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In the rectangular coordinate plane points X and Z lie on the same lin

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In the rectangular coordinate plane points X and Z lie on the same lin [#permalink]

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In the rectangular coordinate plane points X and Z lie on the same line through the origin and points W and Y lie on the same line through the origin. If a^2 + b^2 = c^2 + d^2 and e^2 + f^2 = g^2 + h^2, what is the value of length XZ – length WY?

A. -2
B. -1
C. 0
D. 1
E. 2

[Reveal] Spoiler:
For me the answer should be C ZERO. This is how I arrived to D. Please let me know whether this is correct or not as I don't have an OA.

Distance from all the 4 points from origin can be written as

\(\sqrt{a^2 +b^2}\) + \(\sqrt{e^2 + j^2}\) = \(\sqrt{c^2 + d^2}\) + \(\sqrt{g^2 + h^2}\)

The above will give the answer of zero if we substitute the values from question stem.

Attachment:
Distance.PNG
Distance.PNG [ 4.74 KiB | Viewed 8407 times ]
[Reveal] Spoiler: OA

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In the rectangular coordinate plane points X and Z lie on the same lin [#permalink]

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In the rectangular coordinate plane points X and Z lie on the same line through the origin and points W and Y lie on the same line through the origin. If a^2 + b^2 = c^2 + d^2 and e^2 + f^2 = g^2 + h^2, what is the value of length XZ – length WY?

A. -2
B. -1
C. 0
D. 1
E. 2

Since X and Z lie on the same line through the origin then the distance between X and Z will be equal to the sum of the individual distances of X and Z from the origin: \(\sqrt{c^2 + d^2}+\sqrt{g^2 + h^2}\);

The same way, the distance between W and Y will be equal to the sum of the individual distances of W and Y from the origin: \(\sqrt{a^2 + b^2}+\sqrt{e^2 + f^2}\);

\(XZ-WY=(\sqrt{c^2 + d^2}+\sqrt{g^2 + h^2})-(\sqrt{a^2 + b^2}+\sqrt{e^2 + f^2})=0\).

Answer: C.
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Re: In the rectangular coordinate plane points X and Z lie on the same lin [#permalink]

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New post 24 Feb 2012, 00:24
A quick question: All we know is that the line passes through X, origin, and Z vs. the second line passes through W, origin, and Y. There is no indication that the points are equidistant with respect to the origin. Can we assume this or is there a part of the wording from the original question missing?

The way I approached it:
sqrt ((g-c)^2+(h-d)^2) = sqrt ((a-e)^2+ (b-f)^2)
This simplifies to gc+hd = ae +bf.

Please explain.
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Re: In the rectangular coordinate plane points X and Z lie on the same lin [#permalink]

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mourinhogmat1 wrote:
A quick question: Nowhere in the question does it say that the two points are equidistant right? How can we say that the distance from origins are same? Please explain.


The formula to calculate the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\). Now, if one point is origin, coordinate (0, 0), then the formula can be simplified to: \(D=\sqrt{x^2+y^2}\).

Hence for our original question: a^2+b^2=c^2+d^2 means that points X and W are equidistant from the origin and e^2+f^2=g^2+h^2 means that points Y and Z are equidistant from the origin.

Next, since X and Z lie on the same line through the origin and W and Y lie on the same line through the origin then the distance of line segments XZ and WY is equal (for algebraic proof see above post).

Hope it's clear.
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Re: In the rectangular coordinate plane points X and Z lie on the same lin [#permalink]

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enigma123 wrote:
Attachment:
Distance.PNG
In the rectangular coordinate plane points X and Z lie on the same line through the origin and points W and Y lie on the same line through the origin. If a^2+b^2=c^2+d^2 and e^2+f^2=g^2+h^2, what is the value of length XZ – length WY?

A. -2
B. -1
C. 0
D. 1
E. 2

[Reveal] Spoiler:
For me the answer should be C ZERO. This is how I arrived to D. Please let me know whether this is correct or not as I don't have an OA.

