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Symmetry.

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by lenagmat » Wed Oct 26, 2011 4:44 am
In the number line, is point A and B symmetry about the zero point?

1). The distance from 0 to A is equal to the distance from 1 to B
2). The distance from 0 to A plus the distance from 1 to B is less than 1

For me here is the main question is - the concept symmetry suppose the two distances to be equal or nor?

Becauce if we take statement 2 I do not see that distances are equal? BUT the answer B.

Please, help me.
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Wed Oct 26, 2011 5:49 am
lenagmat wrote:In the number line, is point A and B symmetry about the zero point?

1). The distance from 0 to A is equal to the distance from 1 to B
2). The distance from 0 to A plus the distance from 1 to B is less than 1

For me here is the main question is - the concept symmetry suppose the two distances to be equal or nor?

Becauce if we take statement 2 I do not see that distances are equal? BUT the answer B.

Please, help me.
Edited: Didn't see "number line" - I read it as "x-y plane"

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Mon Oct 31, 2011 8:43 am, edited 1 time in total.
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by GmatMathPro » Wed Oct 26, 2011 10:07 am
Symmetry about a point most commonly comes up in coordinate geometry when you're graphing functions. A function is said to be symmetric about the point (0,0) (the origin) if for every point (a,b) that's on the graph, there is a corresponding point (-a,-b) that is also on the graph. This type of function is called an odd function. y=x^3 is an example of an odd function. Note a few sample points: (2,8) AND (-2,-8) are on the graph. (3,27) AND (-3,-27) are on the graph. Also, if we connect any two corresponding points with a line segment, the midpoint of the segment will be the origin. This is essentially how you should think of it having symmetry with respect to the point (0,0). Both corresponding points lie on a line that goes through the origin, and the distance from each point to the origin is the same.

Symmetry about a point on a number line is defined essentially the same way. If two points are symmetric about zero, they should be equidistant from and on opposite sides of zero. So, to answer your first question, yes, the two distances should be equal.

Finally, remember that this is a a data sufficiency question, and the answer being B doesn't mean you can use statement 2 to prove that the two points are symmetric about zero. It just means that you can use statement 2 to provide a definitive yes or no answer to the question of the points' symmetry about zero.

So, just focusing on statement 2: Assume A and B do have symmetry about zero. Let B=x and A=-x. Hopefully it is clear that if there is any hope of statement 2 being true, it must be true that 0<x<1, so let's make this assumption. The distance from 0 to A is |-x|=x. The distance from 1 to B is |1-x|=1-x. We need the sum of these distances to be less than 1, so: x+1-x<1, which simplifies to 1<1. This is false which implies that the points can't be symmetric about zero if statement 2 is true.

So, you're right that the distances are not equal when you take statement 2 to be true which is precisely why it's sufficient. If statement 2 is true, we know for sure the answer to the original question is no.
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by Brent@GMATPrepNow » Wed Oct 26, 2011 2:24 pm
Ah, my bad - I didn't see "number line."

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by Tarun Khanna » Thu Oct 27, 2011 11:07 am
hi

plz explain why answer is not D because i took examples taking into consideration statement 1 and always got a NO which means the points are not symmetric...can u give an example whereby using statement 1 points A and B can be symmetric about 0..

thanks
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by GmatMathPro » Thu Oct 27, 2011 12:17 pm
Sure. B=0.5 and A=-0.5
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by Tarun Khanna » Fri Oct 28, 2011 9:08 am
Hey thanks GMAT Math Pro...Thanks alot

One more thing...are there any other pair of numbers apart from -0.5 and 0.5 or this is the only pair...

Rgds
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by GmatMathPro » Fri Oct 28, 2011 9:54 am
Tarun Khanna wrote:Hey thanks GMAT Math Pro...Thanks alot

One more thing...are there any other pair of numbers apart from -0.5 and 0.5 or this is the only pair...

Rgds
Tarun
Nope that's it. Here's one way you can approach it: First, note that if A and B are symmetric with respect to zero, the distance from A to zero is the same as the distance from B to zero by definition, so the phrase "The distance from A to zero is the same as the distance from 1 to B" is identical to "The distance from B to zero is the same as the distance from 1 to B" if we are assuming symmetry. With this rephrasing, it is intuitively clear that B must be the midpoint of 0 and 1.

Or, if you prefer an algebraic solution: The distance from B to 0 is |B|. The distance from 1 to B is |1-B|. IF these two are equal, then |B|=|1-B|. This equation is true if B=1-B or B=B-1. Solving the first gives us 2B=1---->B=1/2. The second equation has no solution. Hence, B=0.5 (and A=-0.5) is the only pair of numbers that works.
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by Tarun Khanna » Fri Oct 28, 2011 10:51 am
AWESOME Pete...Thanks a ton :-) I am also struggling alot with coordinate geometry, mixtures and alligations, Probability and P&C...Any suggestions how to improve or any strategies or any suggested concept reading on these? I score 48-49 in the quant section and with these topics correct mayb i can score 51

my problem is i know all concepts and firmulas of the above topics but MOSTLY ALWAYS fail to apply and solve the question

thanks
tarun
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by parul9 » Fri Oct 28, 2011 11:50 am
Not sure if I missed the mention of an answer for this one.
Is it A by any chance?
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by GmatMathPro » Fri Oct 28, 2011 11:52 am
No, it's B.
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by GmatMathPro » Fri Oct 28, 2011 12:17 pm
Tarun Khanna wrote:AWESOME Pete...Thanks a ton :-) I am also struggling alot with coordinate geometry, mixtures and alligations, Probability and P&C...Any suggestions how to improve or any strategies or any suggested concept reading on these? I score 48-49 in the quant section and with these topics correct mayb i can score 51

my problem is i know all concepts and firmulas of the above topics but MOSTLY ALWAYS fail to apply and solve the question

thanks
tarun
Well, there are no easy answers, obviously. Otherwise, we could all just get perfect scores and totally marginalize the GMAT as a means of student evaluation. Also, it's difficult to say what would help you without knowing more about your personal situation, what you've tried, and what you haven't.

That said, here is some general advice:

1. If you're not already doing it, keep an error log. You need to know EXACTLY what mistakes you're making before you can really correct them.

2. If you're consistently struggling with certain areas but are able to do high-level problems in other areas, there may be problems with your foundation in the areas you struggle with. For example, if you know that there are 5C2=10 ways to choose two books out of five to bring on vacation but you're just applying the formula blindly and don't understand why it works, it's going to be difficult for you to do the more advanced counting problems where subtle adjustments have to be made.

3. Try to achieve some depth of understanding on problems from these areas. That may mean looking for different solutions, varying the conditions of the problem to see how it would change the answer, or anything else that would give you a real sense of what's actually going on beneath the surface. Your goal should be to understand the problems so well that you could teach them to somebody else. Somebody who asks lots of questions.

Hope that helps. If you want help coming up with a more personalized plan, feel free to send me a private message.
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