Question

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

Answer 1

ANSWER:

Let the polygon be 'n' sided

Sum of interior angles

Sn = (n - 2)180°

Given

smallest angle ( a ) = 120°

Common difference ( d ) = 5°

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

(n - 2) 180° = n/2[2(120) + (n - 1)5]

(n - 2) 360° = n[240 + 5n - 5]

360n - 720° = 235n + 5n²

360n - 720 - 235n - 5n² = 0

- 5n² - 125n - 720 = 0

Changing the signs

5n² - 125n + 720 = 0

Divide with 5 on both sides

n² - 25n + 144 = 0

Solve the quadratic equation using factorisation

Split the middle term

n² - 16n - 9n + 144 = 0

Taking common

n(n - 16) - 9(n - 16) = 0

(n - 16)(n - 9) = 0

n = 16 ( or ) 9

Hence the polygon has 16(or)9 sides