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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] New post 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] New post 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.

Answer: D.

The question becomes much easier if you just draw a rough sketch:
Attachment:
Reflection2.png
Reflection2.png [ 10.75 KiB | Viewed 5528 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] New post 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] New post 09 Dec 2014, 09:46
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Re: In the rectangular coordinate system above, the line y = x [#permalink] New post 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] New post 16 Feb 2015, 10:12
Expert's post
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] New post 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] New post 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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In the rectangular coordinate system above, the line y = x [#permalink] New post 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)
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