GMATPrep DS equidisant q

[aimhigh715]

In the rectangular coordinate system, are the points (r,s) and (u,v) equidistant from origin?

1) r+S = 1
2) u = 1-r and v = 1-s

[Tud]

I choose C)

Given the short amount of time to answer I just threw some numbers in.

Knowing where (r,s) is doesn't tell us anything about where (u,v) is, so 1) isn't enough on it's own. And statement 2) doesn't rule out using (0,0) for (r,s), which would make them not an equidistance, but (-2,3) for (r,s) would make (u,v) = (3,-2), which would make them equidistant, so 2 alone is insuff.

Continuing to plug stuff in, I selected some basic number to get a range for (r,s). I used (1/3,2/3), (-4,5) and (0,1), since all of there meet the r+s=1 equation.

After pluging these in to get (u,v) I ended up with the following:

(r,s) (u,v)
(1/3,2/3) (2/3,1/3) Equidistant
(-4,5) (5,-4) Equidistant
(0,1) (-1,0) Equidistant

I'd say that both together are suff.. Not terribly mathamatical. Someone else might have a better mathematical explanation.

[gmatguy16]

distance of (u,v) from origin = u^2+v^2..
and distance of (r,s) from origin =r^2+s^2

now u^2+v^2 = (1-r)^2+(1-s)^2
=r^2+s^2-2(r+s)+2
=r^2+s^2 (if r+s=2)..
hence the answer