In the rectangular coordinate system, are the points (r,s) : DS Archive
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# In the rectangular coordinate system, are the points (r,s)

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In the rectangular coordinate system, are the points (r,s) [#permalink]

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04 May 2008, 20:33
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In the rectangular coordinate system, are the points (r,s) and (u,v) equidistant from the origin?
(1) r + s = 1
(2) u = 1 - r and v = 1 - s

The answer in the book says C, but I don't know why the answer isn't B. Please explain. I can't seem to find a number that would make the points equidistant from the origin in B.
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04 May 2008, 20:45
To find the distance (d) from the origin to the point (u,v), d^2 = u^2 + v^2

(1) doesn't provide detail about point (u,v)
(2) d^2 = (1-r)^2 + (1-s)^2
= 1 - 2r + r^2 + 1 -2s + s^2
= r^2 + s^2 + 2 - 2(r + s)

At this point, we don't know enough about r and s to simplify even more but with statement 1 of r + s = 1

d^2 = r^2 + s^2 + 2 - 2(1) = r^2 + s^2
therefore, points (r,s) and (u,v) are equidistant from the origin.

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04 May 2008, 21:59
Please tell me a value for u and v that would allow both to be equidistant from the origin.
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05 May 2008, 10:43
japped187 wrote:
Please tell me a value for u and v that would allow both to be equidistant from the origin.

There are many choices: for example
(r,s) = (-1, 2), (u,v) = (2,-1)
(r,s) = (4, -3), (u,v) = (-3, 4)

The pattern seems to be that point (r,s) and point (u,v) are equidistant from the origin as long as (u,v) = (s, r)
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06 May 2008, 08:15
japped187 wrote:
In the rectangular coordinate system, are the points (r,s) and (u,v) equidistant from the origin?
(1) r + s = 1
(2) u = 1 - r and v = 1 - s

The answer in the book says C, but I don't know why the answer isn't B. Please explain. I can't seem to find a number that would make the points equidistant from the origin in B.

We are looking for whether the (r,s) and (u,v) share the same x-plane, y-plane or diagonal but equal distance.
- if same x-plane, then r = u and v = -s
- if same y-plane, then s = v and u = -r
- if diagonally, then (u,v) = (-s, -r)

Statement (1) alone gives us r + s = 1. Rearrange this to r = 1 - s. We are not given what (u,v). Not sufficient. Eliminate A and D.

Statement (2) alone gives us u = 1 - r and v = 1 - s. Replace (u,v) with (1-r, 1-s). Plot this with (r,s) from stimulus. Not sufficient. Why? Because the vital clue is "rectangular coordinate system". As explained above, the coordinates must be on same x-plane, y-plane or diagonal but equal distance. Eliminate B.

Both Statement (1) and (2) together. Recall in Statement (1), we rearrange to r = 1 - s. Replace in (r,s) will give you (1-s,s). In Statement (2), we have the replacement of (u,v), which is (1-r,1-s).

Now replace r = 1-s in (1-r). You get (1-(1-s)), equals to 1-1+s equals s. Thus, (1-r,1-s) becomes (s,1-s).

We now have (1-s,s) and (s, 1-s) which show that both coordinates are equal distance but opposite of each other. Replace s with any value and you get a diagonal line between the coordinates and equidistance. Eliminate E.

Ans: C
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Jimmy Low, Frankfurt, Germany
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Re: OG DS 140   [#permalink] 06 May 2008, 08:15
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