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

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

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03 Sep 2009, 02:04
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Difficulty:

55% (hard)

Question Stats:

64% (01:05) correct 36% (01:01) wrong based on 198 sessions

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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/in-the-recta ... 92823.html
[Reveal] Spoiler: OA

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

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03 Sep 2009, 02:13
Most discussed GPrep Q

you need to prove

r^2 + s^2 = u^2 + v^2

hence you need both the (1) and (2)

CasperMonday wrote:

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

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03 Sep 2009, 02:17
My solution is:

the distance of the first point from the origin is $$r^2+s^2$$
the distance of the second point is $$u^2+v^2$$
the question is actually asking is $$r^2+s^2=u^2+v^2$$?

(1) $$r+s=1$$
apparently, insufficient. we know nothing about $$u$$ and $$v$$

(2) $$u=1-r$$ and $$v=1-s$$
might be sufficient but we should check by substituting for $$u$$, $$v$$ in the original equation:
$$r^2+s^2=(1-r)^2+(1-s)^2$$
by simplifying the equation we get:
$$r^2+s^2=r^2+s^2-2(r+s)+2$$
this doesn't allow us to make any conclusions. hence, insufficient

by combining (1) and (2), we get that
$$(1-s)^2+s^2=((1-s)^2+s^2-2((1-s)+s)+2$$
simplifying,
$$1-2s+2s^2=1-2s+2s^2$$

but i am wondering if there are other approaches to this problem. thank for contributions.

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

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03 Sep 2009, 02:19
nitya34 wrote:
Most discussed GPrep Q

you need to prove

r^2 + s^2 = u^2 + v^2

hence you need both the (1) and (2)

i was typing my answer right when you posted yours. as i said i am interested in other methods to solve the prob.))

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

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03 Sep 2009, 02:40
same strategy as yours with slight modification

we need to prove r^2+s^2=u^2+v^2

now r^2+s^2=(1-u)^2 + (1-v)^2= 2-2(u+v)+(u^2+v^2)

Now from (1) and (2) u+v=1+1-(r+s) = 2-1=1

hence r^2+s^2=u^2+v^2
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Director
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Re: In the rectangular coordinate system, are the points (r,s) [#permalink]

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01 Oct 2009, 12:17
How can you say $$r^2+s^2$$. it must be sqrt of ($$r^2+s^2$$)........thank god I got it after lot of efforts

CasperMonday wrote:
My solution is:

the distance of the first point from the origin is $$r^2+s^2$$
the distance of the second point is $$u^2+v^2$$
the question is actually asking is $$r^2+s^2=u^2+v^2$$?

(1) $$r+s=1$$
apparently, insufficient. we know nothing about $$u$$ and $$v$$

(2) $$u=1-r$$ and $$v=1-s$$
might be sufficient but we should check by substituting for $$u$$, $$v$$ in the original equation:
$$r^2+s^2=(1-r)^2+(1-s)^2$$
by simplifying the equation we get:
$$r^2+s^2=r^2+s^2-2(r+s)+2$$
this doesn't allow us to make any conclusions. hence, insufficient

by combining (1) and (2), we get that
$$(1-s)^2+s^2=((1-s)^2+s^2-2((1-s)+s)+2$$
simplifying,
$$1-2s+2s^2=1-2s+2s^2$$

but i am wondering if there are other approaches to this problem. thank for contributions.

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

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02 Oct 2009, 07:13

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

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07 Sep 2010, 05:51
3
KUDOS
mrsmarthi wrote:

Hey guyz

I am not master of quant either verbal:) but i did this question really quick.Maybe lucky.

a) R+S=1 (no info about U and V then insufficient)

b) u+r=1, v+s=1 not sufficent

Together

U+R= R+S

U=S

and

V+S=R+S

V=R

if U=S and V=R they are equidistant from the orgin.

PS: I did like that cuz i couldnt remember the main formula however it is more quick
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Re: In the rectangular coordinate system, are the points (r,s) [#permalink]

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07 Sep 2010, 06:09
C.

Find the distance of U,V from origin. Substitute U = 1-R and V = 1-S. You will notice that when R+S=1 this distance equals the distance of R,S from origin.

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

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07 Sep 2010, 06:36
Expert's post
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CasperMonday wrote:

OA
[Reveal] Spoiler:
C

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

Distance between the point A (x,y) and the origin can be found by the formula: $$D=\sqrt{x^2+y^2}$$.

Basically the question asks is $$\sqrt{r^2+s^2}=\sqrt{u^2+v^2}$$ OR is $$r^2+s^2=u^2+v^2$$?

(1) $$r+s=1$$, no info about $$u$$ and $$v$$;

(2) $$u=1-r$$ and $$v=1-s$$ --> substitute $$u$$ and $$v$$ and express RHS using $$r$$ and $$s$$ to see what we get: $$RHS=u^2+v^2=(1-r)^2+(1-s)^2=2-2(r+s)+ r^2+s^2$$. So we have that $$RHS=u^2+v^2=2-2(r+s)+ r^2+s^2$$ and thus the question becomes: is $$r^2+s^2=2-2(r+s)+ r^2+s^2$$? --> is $$r+s=1$$? We don't know that, so this statement is not sufficient.

(1)+(2) From (2) question became: is $$r+s=1$$? And (1) says that this is true. Thus taken together statements are sufficient to answer the question.

Hope it helps.
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Re: In the rectangular coordinate system, are the points (r,s) [#permalink]

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10 Sep 2010, 14:09
1
KUDOS
We need to check whether r^2 + s^2 = u^2 + v^2
=> is $$(r-u)* (r+u) = (v-s)* (v+s)$$ ------------------------------A

Statement 1: Apparently Insufficient as no information about u and v

Statement 2:

Given $$u = 1 - r$$ and $$v = 1 - s$$ ---------------------B

=> $$r+u =1$$ and $$v+s = 1$$------------------------------C

Using equations A and C
we need to check if $$u-r = s-v$$

if $$u+v = r+s$$

if $$u+v = r+s = 1-r + 1-s$$

if $$r+s = 1$$ ----------------------------------D
Not Sufficient.

combine statement 1 and equations D , we can prove the distance is same.
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Re: In the rectangular coordinate system, are the points (r,s) [#permalink]

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21 Oct 2010, 22:53
Was about to post this. I'm glad I checked before I did.

Thanks for the explanations!
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Re: In the rectangular coordinate system, are the points (r,s) [#permalink]

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

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03 Dec 2017, 10:15
CasperMonday 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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/in-the-recta ... 92823.html
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Kudos [?]: 135804 [0], given: 12714

Re: In the rectangular coordinate system, are the points (r,s)   [#permalink] 03 Dec 2017, 10:15
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