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

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.

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

[quote="enigma123"]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

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"

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 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.

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

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.

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

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

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