Distance from all the 4 points from origin can be written as

\(\sqrt{a^2 +b^2}\) + \(\sqrt{e^2 + j^2}\) = \(\sqrt{c^2 + d^2}\) + \(\sqrt{g^2 + h^2}\)

The above will give the answer of zero if we substitute the values from question stem.



Another way to solve this is if I draw a line segment from origin to point W (say w) and origin to point X (say x) will be hypotenuse defined by \(\sqrt{a^2 +b^2}\)= \(\sqrt{w^2}\)
and \(\sqrt{c^2 +d^2}\)= \(\sqrt{x^2}\)

So you will end up with w=x and y=z --> (x+z) -(y+z) =0
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Re: In the rectangular coordinate plane points X and Z lie on the same lin [#permalink]

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New post 01 Apr 2017, 08:59
[quote="enigma123"]
Attachment:
Distance.PNG
In the rectangular coordinate plane points X and Z lie on the same line through the origin and points W and Y lie on the same line through the origin. If a^2+b^2=c^2+d^2 and e^2+f^2=g^2+h^2, what is the value of length XZ – length WY?

A. -2
B. -1
C. 0
D. 1
E. 2

Both the length will be same ( by applying Pythagoras theorem) hence length xz- length wy = 0
hence C
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Re: In the rectangular coordinate plane points X and Z lie on the same lin [#permalink]

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New post 17 Jun 2017, 04:45
enigma123 wrote:
Attachment:
The attachment Distance.PNG is no longer available
In the rectangular coordinate plane points X and Z lie on the same line through the origin and points W and Y lie on the same line through the origin. If a^2+b^2=c^2+d^2 and e^2+f^2=g^2+h^2, what is the value of length XZ – length WY?

A. -2
B. -1
C. 0
D. 1
E. 2

[Reveal] Spoiler:
For me the answer should be C ZERO. This is how I arrived to D. Please let me know whether this is correct or not as I don't have an OA.

Distance from all the 4 points from origin can be written as

\(\sqrt{a^2 +b^2}\) + \(\sqrt{e^2 + j^2}\) = \(\sqrt{c^2 + d^2}\) + \(\sqrt{g^2 + h^2}\)

The above will give the answer of zero if we substitute the values from question stem.


Any set of values can be plugged into this question in order to calculate the result- what this question is basically saying is that absolute value of the x and y coordinates of a and b must be the same as absolute value of the x and y coordinates of c and d; the same holds true for coordinates (e,f) and (g,h).

( l a l , l b l ) = ( l c l , l d l ) ;
( l e l , l f l ) = ( l g l, l h l )

But lastly, this question asks us for the difference between the length of the two diagonals- the two diagonals have the same length so the answer is 0

Hence
"C"
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Re: In the rectangular coordinate plane points X and Z lie on the same lin [#permalink]

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New post 11 Feb 2018, 06:44
Bunuel wrote:
Attachment:
Distance.PNG
In the rectangular coordinate plane points X and Z lie on the same line through the origin and points W and Y lie on the same line through the origin. If a^2+b^2=c^2+d^2 and e^2+f^2=g^2+h^2, what is the value of length XZ – length WY?

A. -2
B. -1
C. 0
D. 1
E. 2

Since X and Z lie on the same line through the origin then the distance between X and Z will be equal to the sum of the individual distances of X and Z from the origin: \(\sqrt{c^2 + d^2}+\sqrt{g^2 + h^2}\);

The same way, the distance between W and Y will be equal to the sum of the individual distances of W and Y from the origin: \(\sqrt{a^2 + b^2}+\sqrt{e^2 + f^2}\);

\(XZ-WY=(\sqrt{c^2 + d^2}+\sqrt{g^2 + h^2})-(\sqrt{a^2 + b^2}+\sqrt{e^2 + f^2})=0\).

Answer: C.



