Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

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

Show Tags

31 Mar 2012, 01:54

2

This post received KUDOS

24

This post was BOOKMARKED

00:00

A

B

C

D

E

Difficulty:

35% (medium)

Question Stats:

68% (01:09) correct 32% (01:35) wrong based on 772 sessions

HideShow timer Statistics

Attachment:

Capture.GIF [ 3.28 KiB | Viewed 25754 times ]

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)

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)

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:

Attachment:

graph.png [ 12.57 KiB | Viewed 25751 times ]

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.

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

Show Tags

01 Aug 2012, 05:56

3

This post received KUDOS

4

This post was BOOKMARKED

teal wrote:

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?

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 \((-y, -x)\).

Attachments

MirrorImage.jpg [ 18.1 KiB | Viewed 25020 times ]

_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

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

Show Tags

02 Jan 2013, 04:09

1

This post received KUDOS

enigma123 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 (1, 4), what are the coordinates of point C?

A. (-4, -1) B. (-1, 4) C. (4, -1) D. (1, -4) E. (4, 1)

Any idea guys how to solve this mathematically?

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...
_________________

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

Show Tags

27 Jun 2013, 23:31

2

This post received KUDOS

2

This post was BOOKMARKED

enigma123 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 (1, 4), what are the coordinates of point C?

A. (-4, -1) B. (-1, 4) C. (4, -1) D. (1, -4) E. (4, 1)

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).

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.
_________________

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

Show Tags

29 Nov 2014, 20:36

EvaJager wrote:

teal wrote:

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?

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 \((-y, -x)\).

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?

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)

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.
_________________

Scott Woodbury-Stewart Founder and CEO

GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions

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

Show Tags

07 Sep 2017, 09:40

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.