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In the rectangular coordinate system above, the line y = x [#permalink]
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31 Mar 2012, 02:54
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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 xaxis 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)
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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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31 Mar 2012, 04:26
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enigma123 wrote: In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the xaxis 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 xaxis is the perpendicular bisector of segment BC then the point C is the mirror reflection of point B around the xaxis, 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 28687 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. Answer: C. Similar questions to practice: inthexycoordinateplaneispointrequidistantfrom143502.htmlintherectangularcoordinatesystemthelineyxisthe132646.htmlthecoordinatesofpointsaandcare03and127769.htmlthelinerepresentedbytheequationy42xisthe127770.htmlifpointacoordinatesare73pointbcoordinatesa141972.htmlintherectangularcoordinatesystemthelineyxisthe88473.htmlintherectangularcoordinatesystemabovethelineyx144774.htmlthelinerepresentedbytheequationy42xistheperpendi87573.htmlHope it helps.
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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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01 Aug 2012, 05:04
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What is the mirror image of a point (x,y) around Yaxis?
Also what is the mirror image of a point (x,y) around line y=x?



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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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01 Aug 2012, 06:56
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teal wrote: What is the mirror image of a point (x,y) around Yaxis? Also what is the mirror image of a point (x,y) around line y=x? The mirror image of \((x,y)\) around the Yaxis 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 \((y, x)\).
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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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02 Jan 2013, 05:09
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enigma123 wrote: In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the xaxis 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...
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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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27 Jun 2013, 23:47



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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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28 Jun 2013, 00:31
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enigma123 wrote: In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the xaxis 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 coordinates 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)> \(\frac {p+1} {2} = \frac {q+4} {2}\) > pq = 3 Also, the slope of the line segment would be 1> \(\frac {q4}{p1} = 1\) > \(p+q = 5\).Thus, on solving, the coordinates of B (4,1). Similarly, as y=0(the xaxis) is the perpendicular bisector of BC, thus, the mid point of BC would like on the xaxis and thus, the y coordinate 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 yaxis(or perpendicular to the x axis). Only option C survives. C.
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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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28 Jun 2013, 05:36
Bunuel wrote: Bumping for review and further discussion*. Get a kudos point for an alternative solution! *New project from GMAT Club!!! Check HEREThe 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.
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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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29 Nov 2014, 21:36
EvaJager wrote: teal wrote: What is the mirror image of a point (x,y) around Yaxis? Also what is the mirror image of a point (x,y) around line y=x? The mirror image of \((x,y)\) around the Yaxis 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?



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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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13 Jul 2016, 07:37
enigma123 wrote: Attachment: Capture.GIF In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the xaxis 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 xaxis is a perpendicular bisector of line segment BC. This means that point C must have the same xcoordinate as point B (3) but the opposite ycoordinate of point B (2). To further elaborate, we can draw a diagram. The answer is D.
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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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22 Oct 2016, 06:06
Hi Bunuel, Is it always true that a perpendicular bisector will have mirror reflection of the ends?



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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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22 Oct 2016, 06:26
NaeemHasan wrote: Hi Bunuel, Is it always true that a perpendicular bisector will have mirror reflection of the ends? Yes, it is always true.
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Re: In the rectangular coordinate system above, the line y = x [#permalink]
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07 Sep 2017, 10:40
I took the approach of eliminating the options.
Since Xaxis 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 xcordinate will be +ve and ycoordinate 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 coordinate of point A.
So,correct answer is C.




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