In the xy-plane, the origin O is the midpoint of line segment PQ. If t

Question

In the xy-plane, the origin O is the midpoint of line segment PQ. If the coordinates of P are (r,s), what are the coordinates of Q ?

A. (r,s)
 B. (s,–r)
 C. (–s,–r)
 D. (–r,s)
 E. (–r,–s)

Bunuel · Accepted Answer

pushpitkc · Answer

Given that one of the points on the line segment(PQ) is P(r,s) and the midpoint is the origin(0,0)
We need to find out the coordinates of the other point Q

Midpoint of 2 points(x1,y1) and (x2,y2) is ((x1+x2)/2,(y1+y2)/2)

Since we have one point P(r,s) x1=r and midpoint is (0,0)
r+x2/2 = 0. This is possible when x2=-r
s+y2/2 = 0. This is possible when y2=-s

Hence point Q has coordinates (-r,-s) which is Option E

sashiim20 · Answer

Given coordinates of P are (r,s) and origin O is midpoint of line segment PQ. Therefore Line PQ falls in I and III quadrants.

Both x and y coordinates will be have same value but negative sign.

Therefore coordinates of Q would be (-r,-s). Answer (E)...

Bunuel · Answer

This can be solved by assuming values. If P(r, s) = P(2, 2) then Q must be at the other end of the diagonal through the origin, so at (-2, -2), which is (-r, -s).

Answer: E.

brandmanrocks · Answer

Hi there. Theoretically, what if point P were on the y-axis. If O is at the origin then wouldn't the other endpoint Q be (-r, s)? For example, if P is (3,0) then Q would be (-3,0). In this case, (r,s) could be (-r,s). This reason I bring this up is that I can find a case (line PQ runs along the axis) in which point Q may not necessarily be equal to (-r,-s) if one of the end points are on an axis. Thanks!

Bunuel · Answer

If P is (3, 0), then yes Q will be (-3, 0) but it's still (-r, -s) because 0 = - 0.

ScottTargetTestPrep · Answer

We can let the coordinates of Q be (x, y). Since the origin (0, 0) is the midpoint of PQ, we have:

(r + x)/2 = 0 and (s + y)/2 = 0

Multiplying each equation by 2, we have:

r + x = 0 and s + y = 0

x = -r and y = -s

Thus, the coordinates of Q are (-r, -s).

Answer: E

EMPOWERgmatRichC · Answer

Hi All,

We're told that the origin O is the MIDPOINT of line segment PQ and that the coordinates of P are (r,s). We're asked for the coordinates of Q. This question can be solved by TESTing VALUES.

IF... Point P = (r,s) = (2,1)... 
then Point Q would be (-2, -1). That equates to (-r,-s).

Final Answer:  E

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

sadikabid27 · Answer

Bunuel  could you please refer to some similar questions.

ngmat12 · Answer

If you draw a point (r,s) in the first quadrant (top-right) of the cartesian plane, you will automatically see that the "opposite" point will be (-r,-s) which makes O the midpoint.

mottagmat · Answer

Sorry I still don't get it.. Why 0 = -0?

LordStark · Answer

all real numbers can be divided into three categories: positive, negative and zero.
check out this  link 
thanks

BrentGMATPrepNow · Answer

Let's sketch a line segment with midpoint at the origin and an endpoint at (r, s) 
 https://i.imgur.com/BxhZmIu.png

We can now travel the SAME DISTANCES (from the origin) to ensure that the origin is the midpoint. 
 https://i.imgur.com/GfUnXHm.png

From here, we can see that (-r, -s) are the coordinates of the other point. 
 https://i.imgur.com/Whm6vfZ.png

Answer: E

Cheers,
Brent

prabsahi · Answer

Hello Brent,

I understand precisely the graph you showed.I had assumed the same.

However,I understand that segment of line PQ can be a simple x axis as well.

_______P(-r,-s)__________________O____________________Q(r,s).

and in this case option (D) since s=0 will always be true.

Upon reflecting further, I see the line PQ can be rotated across Origin and so ycoordinated doesnt necessarily have to be zero.

GMAT sets so beautiful traps .

Please let me know if you see any gap in my thinking

BrentGMATPrepNow · Answer

You're correct to say that the answer COULD be D (in that particular case), but the question is asking for the answer that  must  be true (in all cases).

Cheers,
Brent

prabsahi · Answer

Thanks for confirming ..I guess that was one of the trap and purpose of this question

unraveled · Answer

Note here that point P has coordinate (r,s) where, since both are variables, following possibilities arise: 1) r > 0, s > 0 Ex. r = 2, s = 2 for P1 So Q1 is -r = -2, -s = -2 2) r > 0, s < 0 Ex. r = 2, s = -2 for P2 So Q2 is -r = -2, -s = 2 3) r < 0, s > 0 Ex. r = -2, s = 2 for P3 So Q3 is -r = 2, -s = -2 4) r < 0, s < 0 Ex. r = -2, s = -2 for P4 So Q4 is -r = 2, -s = 2 So you can assume any of that and solve for the point Q which would be exactly opposite to midpoint(0,0). See that midpoint(0,0) is the mirror here. Solving graphically we have PQ line segment.png Answer E.

GMATinsight · Answer

Answer: Option E

Video solution by  GMATinsight

https://www.youtube.com/watch?v=vkeE1GAo2IM

Zabudamous · Answer

Think of the coordinates as:
QI: (+,+)
QII: (-,+)
QIII: (-,-)
QIV:(+,-)

p(r, s) is in QI:(+,+) going the same distance in the opposite direction of the origin would land us in QIII:(-,-) 
so the answer is

*Drawing out the coordinates and labeling them helps.

In the xy-plane, the origin O is the midpoint of line segment PQ. If t

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In the xy-plane, the origin O is the midpoint of line segment PQ. If t

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