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# In the rectangular coordinate system, are the points (a,

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02 Jun 2009, 03:05
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In the rectangular coordinate system, are the points $$(a, b)$$ and $$(c, d)$$ equidistant from the origin?

(1) $$\frac{a}{b} =\frac{c}{d}$$

(2) $$\sqrt{a^2} + \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$
[Reveal] Spoiler: OA

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Last edited by Bunuel on 28 Apr 2015, 03:31, edited 1 time in total.
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Re: In the rectangular coordinate system, are the points (a, [#permalink]

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05 Feb 2011, 12:53
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tinki wrote:
Official explanation says:
"If we know the proportion of a to b is the same as c to d and that |a| + |b| = |c| + |d|, then it must be the case that |a| = |c| and |b| = |d| ?

could someone elaborate how we are supposed to know |a| = |c| and |b| = |d|? a bit vague statement for me

thanks for responses

In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

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

So we are asked whether $$\sqrt{a^2+b^2}=\sqrt{c^2+d^2}$$? Or whether $$a^2+b^2=c^2+d^2$$?

(1) $$\frac{a}{b}=\frac{c}{d}$$ --> $$a=cx$$ and $$b=dx$$, for some non-zero $$x$$. Not sufficient.

(2) $$\sqrt{a^2}+\sqrt{b^2}=\sqrt{c^2} +\sqrt{d^2}$$ --> $$|a|+|b|=|c|+|d|$$. Not sufficient.

(1)+(2) From (1) $$a=cx$$ and $$b=dx$$, substitute this in (2): $$|cx|+|dx|=|c|+|d|$$ --> $$|x|(|c|+|d|)=|c|+|d|$$ --> $$|x|=1$$ (another solution $$|c|+|d|=0$$ is not possible as $$d$$ in (1) given in denominator and can not be zero, so $$d\neq{0}$$ --> $$|c|+|d|>0$$) --> now, as $$|x|=1$$ and $$a=cx$$ and $$b=dx$$, then $$|a|=|c|$$ and $$|b|=|d|$$ --> square this equations: $$a^2=c^2$$ and $$b^2=d^2$$ --> add them: $$a^2+b^2=c^2+d^2$$. Sufficient.

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Re: Difficult Geometry DS Problem [#permalink]

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12 Jan 2011, 18:43
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abmyers wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) $$\sqrt{a^2}+\sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

You can take values to solve this question quickly:

Statement 1: a/b = c/d
(a,b) and (c,d) may be equidistant from the origin e.g. (1, 1) and (-1, -1) or they may not be e.g. (1, 1) and (2, 2). Not sufficient.

Statement 2: $$\sqrt{a^2}+\sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$
That is, |a|+|b|=|c|+|d|
(a,b) and (c,d) may be equidistant from the origin e.g. (1, 3) and (-1, -3) or they may not be e.g. (1, 3) and (2, 2)

Using both together, |a|+|b|=|c|+|d| and a/b = c/d.
This means that if a/c = 1/2, c/d cannot be 2/4 or -3/-6 etc. c/d has to be either 1/2 or (-1)/(-2). Similarly, if a/b = (-1)/2, c/d = (-1)/2 or 1/(-2))
Any pair of such points will be equidistant. Answer (C).
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11 Apr 2010, 00:11
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nsp007 wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) $$\sqrt{a^2}+ \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$

Will post OA later.

In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

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

So we are asked whether $$\sqrt{a^2+b^2}=\sqrt{c^2+d^2}$$? Or whether $$a^2+b^2=c^2+d^2$$?

(1) $$\frac{a}{b}=\frac{c}{d}$$ --> $$a=cx$$ and $$b=dx$$, for some non-zero $$x$$. Not sufficient.

(2) $$\sqrt{a^2}+\sqrt{b^2}=\sqrt{c^2} +\sqrt{d^2}$$ --> $$|a|+|b|=|c|+|d|$$. Not sufficient.

(1)+(2) From (1) $$a=cx$$ and $$b=dx$$, substitute this in (2): $$|cx|+|dx|=|c|+|d|$$ --> $$|x|(|c|+|d|)=|c|+|d|$$ --> $$|x|=1$$ (another solution $$|c|+|d|=0$$ is not possible as $$d$$ in (1) given in denominator and can not be zero, so $$d\neq{0}$$ --> $$|c|+|d|>0$$) --> now, as $$|x|=1$$ and $$a=cx$$ and $$b=dx$$, then $$|a|=|c|$$ and $$|b|=|d|$$ --> square this equations: $$a^2=c^2$$ and $$b^2=d^2$$ --> add them: $$a^2+b^2=c^2+d^2$$. Sufficient.

