+0  
 
0
49
1
avatar

The interior angles of a convex polygon are in an arithmetic progression. If the smallest angle is  $100^{\circ}$ and common difference is  $4^{\circ}$, then find the number of sides.

 Jan 21, 2021
 #1
avatar+116126 
+1

Let  the   first angle  = 100

And let  the greatest angle be   ( 100 + 4(n -1) )   where n  is the  number of sides

 

Sum  of  the  interior  angles  = 180 ( n - 2)

 

So....using the  formula for the sum of an arithmetic progression  we have that

 

(100  +  100 + 4 ( n - 1) ) ( n /2)  =  180 ( n - 2)

 

( 200 + 4n - 4)   ( n/2)   =180 (n  - 2)

 

(196 + 4n) (n /2)   =180 ( n- 2)

 

(98 + 2n)n   = 180 n  -  360

 

98n + 2n^2   = 180n - 360

 

2n^2  - 82n + 360  =  0

 

n^2  - 41n + 180   =  0

 

(n - 45) ( n + 4)  =  0

 

Setting the first  factor to 0  and solving for  n  gives

 

n = 45  = the number of sides

 

 

cool cool cool

 Jan 21, 2021

50 Online Users

avatar
avatar
avatar
avatar
avatar
avatar