In the rectangular coordinate system above, the line y = x : GMAT Problem Solving (PS)
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# In the rectangular coordinate system above, the line y = x

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27 Dec 2012, 04:05
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In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the x-axis is the perpendicular bisector of segment BC (not shown). If the coordinates of point A are (2,3), what are the coordinates of point C ?

(A) (-3,-2)
(B) (-3,2)
(C) (2,-3)
(D) (3,-2)
(E) (2,3)
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In the rectangular coordinate system above, the line y = x [#permalink]

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27 Dec 2012, 04:13
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In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the x-axis is the perpendicular bisector of segment BC (not shown). If the coordinates of point A are (2,3), what are the coordinates of point C ?

(A) (-3,-2)
(B) (-3,2)
(C) (2,-3)
(D) (3,-2)
(E) (2,3)

Since the line y=x is the perpendicular bisector of segment AB, then point B is the mirror reflection of point A around the line y=x, so its coordinates are (3, 2). In any mirror reflection around the line y = x, the x-coordinate and the y-coordinate of a point become interchanged.

The same way, since the x-axis is the perpendicular bisector of segment BC then the point C is the mirror reflection of point B around the y-axis, so its coordinates are 3, -2). In any mirror reflection around the x-axis, the x-coordinate remains the same, and the sign of the y-coordinate changes.

The question becomes much easier if you just draw a rough sketch:
Attachment:

Reflection2.png [ 10.75 KiB | Viewed 9972 times ]
Now, you can simply see that only D can be the correct answer.

Similar questions to practice:
in-the-rectangular-coordinate-system-the-line-y-x-is-the-132646.html
in-the-rectangular-coordinate-system-above-the-line-y-x-129932.html

Hope it helps.
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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30 Oct 2013, 05:25
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From question stem, we can know that, Point C will be in the 4th quadrant (X,-Y), which comes down to option C&D.
B is perpendicular to A (2,3). Hence co-ordinates of B will be (3,2).
C & B are parallel to Y-axis and are on the same line. Hence co-ordinates of C will share same X-co-ordinate. (3,-Y). i.e. Ans. choice D

*I have difficult time understanding Co-ordiante geometry, hence try to solve in the simple way. This was my approach and was able to get ans. in less than a Minute.
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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09 Dec 2014, 09:46
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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16 Feb 2015, 10:03
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The same way, since the y-axis is the perpendicular bisector of segment BC then the point C is the mirror reflection of point B around the y-axis, so its coordinates are (-3, 2)

Hi Bunuel,

Isn't x-axis a perpendicular bisector of line BC and the coordinates of C are (3,-2). Looks like some typos in the explanation?

Thanks,
AJ
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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16 Feb 2015, 10:12
aj0809 wrote:
Quote:
The same way, since the y-axis is the perpendicular bisector of segment BC then the point C is the mirror reflection of point B around the y-axis, so its coordinates are (-3, 2)

Hi Bunuel,

Isn't x-axis a perpendicular bisector of line BC and the coordinates of C are (3,-2). Looks like some typos in the explanation?

Thanks,
AJ

Yes. Edited the typo. Thank you.
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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16 Feb 2015, 19:27
For this question, I am confused. where do I get the position of B and C?
Thanks ...
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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17 Feb 2015, 19:06
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cherryli2015 wrote:
For this question, I am confused. where do I get the position of B and C?
Thanks ...

First of all, this question is not very easy. You should be able to visualize a concept which is not very intuitive to most of us but there are a few OG questions on it. I suggest you to read up on it in the following two posts:
http://www.veritasprep.com/blog/2013/04 ... ry-part-i/
http://www.veritasprep.com/blog/2013/04 ... y-part-ii/

Now, coming back to this question, you are given data to find the positions of B and C.

"the line y = x is the perpendicular bisector of segment AB (not shown)"

You know that point A is (2, 3). We need the mirror image of A on y = x. Imagine drawing a line perpendicular to y = x from A. It will intersect y = x at (2.5, 2.5), a point 0.5 below and 0.5 to the right. So the point B will be 0.5 further to the right and 0.5 down giving us the coordinate (2.5 + .5, 2.5 - .5) i.e. (3, 2).

For C, you are given that
" the x-axis is the perpendicular bisector of segment BC (not shown)"

