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Polygon is not possible

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Polygon is not possible

by sanju09 » Sat Feb 28, 2009 4:05 am
How many sides are there in a convex polygon whose consecutive interior angles increase by 5º each, in a cycle order; with the smallest interior angle being 120º?
A. 12
B. 16
C. 9
D. 15
E. Polygon is not possible
Last edited by sanju09 on Mon Mar 02, 2009 4:29 am, edited 1 time in total.
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by dendude » Sat Feb 28, 2009 11:22 am
The total of the internal angles of a polygon is (n-2)*180 where n represents the # of sides of the polygon

If the internal angles differ by 5 and the smallest angle is 120, then an AP can be set up
a + (a+d) + (a+2d) + ...... (a+(n-1)d) where n will be the total number of terms, d the difference between each term and a the starting term

In this context a = 120 and d = 5
Sum of an AP is given as (n/2)*[2a + (n-1)d]

Now this should be equal to the sum of the internal angles of the polygon (n-2)*180

Equate these two to arrive at, n = 16 or 9.

Sorry I had to use AP's to arrive at the answer. I could'nt think of any other simpler way. If anyone has one pls post.
Last edited by dendude on Sat Feb 28, 2009 3:19 pm, edited 2 times in total.
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by Uri » Sat Feb 28, 2009 2:06 pm
Using A.P., we get the sum of all the interior angle as
n/2 * [(2* 120) + (n-1) * 5] = n/2 * [235 + 5*n]

We know that the sum of the interior angles of a polygon = (n-2) * 180

Equating the above two, we get

n/2 * [235 + 5*n] = (n-2) * 180

or, 5n^2 - 125n +720 = 0

or, n^2 - 25n + 144 = 0

or, (n-16) * (n-9) = 0

so, n = 16 or 9

If n=16, then the biggest angle of the polygon will be 120+(15*5)= 195 and the polygon will not anymore be convex (because a convex polygon will have all its interior angles less than 180 degree). So, the polygon can have 9 sides ( in this case, the biggest angle will be 160 degree).

Ans. (C)

sanju, from the topic of your post, it seems that the OA is (E). please confirm the OA and the source of the problem, if possible.
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how about this?

by jackcrystal » Sat Feb 28, 2009 2:27 pm
largest angle will be less then 180.
Now there are 12 angles from 120 to 175 with difference of 5 degrees.

120
125
130
135
.
..
175


Therefore, 12 sides.
Last edited by jackcrystal on Sat Feb 28, 2009 2:42 pm, edited 1 time in total.
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Re: how about this?

by Uri » Sat Feb 28, 2009 2:31 pm
jackcrystal wrote:largest angle will be 180.
can the largest angle be 180 degree? if it is 180, then where will the vertex be for this angle?
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by dendude » Sat Feb 28, 2009 3:17 pm
jackcrystal: I dont think there's a limit on the largest angle being 180

Uri: Thanks for your post, I did indeed overlook something while I hurriedly solved the quadratic. :shock:
You are right, the quadratic results in n=16 or 9
I've edited my post above

Which now leads me to question, which answer should be chosen?
Sanju can you give us the OA and tell us if the answer choices are indeed what you have posted?
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by Uri » Sat Feb 28, 2009 3:28 pm
dendude wrote:jackcrystal: I dont think there's a limit on the largest angle being 180

Uri: Thanks for your post, I did indeed overlook something while I hurriedly solved the quadratic. :shock:
You are right, the quadratic results in n=16 or 9
I've edited my post above

Which now leads me to question, which answer should be chosen?
Sanju can you give us the OA and tell us if the answer choices are indeed what you have posted?
i have edited my first post after i sent you a PM. and i believe that the answer should be 9. i have also provided the reason for excluding 16.

sanju, the OA is eagerly awaited.
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by dendude » Sat Feb 28, 2009 3:58 pm
Uri wrote:i have edited my first post after i sent you a PM. and i believe that the answer should be 9. i have also provided the reason for excluding 16.
Thanks Uri
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by Uri » Sat Feb 28, 2009 4:33 pm
once again, i am here. this problem seems to be an atom bomb!!! now i feel the problem has to be more visualised than solved mathematically.
How many sides are there in a convex polygon whose consecutive interior angles differ by 5º each; with the smallest interior angle being 120º?
i don't think we can construct any polygon with consecutive angles differing 5 degree. after solving mathematically, i have shown earlier that the polygon can have 9 sides. and as per the problem, the angles will be 120, 125, 130, 135, 140, 145, 150, 155 and 160. now, if we start constructing the polygon with the first angle being 120 degree, then the next angle will be 125 and as we progress, the angles will spread even further. thus ultimately the polygon will not be completed. this will be the same in every case, irrespective of which angle we try to draw first.

let us consider drawing the first side with the angle being 140. now as we progress, the angles will increse. once we reach 160, then the difference between the next angle and this one will not be 5 degree any more, as we are left with the angles 120, 125, 130 and 135. so the condition of the problem will be violated.

i hope, this time i have come up with my final answer to this question!

but to visualise this way, i have simplified the problem a bit, taking clue from another almost similar thread that you created within 5 minutes of creating this one. i have assumed that the angles increase in clockwise or counter-clockwise direction. if it is not so, then the two adjacent angles to 120 deg can each be 125 deg. and then the two adjacent angles to any of the 125 deg angle can be 120 deg, or 130 deg and 120 deg. and this can continue perhaps in endless combination. thus the answer will most probably not be a standard one and the question will ultimately be considered badly worded.
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by sanju09 » Mon Mar 02, 2009 4:46 am
Those who feel that the angles, if taken in increasing order, would eventually leave an opengon; should consider the decreasing order to close it somehow! The polygon is yes possible, one can check it practically with the help of geometrical weapons, and the OA is [spoiler]" it is a nonagon"[/spoiler]; mind the edited version of this question only.

The edited version is:

[spoiler]How many sides are there in a convex polygon whose consecutive interior angles increase by 5º each, in a cycle order; with the smallest interior angle being 120º?[/spoiler]

The question is taken from an NCERT 11 grade text-book published in India, where the original wording is really wooly!
Last edited by sanju09 on Mon Mar 02, 2009 5:30 am, edited 1 time in total.
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by Uri » Mon Mar 02, 2009 5:02 am
well, the answer to the edited qn is 9. but if the previous version, which i have quoted in my earlier post, is considered, then i believe that it will not form a polygon.
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by sanju09 » Mon Mar 02, 2009 5:28 am
Uri wrote:well, the answer to the edited qn is 9. but if the previous version, which i have quoted in my earlier post, is considered, then i believe that it will not form a polygon.
B-) Editing was necessary for the imposition of AP here, E is not the OA in either case.
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