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Re: In the rectangular coordinate system, are the points (a, [#permalink]

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02 Jun 2009, 20:44
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goldeneagle94 wrote:
amolsk11 wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

This singnifies that the sign combination on both sides of = is the same.
i.e. if one of the numbers on LHS is -ve , one of the numbers on RHS has to -ve as well.
However since a,b / c,d can take any value , INSUFFICIENT.

(2) (a^2)^(1/2) + (b^2)^(1/2) = (c^2)^(1/2) + (d^2)^(1/2)
Since we do not know anything about signs of a,b / c,d - INSUFFICIENT.

Using 1 and 2 together ,
using (1) the second statement drills down to
a+b=c+d (the sign combination on both sides of = is the same.)

Since a/b=c/d and a+b=c+d , (a, b) and (c, d) equidistant from the origin and infact represent the same point.

Is there a way we can derive, using these two equations, that the two points are equidistant ?

For the derivation part:

a^2 + b^2 = c^2 + d^2

b^2[(a^2/b^2) + 1] = d^2[(c^2/d^2 +1)] --------> 1

from I

a/b = c/d
squaring both sides
a^2/b^2 = c^2/d^2
putting above in 1 and simplifying

b^2[(c^2/d^2) + 1] = d^2[(c^2/d^2 +1)]
simplifying

b^2 = d^2
or |b| = |d|
=> |a| = |c|

therefore points are equidistant from (0,0) or Origin.
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Re: points equidistant from origin? [#permalink]

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16 Oct 2010, 15:21
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Orange08 wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

a. $$a/b = c/d$$

b. $$\sqrt{a^2}+\sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$

Distance of $$(x,y)$$ from origin is $$\sqrt{x^2+y^2}$$
So we need to answer $$\sqrt{a^2+b^2}=\sqrt{c^2+d^2}$$ ?

(1) a/b=c/d ... doesnt really help in proving or disproving. Insufficient

(2) This is equivalent to saying $$|a|+|b|=|c|+|d|$$. Again insufficient to say anything about the statement we have.

(1+2) a/c=b/d=x say (needs, c,d to be non zero)

a = cx
b = dx

|a|+|b|=|c|+|d|
|cx| - |c| = |d| - |dx|
|c|(|x|-1)=|d|(1-|x|)
(|c|+|d|)(|x|-1)=0
Since c,d are non-zero means |x|=1
So either a=c & b=d OR a=-c or b=-d
Either case a^2=c^2 and b^2=d^2
Hence $$\sqrt{a^2+b^2}=\sqrt{c^2+d^2}$$
Sufficient

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In the rectangular coordinate system, are the points (a, b) [#permalink]

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10 Apr 2010, 22:25
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In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) $$\frac{a}{b}=\frac{c}{d}$$

(2) $$\sqrt{a^2}+ \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$

Last edited by Bunuel on 07 Apr 2012, 07:24, edited 1 time in total.
Edited the question and added the OA
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Re: In the rectangular coordinate system, are the points (a, [#permalink]

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05 Feb 2011, 12:06
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tinki wrote:
Official explanation says:
"If we know the proportion of a to b is the same as c to d and that |a| + |b| = |c| + |d|, then it must be the case that |a| = |c| and |b| = |d| ?

could someone elaborate how we are supposed to know |a| = |c| and |b| = |d|? a bit vague statement for me

thanks for responses

a/b=c/d <---- Equation 1

|a|+|b|=|c|+|d|

Add 1 to both sides of Equation 1

(a+b)/b=(c+d)/d

Now a+b=c+d => 1/b=1/d

Thus, b=d
Similarly,
b/a=d/c
Adding 1 to both sides of above equation:
(b+a)/a=(d+c)/c
a+b=d+c => 1/a=1/c
Therefore a=c
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Re: In the rectangular coordinate system, are the points (a, [#permalink]

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02 Jun 2009, 06:44
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mbaMission wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d
(2) (a^2)^(1/2) + (b^2)^(1/2) = (c^2)^(1/2) + (d^2)^(1/2)

(1) If a/b = c/d, a and b could be 100 and c and d could be 1 or a, b, c and d could be 1. NSF.

