+0  
 
0
125
2
avatar

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
avatar+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

 

 

cool cool cool

 Jan 12, 2021
 #2
avatar+303 
0

I got this answer from https://www.topperlearning.com/answer/the-interior-angles-of-a-polygon-are-in-apif-the-smallest-angle-is-120-degree-and-the-common-difference-is-5-degree-then-the-no-of-sides-in-the-polygo/tk5eo455 so check it out

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

24 Online Users

avatar
avatar
avatar
avatar