GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 25 Aug 2019, 06:43 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. ### Request Expert Reply # In the rectangular coordinate system, are the points (r,s) and (u,v)

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Manager  Joined: 21 Jan 2010
Posts: 141
In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

9
47 00:00

Difficulty:   55% (hard)

Question Stats: 61% (01:54) correct 39% (02:00) wrong based on 704 sessions

### HideShow timer Statistics

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

Originally posted by calvinhobbes on 17 Apr 2010, 06:57.
Last edited by Bunuel on 06 Feb 2019, 04:09, edited 2 times in total.
Renamed the topic.
##### Most Helpful Expert Reply
Math Expert V
Joined: 02 Sep 2009
Posts: 57287
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

15
16
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.

Answer: C.

Hope it helps.
_________________
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 57287
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

2
metallicafan wrote:
Bunuel, I have a question:
How did you know that you had to express the equation in that way?
For example, I expressed (based on clue # 2) in this way:
$$r^2 + s^2 = (1-r)^2 + (1-s)^2$$
So, I obtain:
r + s = 1
The same as clue # 1. How did you know that you had to do in the other way?

Thanks!

Not sure I understand your question. But here is how I solved it:

The question asks: is $$r^2+s^2=u^2+v^2$$?

Then (2) says: $$u=1-r$$ and $$v=1-s$$. So now we can 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.

When combining: 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's clear.
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 57287
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

1
fozzzy wrote:
Bunuel 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

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.

Answer: C.

Hope it helps.

So the formula used here is different from the distance formula of square root of (x2-x1)^2 + (y2-y1)^2

No it's not. 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, coordinates (0, 0), then the formula can be simplified to: $$D=\sqrt{x^2+y^2}$$.

Hope it's clear.
_________________
Math Expert V
Joined: 02 Aug 2009
Posts: 7754
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

1
sakshamchhabra wrote:
chetan2u Bunuel VeritasKarishma

Greetings Experts,

while trying to figure out a shorter approach for this question, I noticed -

St 1. r + s = 1 -----> r = 1 - s OR s = 1 - r

St 2. u = 1 - r and v = 1 - s

from the above statements, we can deduce ---> u = s and v = r

Hence, the points will definitely be equidistant.

Please correct me if I am wrong.

Yes, you are correct in the present state.
_________________
Veritas Prep GMAT Instructor D
Joined: 16 Oct 2010
Posts: 9558
Location: Pune, India
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

1
sakshamchhabra wrote:
chetan2u Bunuel VeritasKarishma

Greetings Experts,

while trying to figure out a shorter approach for this question, I noticed -

St 1. r + s = 1 -----> r = 1 - s OR s = 1 - r

St 2. u = 1 - r and v = 1 - s

from the above statements, we can deduce ---> u = s and v = r

Hence, the points will definitely be equidistant.

Please correct me if I am wrong.

Yes, your logic works and it's great!
Note that r = 1 - s AND s = 1 - r
Since r and s add up to 1, whatever the value of r, value of s will be complementary to that. So r will be 1 - s and s will be 1 - r.
_________________
Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
Manager  Joined: 21 Jan 2010
Posts: 141
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

Awesome. Thanks a bunch Retired Moderator Status: 2000 posts! I don't know whether I should feel great or sad about it! LOL
Joined: 04 Oct 2009
Posts: 1037
Location: Peru
Schools: Harvard, Stanford, Wharton, MIT & HKS (Government)
WE 1: Economic research
WE 2: Banking
WE 3: Government: Foreign Trade and SMEs
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

Bunuel, I have a question:
How did you know that you had to express the equation in that way?
For example, I expressed (based on clue # 2) in this way:
$$r^2 + s^2 = (1-r)^2 + (1-s)^2$$
So, I obtain:
r + s = 1
The same as clue # 1. How did you know that you had to do in the other way?

Thanks!
_________________
"Life’s battle doesn’t always go to stronger or faster men; but sooner or later the man who wins is the one who thinks he can."