(2) If (a^2)^(1/2) + (b^2)^(1/2) = (c^2)^(1/2) + (d^2)^(1/2), then a+b = c+d. In this case, if a = 9 and b = 1, and c =d=5, a+b = c+d is true. but they have different distance.NSF.

From 1 and 2, a must be equal to c and b must be equal to d. So Suff.

Thats C.
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Re: In the rectangular coordinate system, are the points (a, [#permalink]

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03 Jun 2009, 06:24
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in order for the points to be equidistant

a2 + b2 = c2+ d2 -----> eq X

1--> insufficent (has already been discussed)
2--> insufficent (has already been discussed)

combining both,
equation 2 can be written as a+b = c+d

eq1 a/b = c/d
a+b/b = c+d/d, from eq 1 from b=d ----3

similarly, b/a = d/c, a+b/a = d+c/c from eq 1 a = c ----4

plug in values of a, b in eq X... it satifies.

Hence points are equidistant. Answer is C
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09 Sep 2010, 00:55
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amitjash wrote:
I have a question here bunuel. How did you get a = c * x and b= d*x??

Because from this it means that x = b/d = a/c

It's the same: $$\frac{b}{d} = \frac{a}{c}$$ --> $$bc=ad$$--> $$\frac{c}{d}=\frac{a}{b}$$.

Given: $$\frac{a}{b}=\frac{c}{d}=\frac{cx}{dx}$$ (as the ratios are equal then there exist some $$x$$ for which $$a=cx$$ and $$b=dx$$).
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Re: In the rectangular coordinate system, are the points (a, [#permalink]

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05 Feb 2011, 13:27
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Bunuel wrote:
tinki wrote:
Official explanation says:
"If we know the proportion of a to b is the same as c to d and that |a| + |b| = |c| + |d|, then it must be the case that |a| = |c| and |b| = |d| ?

could someone elaborate how we are supposed to know |a| = |c| and |b| = |d|? a bit vague statement for me

thanks for responses

In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

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

So we are asked whether $$\sqrt{a^2+b^2}=\sqrt{c^2+d^2}$$? Or whether $$a^2+b^2=c^2+d^2$$?

(1) $$\frac{a}{b}=\frac{c}{d}$$ --> $$a=cx$$ and $$b=dx$$, for some non-zero $$x$$. Not sufficient.

(2) $$\sqrt{a^2}+\sqrt{b^2}=\sqrt{c^2} +\sqrt{d^2}$$ --> $$|a|+|b|=|c|+|d|$$. Not sufficient.

(1)+(2) From (1) $$a=cx$$ and $$b=dx$$, substitute this in (2): $$|cx|+|dx|=|c|+|d|$$ --> $$|x|(|c|+|d|)=|c|+|d|$$ --> $$|x|=1$$ (another solution $$|c|+|d|=0$$ is not possible as $$d$$ in (1) given in denominator and can not be zero, so $$d\neq{0}$$ --> $$|c|+|d|>0$$) --> now, as $$|x|=1$$ and $$a=cx$$ and $$b=dx$$, then $$|a|=|c|$$ and $$|b|=|d|$$ --> square this equations: $$a^2=c^2$$ and $$b^2=d^2$$ --> add them: $$a^2+b^2=c^2+d^2$$. Sufficient.

your explanation is great as always ! thaaanks
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Re: In the rectangular coordinate system, are the points (a, [#permalink]

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21 Dec 2011, 09:34
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Here's a simpler solution......

We need to prove a^2 + d^2 = c^2 + b^2

Simpler way is to rearrange statement 2 viz. |a| + |b| = |c| + |d| as
1. |a| - |d| = |c| - |b| and then square both sides. We get -
2. a^2 + d^2 - 2*|a|*|d| = c^2 + b^2 - 2*|c|*|b|.

Since ad = bc as per statement 1,

3. |ad| = |bc| => |a|.|d| = |b|.|c| (Rule -> Abs of Product = Product of Abs)

So we can cancel the third term out from both sides of equation 2. to get the desired equation

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Re: In the rectangular coordinate system, are the points (a, [#permalink]

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08 Aug 2012, 02:22
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reagan wrote:
Hi there, i was just wondering if the way i do it is correct!

Statement 1: insuff
Statement 2 : |a|+|b|=|c|+|d|

Our goal is to prove that a^2 + b^2 = c^2 + d^2

(1+2)

Square both sides in stmt 2.

We have a^2 + b^2 + 2|a||b| = c^2 + d^2 + 2|c||d| ----------- *
From one we know that a/b=c/d, therefore their LHS=RHS and therefore, this condition would allow us to cancel out 2|a||b| from LHS and 2|c||d| from equation *.

