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vkb16: Master | Next Rank: 500 Posts; Posts: 122; Joined: Mon Dec 15, 2008 12:03 am; Thanked: 2 times

by vkb16 » Wed Apr 22, 2009 4:20 am

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

oa is C

I read the mgmat explantion, but didnt quite agree with it!
I think its E

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Source: Beat The GMAT — Data Sufficiency | Wed Apr 22, 2009 4:20 am

fleurdelisse: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Tue Feb 03, 2009 2:01 am; Thanked: 1 times

by fleurdelisse » Wed Apr 22, 2009 4:37 am

The distance between any two points in the rectangular coordinate system is given by the square root of the sum of difference between the coordinates of the points.

That is, the coordinates of the origin O are (0,0)

So the distance from O to point (a,b) is:
sqrt((a-0)^2+(b-0)^2))=sqrt(a^2+b^2)
More simply put, you can just draw a rectangle triangle with 0 and point (a,b) as vertexes and apply Pythagorean theorem

The same goes for the distance from the origin to point (c,d), it's:
sqrt(c^2+d^2)

Now for both distances to be equal, we need to have:
sqrt(a^2+b^2)=sqrt(c^2+d^2)
=> a^2+b^2 = c^2+d^2

SO (2) would be sufficient, I don't understand why the answer is C??

Anyone?

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vkb16: Master | Next Rank: 500 Posts; Posts: 122; Joined: Mon Dec 15, 2008 12:03 am; Thanked: 2 times

by vkb16 » Wed Apr 22, 2009 4:46 am

oh my bad! sorry!

the second option is sqrt(a^2) + sqrt(b^2) = sqrt(c^2) + sqrt(d^2)

sorry..

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fleurdelisse: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Tue Feb 03, 2009 2:01 am; Thanked: 1 times

by fleurdelisse » Wed Apr 22, 2009 4:46 am

In this case, 2 is not sufficient since it only tells you that a + b = c+ d

Taking both (1) and (2) together, you would get:

from (2) square both sides:

(a+b)^2 = (c+d)^2
a^2 + 2ab + b^2 = c^2 + 2cd + d^2
but from (1): ab = cd, so you can eliminate them on both sides

This results in:

a^2 + b^2 = c^2 + d^2
exactly what the question is asking for

They are equidistant, so C is your answer

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vkb16: Master | Next Rank: 500 Posts; Posts: 122; Joined: Mon Dec 15, 2008 12:03 am; Thanked: 2 times

by vkb16 » Wed Apr 22, 2009 5:36 am

but from (1): ab = cd, so you can eliminate them on both sides

but from 1 we have aD = bC[/quote]

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fleurdelisse: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Tue Feb 03, 2009 2:01 am; Thanked: 1 times

by fleurdelisse » Wed Apr 22, 2009 8:48 am

right! sorry about that

.

Ok so, let's go back:

we have: a+b = c+d from (2)

multiply both sides of equation by d:

ad + bd = cd + d^2

since ad = bc:

bc + bd = cd + d^2
b(c+d) = d(c+d)

so b=d

and from (1), a = b

The two points are one and the same, so obviously they're equidistant to the origin

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