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How many sides?

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How many sides?

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
The more you suffer before the test, the less you will do so in the test! :)
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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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