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Math

by vaswani.sharan » Sun Apr 15, 2012 11:31 am
The difference between any two consecutive interior angles of a polygon is 5. If the smallest angle is 120 , find the number of the sides of the polygon.
a)5 b)13 c)33 d)7 e)9
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by shubham_k » Sun Apr 15, 2012 11:42 am
The sum of the angles in an n-sided shape is (n-2)*180. We know:

120 + 125 + 130 + ... = (n-2)*180

If we have n angles, the largest angle will be 120 + (n-1)*5 = 115 + 5n. We're adding an evenly spaced series here, so we can find the sum, using the average fomula:

sum = (average)*(number)

Because the series is evenly spaced, the average of the series is equal to the average of the smallest (120) and the largest (115 + 5n). There are n terms, so:

sum = [(120 + 115 + 5n)/2]*n
(n-2)*180 = (235 + 5n)*n/2
360n - 720 = 235n + 5n^2
72n - 144 = 47n + n^2
0 = n^2 - 25n + 144
0 = (n-16)(n-9)
n = 9 or n = 16.

We must discard the n=16 solution, because if n=16, the angles go past 180, and you can't have a 180 degree angle as an interior angle in a polygon- it makes a straight line
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by Neo Anderson » Sun Apr 15, 2012 11:46 am
sum of external angle of a polygon is always 360

external angle is given by 180 - the internal angle

thus the sum of all external angles of this polygon 60+55+50+45+..... = 360

or n/2*(2*60-5*(n-1))=360

on solving for n you get n= 9 or 16

hence the answer e
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by Anurag@Gurome » Sun Apr 15, 2012 5:42 pm
vaswani.sharan wrote:The difference between any two consecutive interior angles of a polygon is 5. If the smallest angle is 120 , find the number of the sides of the polygon.
a)5 b)13 c)33 d)7 e)9
Sum of the interior angles of an n-sided polygon = (n - 2) * 180
It is given that the smallest angle = 120 and difference between any two consecutive interior angles of a polygon is 5.
So, 120 + 125 + 130 + ... = (n - 2) * 180
Value of the largest angle = 120 + (n - 1) * 5 = 120 + 5n - 5 = 115 + 5n

Now sum of the interior angles of n-sides polygon = {[120 + (115 + 5n)]/2} * n
(n - 2) * 180 = [(235 + 5n) * n]/2
2(180n - 360) = (235 + 5n) * n
360n - 720 = 235n + 5n²
5n² - 125n + 720 = 0
n² - 25n + 144 = 0
(n - 9)(n - 16) = 0 implies n = 9 or 16
n = 16 is not possible as in this case an interior angle in a polygon is a straight line, which is not possible. Hence [spoiler]n = 9[/spoiler].

The correct answer is E.
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