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

Question

Capture.GIF 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 (1, 4), what are the coordinates of point C? A. (-4, -1) B. (-1, 4) C. (4, -1) D. (1, -4) E. (4, 1)

Bunuel · Accepted Answer

Since the line y=x is the perpendicular bisector of segment AB, then the point B is the mirror reflection of point A around the line y=x, so its coordinates are (4, 1). 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 x-axis, so its coordinates are (4, -1). Answer: C. The question becomes much easier if you just draw rough sketch of the diagram: graph.png Now, you can simply see that options A, B, and D (blue dots) just can not be the right answers. As for option E: point (4, 1) coincides with point B, so it's also not the correct answer. Only answer choice C remains. Answer: C. Similar questions to practice: in-the-xy-coordinate-plane-is-point-r-equidistant-from-143502.html in-the-rectangular-coordinate-system-the-line-y-x-is-the-132646.html the-coordinates-of-points-a-and-c-are-0-3-and-127769.html the-line-represented-by-the-equation-y-4-2x-is-the-127770.html if-point-a-coordinates-are-7-3-point-b-coordinates-a-141972.html in-the-rectangular-coordinate-system-the-line-y-x-is-the-88473.html in-the-rectangular-coordinate-system-above-the-line-y-x-144774.html the-line-represented-by-the-equation-y-4-2x-is-the-perpendi-87573.html Hope it helps.

EvaJager · Answer

The mirror image of  (x,y)  around the Y-axis is  (-x,y) .

For the second question:
Assume we have a point P(a,b) and we want to find its mirror image around the line  y = -x . 
Let's denote the point we seek by Q(A,B). See the attached drawing.

The equation of the line passing through P and perpendicular to the line  y = -x  is  y - b = x - a , or  y = x + b - a .
Since Q is also on this line, we have  B = A + b - a , from which   A - B = a - b  .
The middle point of the line segment PQ (denoted by M) is also on the line  y = -x , therefore  \frac{a+A}{2}=-\frac{b+B}{2} , or   A + B = -a - b  .
Solving for A and B, we find that   A = -b, B =-a  .

Therefore, the mirror image of  (x,y)  around the line  y = -x  is  (-y, -x) .

teal · Answer

What is the mirror image of a point (x,y) around Y-axis?

Also what is the mirror image of a point (x,y) around line y=-x?

mbaiseasy · Answer

For me, the best approach to this question is to draw and estimate AB and BC lines. Doing so obviously shows that: 
(a) y has negative coordinates... Thus, eliminate B and E.
(b) x has positive coordinates beyond 2. Thus, eliminate A and D.

Answer: C

Or, if you want to be really sure... We can get the line perpendicular x=y.
(a) get negative reciprocal of slope of y=x which is m=1. Thus, reciprocal is m=-1. Perpendicular line: y=-x + b
(b) calculate b using A coordinates: y = -x + b ==> 4 = -1 + b ==> b = 5
(c) get the pt. of intersection. -x + 5 = x ==> x = 2.5
(d) get y=-(2.5) + 5 --> y = 2.5

So, obviously... C would have negative for y coordinate and x > 2.5... only C fits the bill

Answer: C

But still, drawing should suffice...

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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mau5 · Answer

Let the co-ordinates of B = (p,q). As x=y is the perpendicular bisector of the line segment AB, thus the middle point of AB will lie on x=y itself. Thus, for A(1,4) and B(p,q)-->

\frac {p+1} {2} = \frac {q+4} {2}  --> p-q = 3

Also, the slope of the line segment would be -1-->  \frac {q-4}{p-1} = -1  -->  p+q = 5 .Thus, on solving, the co-ordinates of B (4,1).

Similarly, as y=0(the x-axis) is the perpendicular bisector of BC, thus, the mid point of BC would like on the x-axis and thus, the y co-ordinate of C has to be -1. Thus, only A and C survive. Again, the slope of line segment BC has to be undefined as it is parallel to the y-axis(or perpendicular to the x axis).

Only option C survives.

C.

Narenn · Answer

The line Y = X always makes 45 deg angle with the X axis and has slope 1

Since the line Y = X is perpendicular to line AB, The slope of AB must be -1 --------

Hereinafter, Even if we are not familiar with 'mirror image' concept we can draft the following figure and can check the answer options.

https://lh4.googleusercontent.com/-xfj7XL52xHE/Uc2AA-3HNUI/AAAAAAAAAIg/_hI3H4q3eXY/s578/co-od%2520figure%25202.png

Since X axis itself is bisector of line BC we can deduce that X value of C can not be negative. Eliminate A, B

We also know Y value of C can not be positive. Eliminate E

Now consider the point A(1,4) This point is on the line that has slope -1, so the X value of its opposite end (i.e. X of B and C also) must be greater than 1. So Choice D (1, -4) Can not be the location of C. Eliminate.

Only option left is C, which is the Answer.

vietnammba · Answer

Hi I have one question. What if the question ask to find the mirror image of (x,y) around the line y = 2x + 3 for example, how to solve it?

ScottTargetTestPrep · Answer

This problem does contain a diagram that looks like the following:

https://i.imgur.com/6JhukFx.png

We are given that the line y = x is a perpendicular bisector of line segment AB. This indicates that point B is a reflection of point A across the line y = x. When we reflect a point (a,b) across the line y = x, the reflected point has the coordinates reversed: (b,a). Thus, since point A is at (2,3), point B must be (3,2).
 
We are next given that the x-axis is a perpendicular bisector of line segment BC. This means that point C must have the same x-coordinate as point B (3) but the opposite y-coordinate of point B (-2). To further elaborate, we can draw a diagram.

https://i.imgur.com/ClEqonc.png

The answer is D.

NaeemHasan · Answer

Hi  Bunuel ,
Is it always true that a perpendicular bisector will have mirror reflection of the ends?

abhimahna · Answer

Yes, it is always true.

BloomingLotus · Answer

I took the approach of eliminating the options.

Since X-axis is perpendicular bisector of line BC, it means that B will line in I quadrant and C in IV quadrant . This means for point C x-cordinate will be +ve and y-co-ordinate will be negative. With this finding , remove options A,B and E

Now as per option D if point C is (1,-4) then point B will be (1,+4). But (1,4) is co-ordinate of point A.

So,correct answer is C.

babloorudra · Answer

Hello Buenel, I am a bit confused between Mirror reflection and when the angles are shifted by 90*. As per this question: https://gmatclub.com/forum/in-the-figur ... 85154.html when 2 lines are perpendicular, the 2 coordinates are same but in reverse order. But in this question, the coodinates are same but the sign of y changes. Can you please explain where I am going wrong

Vijay_K · Answer

I know this is hard to accept but you can solve it without taking your pen.

The coordinates of point C should lie on the IV Quadrant  since it is given that the x-axis is the perpendicular bisector of segment BC . From that we can eliminate A,B and E and it leads to option C and D. We know that point A is (1,4) and B should be right side of point A becoz of perp bisector, so answer shouldn't be D(1,-4). That makes our final answer as C

Why D is not the answer?  
When you try to draw randomly B on the plane,  you have to mark the point B on the right side since it is perpendicular bisector to the line y=x and also we know that B doesn't lie on the point where x co-ordinate value is 1 for the same reason . From that we know, x co-ordinate of B should be greater than 1, which means the value can be 2,3,4,5.... Now check option D in which the x co-ordinate is 1. so we can eliminate D.

You may ask how did you find point C's x-value from point B's x-value? In question stem, we are given that B and C are perp bisector to x axis which means their x value is same.

I hope the answer is clear now. Happy to help!

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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