Angles
Geometry
Level
3
The smallest interior angle of a particular convex polygon T has measure 120 degrees. From the angle that is 120 degrees, each one moving clockwise is 5 degrees bigger than the angle before. How many sides does this polygon have?
-NCML, Terry Yu
There are 2 answers!
9
not possible
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1 solution
For n=number of sides
1 8 0 n − 3 6 0 = n × 2 ( 1 2 0 + ( 1 2 0 + 5 ( n − 1 ) ) )
3 6 0 n − 7 2 0 = 5 n 2 + 2 3 5 n
5 n 2 − 1 2 5 + 7 2 0 = 0
n 2 − 2 5 n + 1 4 4 = 0
After you plug that in to the quadratic formula, you get a 9 sided polygon and a 15 sided polygon (the quadratic equation says it's 16 but remember when the angle is 180 degrees that is 2 sides in the quadratic equation but 1 in a drawing). But wait! The problem states a Convex polygon, and the polygon with 16 sides is not a convex polygon (as a polygon with 12 or more sides has angles more than 180 degrees). Therefore, the answer is 9 .