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

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?

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...

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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.

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 --------[The product of the slopes of a line and its perpendicular is always -1]

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



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.

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?

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



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.



The answer is D.

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

Yes, it is always true.

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.

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

VeritasKarishma Bunuel or any other expert

please can you explain the concept of mirror reflection

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