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Intern  Joined: 17 Mar 2011
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Re: In the rectangular coordinate system, are the points (a,  [#permalink]

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

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1
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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1
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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Re: ManhttanGMAT Practice CAT  [#permalink]

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Bunuel wrote:
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.

Hi Bunnel
i am not understanding how $$\sqrt{a^2}+\sqrt{b^2}=\sqrt{c^2} +\sqrt{d^2}$$ --> $$|a|+|b|=|c|+|d|$$
have come?
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Re: ManhttanGMAT Practice CAT  [#permalink]

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mun23 wrote:
$$\sqrt{a^2}+\sqrt{b^2}=\sqrt{c^2} +\sqrt{d^2}$$ --> $$|a|+|b|=|c|+|d|$$
have come?

By convention, only positive roots are considered (at least in GMAT)

$$\sqrt{4} = 2$$ (and not -2)

Similarly, $$\sqrt{a^2} = |a|$$ i.e. only the positive value
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Posts: 112
Re: coordinate geometry  [#permalink]

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2
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, b)  [#permalink]

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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

basically we need to prove
$$a^2+b^2 = c^2+d^2$$

stmnt 1:
$$\frac{a}{b} = \frac{c}{d}$$

1/2 = 3/6 ==>using this we can prove NO
1/2 = 1/2 ==>using these numbers we can prove yes

hence insufficient
statement 2:
$$\sqrt{(a^2)} + \sqrt{(b^2)} = \sqrt{(c^2)} + \sqrt{(d^2)}$$or$$|a| + |b| = |c| + |d|$$

putting $$a=b=c=d=1$$..we will get YES.
Putting $$a= 1, b=2, c= 0, d=3$$...we will get NO.
HENCE INSUFFICIENT.

combining.
from 2nd statement: $$|a| + |b| = |c| + |d|$$
rearrange:
$$|a| - |d| = |c| - |b|$$ and then square both sides.
$$a^2 + d^2 - 2*|a|*|d| = c^2 + b^2 - 2*|c|*|b|.$$

Since $$ad = bc$$ as per statement 1
we can cancel few things:
$$a^2 + d^2 - 2*|a|*|d| = c^2 + b^2 - 2*|c|*|b|.$$

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

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nsp007 wrote:
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) $$\sqrt{a^2}+ \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$

1. ad-bc=0 means that the vector pointing to (c,d) is a scalar multiple of the vector pointing to (a,b). In other words they are collinear with the origin. If they are equidistant to the origin, the scalar will be 1 or -1. But given this information it could be any scalar. Not sufficient.

2. |a|+|b|=|c|+|d|. This means that (a,b) and (c,d) both lay on some square centered at the origin. But they could be anywhere on the square. Not sufficient.

1 and 2. (a,b) and (c,d) are collinear with the origin and both lay on a square centered at the origin. Therefore they must be the same distance from the origin. Sufficient. C
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Re: In the rectangular coordinate system, are the points (a,  [#permalink]

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(Don't read my "solution" I am studying and got this one wrong. )

The first statement is telling me that the ratio between the x and y value are the same for both points.

This could be (1,3) or (10,30) where the second one is obviously further away from the origin. Hence insufficient.

Statement (2) I cannot fully understand intuitively. So I'll plug in some numbers.

0,1 =1
1/4,1/4 = 1/2+1/2=1
distance from origin in the second case = (1/2)^2+(1/2)^2=(2/4) square root of (1/2) =/= 1.

So it differs.

However, by using the same numbers I could make it work. So I can rule out both and go with E.

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

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This was my view:

You see immediately that both statements alone are not sufficient. So start evaluating the combination of both:

Statement A says that that the points are on the same line through the origin. Statement B says that the different "possible" points all have an equal distance to the x-axis, but also to the y-axis. Therefore, combining both options say that either (a,b) = (c,d) or (a,b) = (-c, -d). So equidistant from origin.
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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.

Bunuel, how did you quickly determine that (2) was Not Sufficient?
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Re: In the rectangular coordinate system, are the points (a,  [#permalink]

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TooLong150 wrote:
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.

Bunuel, how did you quickly determine that (2) was Not Sufficient?

You can do this with number plugging. The question asks whether $$a^2+b^2=c^2+d^2$$ and (2) says that $$|a|+|b|=|c|+|d|$$. If a = b = c = d = 0, then the answer would be YES but if a = 0, b = 2, c = 1 and d = 1, then the answer would be NO.
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Re: In the rectangular coordinate system, are the points (a, b)  [#permalink]

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Bunuel Sir,

I Didn't understand this:-

given a/b=c/d=cx/dx

(as the ratios are equal then there exist some for which d=cx and b=dx)
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Re: In the rectangular coordinate system, are the points (a, b)  [#permalink]

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manojpandey80 wrote:
Bunuel Sir,

I Didn't understand this:-

given a/b=c/d=cx/dx

(as the ratios are equal then there exist some for which d=cx and b=dx)

a/b = c/d = cx/dx

For example: 16/32 = 2/4 = (2*8)/(4*8).
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Re: In the rectangular coordinate system, are the points (a,  [#permalink]

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mbaMission wrote:
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) $$\sqrt{a^2} + \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$

C

this question is pretty weid. is this question from gmatprep?
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Re: In the rectangular coordinate system, are the points (a,  [#permalink]

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thangvietnam wrote:
mbaMission wrote:
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) $$\sqrt{a^2} + \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$

C

this question is pretty weid. is this question from gmatprep?

Yes, it's a GMAT Prep question.
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Re: In the rectangular coordinate system, are the points (a,  [#permalink]

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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.

the key point here is to know that a=xc
this is simple but tricky

wonderful explanation of this problem . thank you , Bunnu
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Re: In the rectangular coordinate system, are the points (a,  [#permalink]

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Bunuel wrote:
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.

Hi y'all,

I'm a newbie here.
can you please review my method?

I was disagreeing that only (2) is not sufficient.
I use Trigonometry to solve (2). The (x,y) must make a right triangle with x-axis.
Thus, the distance between the dot and the origin must be the hypotenuse. And (2) seems to be the equation for the hypotenuse.
If the 2 triangle are equal, then the distance must be equal.

Or am I missing something...?
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In the rectangular coordinate system, are the points (a,  [#permalink]

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mbaMission wrote:
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) $$\sqrt{a^2} + \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$

Asked: In the rectangular coordinate system, are the points $$(a, b)$$ and $$(c, d)$$ equidistant from the origin?
Q. a^2 + b^2 = c^2 + d^2 ?

(1) $$\frac{a}{b} =\frac{c}{d}$$
NOT SUFFICIENT

(2) $$\sqrt{a^2} + \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$
|a| + |b| = |c| + |d|
Squaring both sides
a^2 + b^2 + 2|a||b| = c^2 + d^2 + 2 |c||d|
NOT SUFFICIENT

(1) + (2)
(1) $$\frac{a}{b} =\frac{c}{d}$$

(2) $$\sqrt{a^2} + \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}$$
a^2 + b^2 + 2|a||b| = c^2 + d^2 + 2 |c||d|
NOT SUFFICIENT

IMO E
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