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In the rectangular coordinate system above, the line y = x

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

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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)
[Reveal] Spoiler: OA
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In the rectangular coordinate system above, the line y = x [#permalink]

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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.

Answer: D.

The question becomes much easier if you just draw a rough sketch:
Attachment:
Reflection2.png
Reflection2.png [ 10.75 KiB | Viewed 8886 times ]
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.
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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New post 30 Oct 2013, 06:25
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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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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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New post 16 Feb 2015, 11:03
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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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New post 16 Feb 2015, 11:12
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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New post 16 Feb 2015, 20:27
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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New post 17 Feb 2015, 20:06
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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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In the rectangular coordinate system above, the line y = x [#permalink]

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New post 16 Jul 2015, 06:01
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)
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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New post 28 Sep 2015, 04:45
Bunuel wrote:
Image
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
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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New post 28 Sep 2015, 21:01
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kunal555 wrote:
Bunuel wrote:
Image
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/
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Re: In the rectangular coordinate system above, the line y = x [#permalink]

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New post 25 May 2016, 05:14
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] 25 May 2016, 05:14
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