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800 lvl problem.

by abcdefg » Fri Jul 10, 2009 6:12 am
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) Square root of (a^2) + Square root of (b^2) = Square root of (c^2) + square root of (d^2).

The correct answer is C . Just wondering if anybody has some good insight on how to solving this problem using shortcuts. It would take me over 5 minutes to logic out all the scenarios.
Last edited by abcdefg on Fri Jul 10, 2009 6:41 am, edited 1 time in total.
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by mike22629 » Fri Jul 10, 2009 6:26 am
First of all, unless im mistaken the answer is B, assuming you wrote the question down wrong (which im pretty sure that you did)

sqrt of a^2 + sqrt of b^2 = a + b

I assume you mean sqrt (a^2+ b^2). Am I right with this assumption?

Assuming I am right, here is my reasoning:

What is the question asking?

A,B equidistant from origin to C,D

What does that mean? Use pythagoreom thereom (a^2 + b^2 = c^2)
(c = distance from origin)



So question is asking...
does sqrt of (a^2 + b^2) = sqrt of (c^2 + d^2)?

This is exactly what Statement B tells you


For A.) Can not see how that is sufficient

If A=8, B=2, D = 4, C = 1

point (8,2) is certainly not equidistant from origin compared to (4,1)

Can you double check question?
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by sreak1089 » Fri Jul 10, 2009 6:33 am
Correct me if I am wrong, IMO answer is B.

stmt # 1 says:
a/b = c/d
Lets take coordinates (a,b) that satisfies stmt # 1
(2,4) & (c,d) = (-2,-4) In this case, (a,b) & (c,d) are indeed
equidistant.

However, lets take coordinates (a,b) = (2, 4) & (c,d) = (4,8) such that
Distance of (a,b) from origin would be sqrt(20) and distance of (c,d)
would be sqrt(80)

Hence stmt # 1 NOT SUFFICIENT

stmt # 2 says:

a^2 + b^2 == c^2 + d^2 which really implies that both the coordinates
are equidistant from the origin if you apply the distance formula

Hence stmt # 2 SUFFICIENT.

Ans in my opinion B
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by abcdefg » Fri Jul 10, 2009 6:42 am
Sorry I made the mistake. I've corrected it in my original post. The right answer is C .

To find the distance from the origin, we simply take the square root of the sum of the squared x- and y-coordinates, i.e. .

(1) INSUFFICIENT: This simply tells us that the proportions between the x- and y-coordinates of both points are the same. E.g. take a = 5, b = 10, c = 6 and d = 12. The proportions are the same but the coordinate points are not the same distance from the origin. Conversely, if a = 5, b = 10, c = -5 and d = -10, then the proportions are equal and the coordinate points are the same distance from the origin.

(2) INSUFFICIENT: By simplifying the expression, we get |a| + |b| = |c| + |d|. This is not enough to tell if the points are equidistant. E.g. take a = 11, b = 1, c = 6 and d = 6. The expression |a| + |b| = |c| + |d| is true but the coordinate points are not the same distance from the origin. Conversely, if a = -6, b = 6, c = 6 and d = 6, then the given expression is true and the coordinate points are the same distance from the origin.

(1) AND (2) SUFFICIENT: Together the statements are sufficient. Why? 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|. Plugging this into our distance formula, we get:

= ; Plug in |a| = |c| and |b| = |d| to get:
=

This is enough to show that the two points are equidistant.

The correct answer is C.
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by sreak1089 » Fri Jul 10, 2009 6:58 am
oh yeah .. you are right I didn't read qn properly
it was sqrt(a^2) + sqrt(b^2) ..................
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by abcdefg » Fri Jul 10, 2009 7:06 am
are there any cues or shortcuts I can take to do this question?
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by prindaroy » Thu Jul 16, 2009 3:28 pm
To prove they are equidistant we must show that

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

So,

we know that 1 and 2 alone are not sufficient.

but together we know that, ad = bc

we also know that a + b = c + d, since sqrt(a^2)+sqrt(b^2)=sqrt(c^2)+sqrt(d^2)

Now, a + b = c + d becomes a - d = c - b

Square both sides to get;

a^2 - 2ad - d^2 = c^2 - 2bc +b^2

ad = bc, so 2ad and 2bc cancel out; and we have that a^2 - b^2 = c^2 - d^2,

so it must be true that a^2 + b^2 = c^2 + d^2

hence they are equidistant since a/b = c/d
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by real2008 » Fri Jul 17, 2009 12:20 pm
abcdefg wrote:Sorry I made the mistake. I've corrected it in my original post. The right answer is C .

To find the distance from the origin, we simply take the square root of the sum of the squared x- and y-coordinates, i.e. .

(1) INSUFFICIENT: This simply tells us that the proportions between the x- and y-coordinates of both points are the same. E.g. take a = 5, b = 10, c = 6 and d = 12. The proportions are the same but the coordinate points are not the same distance from the origin. Conversely, if a = 5, b = 10, c = -5 and d = -10, then the proportions are equal and the coordinate points are the same distance from the origin.

(2) INSUFFICIENT: By simplifying the expression, we get |a| + |b| = |c| + |d|. This is not enough to tell if the points are equidistant. E.g. take a = 11, b = 1, c = 6 and d = 6. The expression |a| + |b| = |c| + |d| is true but the coordinate points are not the same distance from the origin. Conversely, if a = -6, b = 6, c = 6 and d = 6, then the given expression is true and the coordinate points are the same distance from the origin.

(1) AND (2) SUFFICIENT: Together the statements are sufficient. Why? 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|. Plugging this into our distance formula, we get:

= ; Plug in |a| = |c| and |b| = |d| to get:
=

This is enough to show that the two points are equidistant.

The correct answer is C.
I need a clarification:

As per your explanation I understand square root (x^2) is x and not -x.
Is my understanding correct?
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