My Integrated Reasoning Logbook / Diary: http://gmatclub.com/forum/my-ir-logbook-diary-133264.html

GMAT Club Premium Membership - big benefits and savings
SVP  G
Status: Top MBA Admissions Consultant
Joined: 24 Jul 2011
Posts: 1878
GMAT 1: 780 Q51 V48 GRE 1: Q800 V740 Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

(r,s) and (u,v) will be equidistant from the origin when
r^2 + s^2 = u^2 + v^2

Using statement (1), r+s=1 gives us no information about u and v and so is insufficient.
Using statement (2), u = 1-r and v=1-s
=> r^2 + s^2 = (1-r)^2 + (1-s)^2
=> 2r + 2s - 2 = 0
or r + s = 1, which may or may not be true. Insufficient.

Combining (1) and (2) is clearly sufficient.

(C) is the answer.
_________________
GyanOne [www.gyanone.com]| Premium MBA and MiM Admissions Consulting

Awesome Work | Honest Advise | Outstanding Results

Reach Out, Lets chat!
Email: info at gyanone dot com | +91 98998 31738 | Skype: gyanone.services
Intern  Joined: 22 Jan 2012
Posts: 5
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

I think it is a simple way to pick up values to solve this question because it is clear that each statement is not sufficient. For example;

for r=2, s=-1 we have u=-1, v=2 or for r=1, s=0 we have u=0, v=1 and so on. Therefore only if we know both statements, we can talk about the distance. So, the answer is C.
Math Expert V
Joined: 02 Sep 2009
Posts: 57287
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

ustureci wrote:
I think it is a simple way to pick up values to solve this question because it is clear that each statement is not sufficient. For example;

for r=2, s=-1 we have u=-1, v=2 or for r=1, s=0 we have u=0, v=1 and so on. Therefore only if we know both statements, we can talk about the distance. So, the answer is C.

This is not a good question for number picking. Notice that variables are not restricted to integers only, so r+s=1, u=1-r and v=1-s have infinitely many solutions for r, s, u and v.
_________________
Manager  Joined: 28 Apr 2011
Posts: 97
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

best approch is imagine point to be on circumference of same circle.

Now radius of circle = use distance formula

so use equations in this logic. and get C Manager  Status: mba here i come!
Joined: 07 Aug 2011
Posts: 198
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

question: $$r^2+s^2=u^2+v^2?$$

(a) insufficient because there's no info about u,v.
(b) insufficient. plug in numbers to see if it holds: find a 'yes' and then find a 'no'.

$$(r,s)=(u,v)=(\frac{1}{2},\frac{1}{2})$$ -------> 'yes' points are equidistant
$$(r,s)=(0,0$$, then $$(u,v)=(1,1)$$ -------> 'no' points are not equidistant

(c) together we can even prove it algebraically.
from (1) $$s=1-r$$ and from (2) $$u=1-r$$. so, $$s=u$$
likewise, from (1) $$s=1-r$$ and from (2) $$s=1-v$$. so, $$r=v$$

ans: C
_________________
press +1 Kudos to appreciate posts
Director  Joined: 29 Nov 2012
Posts: 725
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

Bunuel 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

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.

Answer: C.

Hope it helps.

So the formula used here is different from the distance formula of square root of (x2-x1)^2 + (y2-y1)^2
Intern  Joined: 02 Mar 2010
Posts: 19
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

1) Not Suff as no info about u & v.
2) Not suff as 4 variables and 2 equations.

(1) and (2) combined:
From Statement (1), r =(1-s) = v by definition given in statement (2); and similarly s=(1-r)=u by definition given in statement (2).
Therefore s=u and r=v. Hence (r,s) and (u,v) represent same point and so have the same distance from origin. SUFF. Correct answer = C.
Manager  B
Joined: 09 Mar 2018
Posts: 56
Location: India
Schools: CBS Deferred "24
Re: In the rectangular coordinate system, are the points (r,s) and (u,v)  [#permalink]

### Show Tags

chetan2u Bunuel VeritasKarishma

Greetings Experts,

while trying to figure out a shorter approach for this question, I noticed -

St 1. r + s = 1 -----> r = 1 - s OR s = 1 - r

St 2. u = 1 - r and v = 1 - s

from the above statements, we can deduce ---> u = s and v = r

Hence, the points will definitely be equidistant.

Please correct me if I am wrong. Re: In the rectangular coordinate system, are the points (r,s) and (u,v)   [#permalink] 06 Jun 2019, 11:08
Display posts from previous: Sort by

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

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

#### MBA Resources  