+0

# polygon

0
125
2

The interior angles of a convex polygon are in arithmetic progression. The smallest angle is 120 degrees and the common difference is 5 degrees.  Find the number of sides of the polygon.

Jan 12, 2021

#1
+117447
+2

Let  the first term  be 120

Let the last term  = 120 + 5(n - 1)

The sum of the interior angles  =  (number of sides - 2) 180

Sum  of the interior angles  =     [ 1st term +  last term ] (number of terms/2 ]

First term  =120

Last term = 120 + 5(n - 1)

So

[120  +  120 + 5(n-1) ]  (n /2)  =   (n - 2) 180

[ 240 + 5n  - 5 ]  n    =  (n - 2) 360

[ 235 + 5n ] n =  (n - 2)360

235n +  5n^2  =  360n - 720      rearrange   as

5n^2  - 125n +  720  = 0         divide through  by  5

n^2  - 25n  +  144  =  0      factor as

(n - 16) ( n - 9)  =  0

We have two

Set each factor to 0  and solve for n

n =  16  or  n  = 9

We have two possibilities

n =  9      or      n   = 16

Jan 12, 2021
#2
+303
0

Smallest angle=120degrees

Common difference=5

A P is 120, 125, 130,..

The sum of interior angles of a polygon= (n-2)180

Hence Sum of n terms of an A P = (n-2)180

n/2 {2.120+(n-1)5} = 180(n-2)

5n2 -125n +720 = 0

n2 -25n +144=0

n=9 or 16

Hence number of sides can be 9 or 16

-hihihi

😎😎😎

Jan 12, 2021