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
a)5 b)13 c)33 d)7 e)9
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