A line perpendicular to x axis will be x = 0 i.e. vertical line. So C's x coordinate will be the same as B's x coordinate. Since B is 2 above the x axis, C will be 2 below the x axis. So C will be at (3, -2)
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Director Joined: 10 Mar 2013 Posts: 608 Location: Germany Concentration: Finance, Entrepreneurship GMAT 1: 580 Q46 V24 GPA: 3.88 WE: Information Technology (Consulting) Followers: 15 Kudos [?]: 266 [0], given: 200 In the rectangular coordinate system above, the line y = x [#permalink] ### Show Tags 16 Jul 2015, 05:01 YX is bisector of AB --> AP=PB, A(2,3) and B(3,2). X axis is bisector of BC means BX=XC, BX=2, XC=-2 --> C(3,-2) (D) Attachments PS202.png [ 10.08 KiB | Viewed 4617 times ] _________________ When you’re up, your friends know who you are. When you’re down, you know who your friends are. Share some Kudos, if my posts help you. Thank you ! 800Score ONLY QUANT CAT1 51, CAT2 50, CAT3 50 GMAT PREP 670 MGMAT CAT 630 KAPLAN CAT 660 Manager Joined: 29 Jul 2015 Posts: 161 Followers: 0 Kudos [?]: 132 [0], given: 59 Re: In the rectangular coordinate system above, the line y = x [#permalink] ### Show Tags 28 Sep 2015, 03:45 Bunuel wrote: In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the x-axis is the perpendicular bisector of segment BC (not shown). If the coordinates of point A are (2,3), what are the coordinates of point C ? (A) (-3,-2) (B) (-3,2) (C) (2,-3) (D) (3,-2) (E) (2,3) Since the line y=x is the perpendicular bisector of segment AB, then point B is the mirror reflection of point A around the line y=x, so its coordinates are (3, 2). In any mirror reflection around the line y = x, the x-coordinate and the y-coordinate of a point become interchanged. The same way, since the x-axis is the perpendicular bisector of segment BC then the point C is the mirror reflection of point B around the y-axis, so its coordinates are 3, -2). In any mirror reflection around the x-axis, the x-coordinate remains the same, and the sign of the y-coordinate changes. Answer: D. The question becomes much easier if you just draw a rough sketch: Attachment: Reflection2.png Now, you can simply see that only D can be the correct answer. Answer: D. Similar questions to practice: in-the-rectangular-coordinate-system-the-line-y-x-is-the-132646.html in-the-rectangular-coordinate-system-above-the-line-y-x-129932.html Hope it helps. Hi Bunuel I solved this question with slope intercept form and took more than 2 minutes. Can you please explain how did you conclude mirror reflection thing. It would be really helpful if you could provide theory or link to theory which explains the concept of points and their mirror reflection with respect to lines, axes and quadrants. Thanks Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7125 Location: Pune, India Followers: 2137 Kudos [?]: 13674 [2] , given: 222 Re: In the rectangular coordinate system above, the line y = x [#permalink] ### Show Tags 28 Sep 2015, 20:01 2 This post received KUDOS Expert's post kunal555 wrote: Bunuel wrote: In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the x-axis is the perpendicular bisector of segment BC (not shown). If the coordinates of point A are (2,3), what are the coordinates of point C ? (A) (-3,-2) (B) (-3,2) (C) (2,-3) (D) (3,-2) (E) (2,3) Since the line y=x is the perpendicular bisector of segment AB, then point B is the mirror reflection of point A around the line y=x, so its coordinates are (3, 2). In any mirror reflection around the line y = x, the x-coordinate and the y-coordinate of a point become interchanged. The same way, since the x-axis is the perpendicular bisector of segment BC then the point C is the mirror reflection of point B around the y-axis, so its coordinates are 3, -2). In any mirror reflection around the x-axis, the x-coordinate remains the same, and the sign of the y-coordinate changes. Answer: D. The question becomes much easier if you just draw a rough sketch: Attachment: Reflection2.png Now, you can simply see that only D can be the correct answer. Answer: D. Similar questions to practice: in-the-rectangular-coordinate-system-the-line-y-x-is-the-132646.html in-the-rectangular-coordinate-system-above-the-line-y-x-129932.html Hope it helps. Hi Bunuel I solved this question with slope intercept form and took more than 2 minutes. Can you please explain how did you conclude mirror reflection thing. It would be really helpful if you could provide theory or link to theory which explains the concept of points and their mirror reflection with respect to lines, axes and quadrants. Thanks Here are 3 posts which explain this concept: http://www.veritasprep.com/blog/2013/04 ... ry-part-i/ http://www.veritasprep.com/blog/2013/04 ... y-part-ii/ http://www.veritasprep.com/blog/2013/05 ... -part-iii/ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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25 May 2016, 04:14
after a bit of sketching you can easily reason out the solution. u understand that C lies in quadrant IV hence its coordinates are positive x and negative y. You kick A and B and then you kick out E. Between C and D: point C lies a bit further than point A hence they cannot share the same x coordinate. Then C out.
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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21 Nov 2016, 11:11
Bunuel wrote:

In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the x-axis is the perpendicular bisector of segment BC (not shown). If the coordinates of point A are (2,3), what are the coordinates of point C ?

(A) (-3,-2)
(B) (-3,2)
(C) (2,-3)
(D) (3,-2)
(E) (2,3)

Since the line y=x is the perpendicular bisector of segment AB, then point B is the mirror reflection of point A around the line y=x, so its coordinates are (3, 2). In any mirror reflection around the line y = x, the x-coordinate and the y-coordinate of a point become interchanged.

The same way, since the x-axis is the perpendicular bisector of segment BC then the point C is the mirror reflection of point B around the y-axis, so its coordinates are 3, -2). In any mirror reflection around the x-axis, the x-coordinate remains the same, and the sign of the y-coordinate changes.

The question becomes much easier if you just draw a rough sketch:
Attachment:
Reflection2.png
Now, you can simply see that only D can be the correct answer.

Similar questions to practice:
in-the-rectangular-coordinate-system-the-line-y-x-is-the-132646.html
in-the-rectangular-coordinate-system-above-the-line-y-x-129932.html

Hope it helps.

Bunuel, thanks for such an elegant solution
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Re: In the rectangular coordinate system above, the line y = x   [#permalink] 21 Nov 2016, 11:11
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