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# 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  [#permalink]

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17 Jun 2017, 09:47
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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)
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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23 Mar 2018, 05:29
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In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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17 Jun 2017, 10:02
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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
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In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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17 Jun 2017, 09:51
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AbdurRakib wrote:
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)

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)...
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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17 Jun 2017, 12:46
3
AbdurRakib wrote:
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)

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).

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

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19 Jul 2017, 10:39
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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!
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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19 Jul 2017, 20:55
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brandmanrocks wrote:
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!

If P is (3, 0), then yes Q will be (-3, 0) but it's still (-r, -s) because 0 = - 0.
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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15 Nov 2017, 15:53
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AbdurRakib wrote:
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)

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).

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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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20 Nov 2017, 17:03
2
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).

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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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23 Mar 2018, 03:18
1
Bunuel could you please refer to some similar questions.
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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18 Oct 2018, 04:27
AbdurRakib wrote:
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)

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.
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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27 Feb 2019, 06:02
Bunuel wrote:
brandmanrocks wrote:
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!

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

Sorry I still don't get it.. Why 0 = -0?
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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01 Mar 2019, 17:21
mottagmat wrote:
Bunuel wrote:
brandmanrocks wrote:
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!

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

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

all real numbers can be divided into three categories: positive, negative and zero.
thanks
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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16 Jun 2019, 08:24
3
Top Contributor
AbdurRakib wrote:
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)

Let's sketch a line segment with midpoint at the origin and an endpoint at (r, s)

We can now travel the SAME DISTANCES (from the origin) to ensure that the origin is the midpoint.

From here, we can see that (-r, -s) are the coordinates of the other point.

Cheers,
Brent
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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19 Jul 2019, 07:58
GMATPrepNow wrote:
AbdurRakib wrote:
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)

Let's sketch a line segment with midpoint at the origin and an endpoint at (r, s)

We can now travel the SAME DISTANCES (from the origin) to ensure that the origin is the midpoint.

From here, we can see that (-r, -s) are the coordinates of the other point.

Cheers,
Brent

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
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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19 Jul 2019, 08:06
Top Contributor
prabsahi wrote:
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

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
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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19 Jul 2019, 08:09
GMATPrepNow wrote:
prabsahi wrote:
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

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

Thanks for confirming ..I guess that was one of the trap and purpose of this question
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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t  [#permalink]

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06 Mar 2020, 21:44
AbdurRakib wrote:
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)

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
Attachment:

PQ line segment.png [ 114.43 KiB | Viewed 2915 times ]

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Re: In the xy-plane, the origin O is the midpoint of line segment PQ. If t   [#permalink] 06 Mar 2020, 21:44