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How many sides?

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by ronaldramlan » Sun Aug 07, 2011 5:03 am
The measures of the interior angles in a polygon are consecutive integers. The smallest angle measures 136 degrees. How many sides does this polygon have?

A) 8
B) 9
C) 10
D) 11
E) 13

There you go ...
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by GmatKiss » Sun Aug 07, 2011 5:15 am
IMO:E
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by Anurag@Gurome » Sun Aug 07, 2011 5:38 am
ronaldramlan wrote:The measures of the interior angles in a polygon are consecutive integers. The smallest angle measures 136 degrees. How many sides does this polygon have?

A) 8
B) 9
C) 10
D) 11
E) 13
Say, the number of sides of the polygon is n.
Hence, sum of the interior angles of the polygon = (n - 2)*180 degrees

Smallest angle is 136 degrees and all the interior angles are consecutive integers. Hence, the measures of the interior angles of the polygon in degrees are 136, (136 + 1), (136 + 2), ..., and (136 + n - 1).

Hence, sum of the angles = [136 + (136 + 1) + (136 + 2) + ... + (136 + n - 1)] = 136n + n(n - 1)/2

Thus, 136n + n(n - 1)/2 = (n - 2)*180
----> 272n + n² - n = 360n - 720
----> n² -89n + 720 = 0
----> (n - 9)(n - 80) = 0

Now, if n is equal to 80, then some of the interior angles of the polygon will be greater than 180. Hence, only possible value of n is 9.

The correct answer is B.
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by Ozlemg » Sun Aug 07, 2011 7:44 am
Woow!!! Thank you indeed!
Mentality of the problem is simple; and the explanation of the answer is easy to follow but construstion of the equations is hell hard! I think this can be sharpen by practice and somehow, developing an ability to see the big picture...


Thank you again.
Ozlem
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by goalevan » Sun Aug 07, 2011 2:07 pm
I avoided the quadratic and approximated by taking [(n-2)*180]/n] = 140
180n - 360 = 140n
40n = 360
n = 360/40
n = 9

Then listed the numbers to ensure it worked with 136 as the smallest integer:

136, 137, 138, 139, 140, 141, 142, 143, 144

B
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by mk101 » Mon Aug 08, 2011 8:02 am
ronaldramlan wrote:The measures of the interior angles in a polygon are consecutive integers. The smallest angle measures 136 degrees. How many sides does this polygon have?

A) 8
B) 9
C) 10
D) 11
E) 13

There you go ...
Another approach to this question -
sum of interior angles of a polygon = (n-2)180 - this must have "0" at its unit's place
the smallest angle is 136, which has 6 at the units place - Now you have to add the units place of consecutive digits i.e. - 6 + 7 + 8 + 9 + 0 +1 +2 +3 +4 = gives a 0 at the units place.

check the options if 9 sides exist as one of the answers. Ans b
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