rxs0005 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
OFFICIAL SOLUTION
We are told that the smallest angle measures 136 degrees--this is the first term in the consecutive set. If the polygon has S sides, then the largest angle--the last term in the consecutive set--will be (S - 1) more than 136 degrees.
The sum of consecutive integers = (Average Term) * (# of Term)= \(\frac{First + Last}{2}\) * (# of Terms).
Given that there are S terms in the set, we can plug in for the first and last term as follows:
\(\frac{136 + 136 + (S-1)}{2} * S\) = sum of the angles in the polygon.
We also know that the sum of the angles in a polygon = 180 (S-2) where S represents the number of sides.
Therefore: \(180(S-2) = \frac{136+136+(S-1)}{2} * S\). We can solve
for S by cross-multiplying and simplifying as follows:
\(2(180)(S-2) = [272 + (S-1)] S\)
\(360S - 720 = (271 + S)S\)
\(360S - 720 = 271S + S^2\)
\(S^2 - 89S + 720 = 0\)
A look at the answer choices tells you to try (S - 8), (S - 9), or (S - 10) in factoring.
As it turns out (S-9)(S-80)=0, which means S can be 9 or 80. However S cannot be 80 because this creates a polygon with angles greater than 180.
Therefore S equals 9; there are 9 sides in the polygon.
The correct answer is
B.