Please tell me that i am correct! =)

Reagan

Absolutely, I found this approach much better. Infact I am wondering why we are targetting absolute values in the equation.

(a+b)^2 = a^2+b^2+2ab [no absolute |a|, |b| needed]

From
1) we know ab = cd
2) we know a + b = c + d. and hence (a+b)^2 = (c+d)^2

Combining 1) & 2) we can very well see that a^2+b^2 = c^2+d^2
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Re: In the rectangular coordinate system, are the points (a, b) [#permalink]

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15 Jan 2013, 07:45
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1+2:

1) Ensures that the lines joining each of the two points to origin have same slope.
2) Ensures that absicca and ordinates correspondingly have equal magnitudes.
{If one line has points (a,b) then the other line will have coordinates (ak,bk) but (2) ensures that |k| = 1.}

Hence C.

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15 Aug 2013, 18:29
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sudharsansuski wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) (a^2)^0.5 + (b^2)^0.5 = (c^2)^0.5 + (d^2)^0.5

Origin on coordinate system is (0,0). The question is if distance between (0,0) to (a,b) is same as distance between (0,0) to (c,d)

Case 1: - a/b = c/d

let a=7, b= 14 then a/b = 1/2.

So c/d fraction has to be 1/2. i.e. c can be 2 and d can be 4. or c can be 6 & d can be 12 or c can be 7 & d can be 14. So distance between origin can either be same to (c,d) or different from (a,b). Clearly insufficient

Case 2:- (a^2)^0.5 + (b^2)^0.5 = (c^2)^0.5 + (d^2)^0.5

translates to "a + b = c + d"

let (a,b) = (4,4) and (c,d) = (4,4). Then it satisfies the condition a+b = c+d. Also distance from origin to (a,b) is same as (c,d).

let (a,b) = (5,3) and (c,d) = (4,4). Then it satisfies the condition a+b = c+d. And distance from origin to (a,b) is not same as (c,d).

Insufficient.

Lets take 1 & 2 together

we have a/b = c/d.......so a= bc/d--- Eq (1)

we also have a + b = c + d

substitute a=bc/d from Eq(1)

bc/d + b = c + d

b(c/d + 1) = c + d

b(c + d) = d(c+d)

so b = d

similarly we get a=b.

so taking 1 & 2 together, the distance between origin to both points are equal.
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Re: In the rectangular coordinate system, are the points (a, [#permalink]

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02 Jun 2009, 11:13
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

This singnifies that the sign combination on both sides of = is the same.
i.e. if one of the numbers on LHS is -ve , one of the numbers on RHS has to -ve as well.
However since a,b / c,d can take any value , INSUFFICIENT.

(2) (a^2)^(1/2) + (b^2)^(1/2) = (c^2)^(1/2) + (d^2)^(1/2)
Since we do not know anything about signs of a,b / c,d - INSUFFICIENT.

Using 1 and 2 together ,
using (1) the second statement drills down to
a+b=c+d (the sign combination on both sides of = is the same.)

Since a/b=c/d and a+b=c+d , (a, b) and (c, d) equidistant from the origin and infact represent the same point.
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08 Sep 2010, 01:22
Bunuel, had condition (2) simply said a+b=c+d (instead of the squares and square root) how would have the answer changed?

Also in the current question - is a^2+b^2=c^2+d^2?
When I square both sides of (2), I get a^2+b^2+2sqrt(a^2b^2)=c^2+d^2+2sqrt(c^2d^2)
so if ab=cd then this is satisfied. however (1) only gives me ad=bc, how do I infer ab=cd from that? I am following a different process, but I should end up with the same answer. Not sure where am I wrong?
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09 Sep 2010, 00:42
I have a question here bunuel. How did you get a = c * x and b= d*x??

Because from this it means that x = b/d = a/c

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Re: points equidistant from origin? [#permalink]

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16 Oct 2010, 14:52
(1) knowing these proportions does not help me solve it, because for example if 3/1 = 9/3 , point (a,b) will be closer to the origin than point (b,c)

(2) this statement tells that |a|+|b|=|c|+|d| , which is still not sufficient because we lack information about the correlation between |a| and |b|,and |c|and|d|.

But if we combine the two statements together we will have this correlation from statement (1) and then both statements taken together will be sufficient.

Re: points equidistant from origin?   [#permalink] 16 Oct 2010, 14:52

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