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Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by sana.noor » Wed Jul 24, 2013 10:33 am
There is an 8-polygons. what is the number of the diagonals of the 8-polygons ?
A. 20 B. 27 C. 32 D. 43 E. 45

8-polygon?
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by [email protected] » Wed Jul 24, 2013 11:13 am
Hi sana.noor,

I assume that you're talking about an 8-SIDED polygon (eg. an octagon; "stop" sign, etc.).

You won't need any special math to answer this question, just draw a picture. I'm going to get you started, then I want you to try to finish the task:

1) Draw the picture.
2) Pick one point. Now how many diagonals can you draw from that point?
3) Count them up.
4) Now pick the next point and repeat the process.
5) Be careful that you don't count a diagonal twice (a diagonal from point A to D is the SAME line as from D to A, so you only count this line ONCE).

You'll end up with the correct answer.

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by satinder kaur » Thu Jul 25, 2013 7:55 am
To find the number of diagonals of a polygon.
Since its a eight sided polygon having eight vertices and diagonal forms by connecting two vertices.

hence : 8C2 - 8(number of sides)
where 8C2 will give all possible values.

Answer is : 20 (A)

Thanks.
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by [email protected] » Thu Jul 25, 2013 10:10 am
Hi satinder kaur,

Unfortunately, you can't use the combination formula in the way that you've described:

8c2 = 56/2 = 28 NOT 20 (the correct answer)

8c2 would give you all of the possible PAIRS of points, but not every pair is a diagonal. If you draw an 8-sided shape and pick a point, then there are 7 OTHER points, but only 5 of them would form diagonals (2 of the points are on straight lines with the point that you started with, and as such are NOT considered diagonals).

I've always been a proponent of solving a problem in the easiest way possible. No special math is required here - an 8 year old can draw a picture with a crayon and answer this question, which means that you can too!

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by sandmandreams » Fri Aug 16, 2013 12:49 am
Hi Rich

Why is the combination solution not possible?

Isn't it basically 8!/(2!8!) for all line combinations, and then deduct all the sides, in this case 8?

I'm asking because I try to avoid listing down possibilities since the time pressure gets to me and I tend to overlook or re-list other possibilities.
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by [email protected] » Fri Aug 16, 2013 4:18 pm
Hi sandmandreams,

The reason why you can't use the combination formula (as is) on this question is because NOT every set of 2 points creates a diagonal. If you draw an 8-sided shape, and pick one of the points, you'll notice that 2 points (out of the 8) are ALREADY CONNECTED to that point, so they can't be diagonals that run through the shape.

As a simpler example, draw a square. Notice that the number of diagonals is NOT 4c2? The same reason applies here.

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by sandmandreams » Sat Aug 17, 2013 1:55 am
Hi rich, i get that point. But what I meant was: isn't subtracting the # of sides basically taking out the points that are already connected?

Sorry for pestering regarding this, but I just wanted to check in case of the unlikely event of a similar problem but with a 20-sided polygon instead (for example).

Thanks in advance.
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by [email protected] » Sat Aug 17, 2013 2:35 pm
Hi sandmandreams,

Yes, as long as you're sure to REMOVE those 8 sides, then that approach will get you the correct answer. By itself though, the combination formula does NOT provide the correct answer. If this were on the actual GMAT, on this type of question, it would be likely that one of the answers would be 28, which would catch some people who weren't perfectly clear on how the high-level math works (and thus they'd be wrong and not know it).

Since the answers are relatively small and drawing the picture/lines wouldn't take much time, I'm going to reiterate the simplicity of the approach that I mentioned earlier. On Test Day, you want to solve problems in the easiest way possible, so that you don't get hung up on long-winded math (and thus burn through your 75 minute clock) and you keep your stress level low.

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