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

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

27 Dec 2012, 05:05

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C

D

E

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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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Bunuel

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

27 Dec 2012, 05:13

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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:

Now, you can simply see that only D can be the correct answer.

Answer: D.

Similar questions to practice:
https://gmatclub.com/forum/in-the-rectan ... 32646.html
https://gmatclub.com/forum/in-the-rectan ... 29932.html

Hope it helps.

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

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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KarishmaB

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

Updated on: 17 Oct 2022, 01:45

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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.

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)

Originally posted by KarishmaB on 17 Feb 2015, 20:06.
Last edited by KarishmaB on 17 Oct 2022, 01:45, edited 1 time in total.

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KarishmaB

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

28 Sep 2015, 21:01

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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:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2013/04 ... ry-part-i/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2013/04 ... y-part-ii/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2013/05 ... -part-iii/

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KarishmaB

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

18 May 2020, 23:34

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adkikani wrote:

Hi VeritasKarishma

For this blog, could you explain how did you deduce a 30-60-90 triangle from only knowing that radius of circle / hypotenuse = 2? The length of minute clock is fixed, so if I join center to number 9 and center to number 12, both lengths of both these segments will be the same. Right?

When the minute hand is at 9 to when it goes to 12, it sweeps a 90 degree angle. This means that from 9 to 10, it sweeps 30 degrees, from 10 to 11, it sweeps another 30 degrees and from 11 to 12, it sweeps yet another 30 degrees - making a total of 90 degrees.
Hence, from 9 to 10, the angle would be 30 degrees, making the shown triangles 30-60-90. Since hypotenuse is 2, the other two sides must be 1 and sqrt(3).

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

16 Jul 2015, 06:01

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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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kunal555

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

28 Sep 2015, 04:45

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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

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

11 Apr 2018, 05:53

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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:

Now, you can simply see that only D can be the correct answer.

Answer: D.

Similar questions to practice:
https://gmatclub.com/forum/in-the-rectan ... 32646.html
https://gmatclub.com/forum/in-the-rectan ... 29932.html

Hope it helps.

Bunuel, why we didn't extend line AB more to have some other co-ordinates for B? Say, if AB cuts line xy at point P perpendicularly, then is it necessary that AP = PB?
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Bunuel

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

11 Apr 2018, 06:02

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QZ 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:

Now, you can simply see that only D can be the correct answer.

Answer: D.

Similar questions to practice:
https://gmatclub.com/forum/in-the-rectan ... 32646.html
https://gmatclub.com/forum/in-the-rectan ... 29932.html

Hope it helps.

Bunuel, why we didn't extend line AB more to have some other co-ordinates for B? Say, if AB cuts line xy at point P perpendicularly, then is it necessary that AP = PB?

A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°.

So, if you extend AB, y = x will no longer be a bisector of segment AB.
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cherryli2015

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

16 Feb 2015, 20:27

For this question, I am confused. where do I get the position of B and C?
Thanks ...

S

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

21 Apr 2019, 05:38

VeritasKarishma wrote:

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:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2013/04 ... ry-part-i/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/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)

Hi, i could not understand adding up 0.5 part. Can you please explain in detail?

Posted from my mobile device

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adkikani

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

15 May 2020, 22:20

Hi VeritasKarishma

For this blog, could you explain how did you deduce a 30-60-90 triangle from only knowing that radius of circle / hypotenuse = 2? The length of minute clock is fixed, so if I join center to number 9 and center to number 12, both lengths of both these segments will be the same. Right?
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bones2020

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

25 May 2020, 15:17

One of these problems where looking at the answer choices before tackling the problem is really helpful and efficient.

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