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

Question

https://gmatclub.com/forum/download/file.php?id=16635 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 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. Distance.PNG

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=16635

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.

mourinhogmat1 · Answer

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.

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.

Bunuel · Answer

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.

mal208213 · Answer

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

daboo343 · Answer

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

Nunuboy1994 · Answer

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"

dave13 · Answer

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 !

Bunuel · Answer

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 .

dave13 · Answer

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!

DSBlender · Answer

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

First off, if you are given a point (X,Y) on a coordinate plane, you can make a right triangle from the origin. In this problem, we are asked to find the difference of the length of two lines. We can use the Pythagorean Theorom to find the hypotenuse (distance from the origin) of each point on the coordinate plane. Since we know that a^2 + b^2 = c^ + d^2 and e^2 + f^2 = g^2 + h^2, we know that the distance from the origin is the same for the points in Q1 (W) and Q2 (X), and the distance from the origin is the same for the points in Q3 (Y) and Q4 (Z). This problem is essentially asking us the difference in lengths between XZ and WY. We don't need to know exact lengths, but for instance, if X and Z are 2 from the orign, and W and Y and 1 from the origin, the difference between XZ and WY is equal to 0. This is because no matter what the actual points on the coordinate plane are, we know that each line is the same length, because each line has the same distance from the origin. Answer C is correct.

saby1410 · Answer

Bunuel   VeritasKarishma  
 why can't we find length using distance formula directly between points X and Z. why find distance form X to origin ?

KarishmaB · Answer

We don't have any equations giving us relations with (c - g)^2, (d - h)^2 etc. Since all equations have distance from origin, those are what we need to consider.

NamaySharma · Answer

This is my solution.
So a^2 + b^2 = c^2 + d^2 and e^2 + f^2 = g^2 + h^2
We square root on both sides for each equation and we get
Distance from origin for X = Distance from origin for W--------(1)
Distance from origin for Z = Distance from origin for Y---------(2)

Add One and Two and we get:
D of XZ = D of WY
Therefore when subtracted the answer = 0

NamaySharma · Answer

We don't hav the values. Thus this is the approach we use.
So a^2 + b^2 = c^2 + d^2 and e^2 + f^2 = g^2 + h^2
We square root on both sides for each equation and we get
Distance from origin for X = Distance from origin for W--------(1)
Distance from origin for Z = Distance from origin for Y---------(2)

Add One and Two and we get:
D of XZ = D of WY
Therefore when subtracted the answer = 0

bumpbot · Answer

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

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