Hello Bunuel :-)
if both lines lie through point of origin, that means their coordinates are (0;0) Right ? BUT we still dont know the LENGTHS of these two lines? How can you assume the lines are of the same lengths :? I would appreciate your explanation
Thanks ! :)
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Re: In the rectangular coordinate plane points X and Z lie on the same lin [#permalink]

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New post 11 Feb 2018, 07:53
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dave13 wrote:
Bunuel wrote:
Attachment:
Distance.PNG
In the rectangular coordinate plane points X and Z lie on the same line through the origin and points W and Y lie on the same line through the origin. If a^2+b^2=c^2+d^2 and e^2+f^2=g^2+h^2, what is the value of length XZ – length WY?

A. -2
B. -1
C. 0
D. 1
E. 2

Since X and Z lie on the same line through the origin then the distance between X and Z will be equal to the sum of the individual distances of X and Z from the origin: \(\sqrt{c^2 + d^2}+\sqrt{g^2 + h^2}\);

The same way, the distance between W and Y will be equal to the sum of the individual distances of W and Y from the origin: \(\sqrt{a^2 + b^2}+\sqrt{e^2 + f^2}\);

\(XZ-WY=(\sqrt{c^2 + d^2}+\sqrt{g^2 + h^2})-(\sqrt{a^2 + b^2}+\sqrt{e^2 + f^2})=0\).

Answer: C.



Hello Bunuel :-)
if both lines lie through point of origin, that means their coordinates are (0;0) Right ? BUT we still dont know the LENGTHS of these two lines? How can you assume the lines are of the same lengths :? I would appreciate your explanation
Thanks ! :)


The coordinates of X and Z are not (0, 0). X is in the second quadrant and Y is in fourth quadrant.

Points X and Z lie on the same line through the origin means that X, Y and the origin lie on the same straight line.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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In the rectangular coordinate plane points X and Z lie on the same lin [#permalink]

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New post 14 Feb 2018, 10:44
Bunuel wrote:
dave13 wrote:
Bunuel wrote:
Image
In the rectangular coordinate plane points X and Z lie on the same line through the origin and points W and Y lie on the same line through the origin. If a^2+b^2=c^2+d^2 and e^2+f^2=g^2+h^2, what is the value of length XZ – length WY?

A. -2
B. -1
C. 0
D. 1
E. 2

Since X and Z lie on the same line through the origin then the distance between X and Z will be equal to the sum of the individual distances of X and Z from the origin: \(\sqrt{c^2 + d^2}+\sqrt{g^2 + h^2}\);

The same way, the distance between W and Y will be equal to the sum of the individual distances of W and Y from the origin: \(\sqrt{a^2 + b^2}+\sqrt{e^2 + f^2}\);

\(XZ-WY=(\sqrt{c^2 + d^2}+\sqrt{g^2 + h^2})-(\sqrt{a^2 + b^2}+\sqrt{e^2 + f^2})=0\).

Answer: C.



Hello Bunuel :-)
if both lines lie through point of origin, that means their coordinates are (0;0) Right ? BUT we still dont know the LENGTHS of these two lines? How can you assume the lines are of the same lengths :? I would appreciate your explanation
Thanks ! :)


The coordinates of X and Z are not (0, 0). X is in the second quadrant and Y is in fourth quadrant.

Points X and Z lie on the same line through the origin means that X, Y and the origin lie on the same straight line.



H Bunuel

say X(-3;4) ; W (3;4) and Y(-3;-4), Z(3;-4)

So as per formula of distance between two points i get

D = XW = \(\sqrt{(3-(-3))^2 + (4-4)^2}\) = 3
D = YZ = \(\sqrt{(3-(-3)^2+(-4-(-4))^2}\)= 3

XW -YZ = 3-3 =0
Is my understabding correct now ? :)

thanks!
In the rectangular coordinate plane points X and Z lie on the same lin   [#permalink] 14 Feb 2018, 10:44
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