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

Question

https://gmatclub.com/forum/download/file.php?id=19056 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) Reflcetion.png

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=19056 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: https://gmatclub.com/forum/download/file.php?id=19057 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. Reflection2.png

jalpagandhi · Answer

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.

cherryli2015 · Answer

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

KarishmaB · Answer

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)

BrainLab · Answer

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)

kunal555 · Answer

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

KarishmaB · Answer

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/

shasadou · Answer

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.

AkshdeepS · Answer

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?

Bunuel · Answer

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.

sach24x7 · Answer

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

Posted from my mobile device

adkikani · Answer

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?

KarishmaB · Answer

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).

napolean92728 · Answer

Well, I prefer taking the approach of reflection here as y=x is perp. bisector of AB therefore , point B is obtained by interchanging x and y co-ordinates of A. Hence, B : ( 3 , 2 ), Since X-axis ( y = 0 ) is perp. bisector of BC therefore C is obtained by keeping x co-ordinate of C same as B and negating y-co-ordinate of B. Hence C : ( 3, -2 ) . Therefore Ans is D. ( 3, -2 )

siddhantvarma · Answer

Bunuel   KarishmaB  Can a question like this be expected in GMAT FE?

Bunuel · Answer

I've never seen any coordinate geometry questions like that in GMAT Prep mocks or the new Official Guides, so it's unlikely you'll encounter them on the test.

sujoykrdatta · Answer

Note:
 
1. When you reflect a point about the line y=x, the coordinates swap, so (a,b) becomes (b,a)

2. When you reflect a point about the line y=-x, the coordinates swap and negate, so (a,b) becomes (-b,-a)

3. When you reflect a point about the line y=0 (X axis) the y-coordinate is negated, so (a,b) becomes (a,-b)

4. When you reflect a point about the line x=0 (Y axis) the x-coordinate is negated, so (a,b) becomes (-a,b)

The reflection of A (2,3) about the line y=x is the point B (3,2)
The reflection of B (3,2) about the X-axis is the point C (3,-2)

Ans D

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

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

Prep Toolkit

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