Question

The interior angles of a polygon are in A.P. The smallest angle is 120 degree and the common difference is 5degree. Find the number of sides of the polygon.

Answer 1

Answer:

Smallest angle=120degree

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)

5n^2 -125n +720 = 0

n^2 -25n +144=0

n=9 or 16

hence number of sides can be 9 or 16

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Answer 2

SOLUTION

Suppose their are n sides of the polygon. The sum of the interior angles is given by

Suppose their are n sides of the polygon. The sum of the interior angles is given bysn= (2n-4)π/2 = (n-2)π

Since, interior angles forms an A.P. with

a= 120°

d= 5°

sn= n/2[ 2a+(n-1)d]

=) sn= n/2[2×120° +(n-1)5°]

=) (n-1)180° =n/2[240° +(n-1)5°] [from1]

=) (n-2)× 360° = n(5n+235°)

=) n^2 -25 +144 = 0

=) n= 9,16

when n= 16, then the last angle

an= a+(n-1)d

=) 120° +(16-1) 5°

=) 120° + 15×5°

=) 120°+ 75°

=) 195°, which is not possible.

Hence, required value of n is 9.

hope it helps ☺️