SOLUTION: The interior angles of a polygon have measures 170 degrees, 160 degrees, 150 degrees, 140 degrees... down to some smallest angle. The numbers 170 degrees, 160 degrees, 150 degrees,

Algebra ->  Polygons -> SOLUTION: The interior angles of a polygon have measures 170 degrees, 160 degrees, 150 degrees, 140 degrees... down to some smallest angle. The numbers 170 degrees, 160 degrees, 150 degrees,      Log On


   

Question 174420: The interior angles of a polygon have measures 170 degrees, 160 degrees, 150 degrees, 140 degrees... down to some smallest angle. The numbers 170 degrees, 160 degrees, 150 degrees, 140 degrees are numbers of an Arithmetic Progression. Find the number of sides of the polygon.
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
The interior angles of a polygon have measures 170 degrees, 160 degrees, 150 degrees, 140 degrees... down to some smallest angle. The numbers 170 degrees, 160 degrees, 150 degrees, 140 degrees are numbers of an Arithmetic Progression. Find the number of sides of the polygon. 

First term = a%5B1%5D=170, difference = d=-10

Formula for the sum of the first n terms:

S%5Bn%5D=+%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%29

Formula for the sum of the interior angles of an 
n-sided polygon:

sum+=+%28n-2%29%2A180

Set the right sides of the formulas equal:

%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%29=%28n-2%29%2A180

%28n%2F2%29%282%28170%29%2B%28n-1%29%28-10%29%29=%28n-2%29%2A180

%28n%2F2%29%28340-10n%2B10%29=180n-360%29

Multiply both sides by 2:

%28n%29%28340-10n%2B10%29=360n-720%29

Distribute:

340n-10n%5E2%2B10n=360n-720

Combine like terms:

350n-10n%5E2=360n-720

Get 0 on the right:

-10n-10n%5E2%2B720=0

Rearrange in descending order:

-10n%5E2-10n%2B720=0

Divide through by -10:

n%5E2%2Bn-72=0

Factor left side:

%28n%2B9%29%28n-8%29=0

Use zero-factor property:

   

Discard the negative answer.

It's an 8-sided polygon, or octagon.

Edwin