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

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18 Feb 2012, 16:28

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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?

Re: In the rectangular coordinate plane points X and Z lie on [#permalink]

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18 Feb 2012, 17:17

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Distance.PNG [ 4.74 KiB | Viewed 4348 times ]

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: \(\sqrt{a^2 + b^2}+\sqrt{e^2 + f^2}\);

Re: In the rectangular coordinate plane points X and Z lie on [#permalink]

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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.

Re: In the rectangular coordinate plane points X and Z lie on [#permalink]

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24 Feb 2012, 01:15

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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).

Re: In the rectangular coordinate plane points X and Z lie on [#permalink]

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30 May 2013, 20:29

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?

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

Re: In the rectangular coordinate plane points X and Z lie on [#permalink]

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29 Oct 2014, 08:28

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Re: In the rectangular coordinate plane points X and Z lie on [#permalink]

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15 Nov 2015, 14:27

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