -1 and since d is positive, d can be 1. So the vertex becomes (-(-1), 1) i.e. (1, 1).

Very Tricky Question. Thanks for the explanations Mike/Bunuel.

I almost answered B, because it would match the quadrant; however, when you look at the final piece to the pattern; that is, (-c, d) you realize -5 can not be -c, because
-(-5) = 5, which contradicts the given "c<0"

After understanding what the question asks, The first thing you should do is list the givens:
(c,d)
(c,-d)
(-c,-d)
(???) missing fourth point not given.
Analyzing the points given, you see a pattern of c and d both positive, one positive and one negative, and both negative, the only pattern left is one negative and one positive.
(c,-d) is already given; therefore the missing fourth point is (-c, d)

(-c,d) cannot be B, because, again, when you plug -5 into (-c,d) you get (-(-5), d) which contradicts "c<0"
Now, B and A are both eliminated. You know it cannot be an answer where d is a negative number, so C and D are also eliminated.

Therefore, the only remaining answer is E

Bottomline: You should be able to recognize the pattern, and thereafter eliminate answer choices .

Hi,

I am a bit confused here to understand how did you come across, Vertex #2 = (c, -d) is in QIII (that is, both x & y negative) while you are saying Vertex #1 = (c, d) is in QII (that is, negative x and positive y)

Because, for D to become negative, the original value of D should be a positive value with the condition given d > 0.

So do we have to consider the original value of the points or the value after multiplying +/-?

Can you please explain this.

mihiri

Hi babloorudra,

When working through a GMAT question, you have to be sure to answer the question that is ASKED. You correctly determined that the missing co-ordinate would be in the first Quadrant, but the question that was asked did NOT instruct you to 'modify' any of the answer choices - it merely asked which of the 5 answers was in the SAME Quadrant. With certain GMAT questions, you'll find that one (or more) of the wrong answers is the result of a little mistake that someone might make - so that's all the more reason to take lots of notes, be precise with your work and make sure that you've actually answered the specific question that was asked.

GMAT assassins aren't born, they're made,
Rich