Question

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

(a) 8 

(b) 9 

(c) 12 

(d) 19

Answer 1

(b) 9

Let the polygon have n sides. 

Then, sum of its interior ∠s (S_n) = (2n – 4) rt. ∠s 

= (n – 2) × 180°                           ...(i) 

Number of sides of the polygon = number of interior angles of the polygon = n.

The interior angles form an A.P. with first term = 120° and common difference 5°.

∴ S_n = \(\frac{n}{2}\) [2 × 120° + (n – 1) × 5°]

= \(\frac{n}{2}\) [240° + 5n – 5°]                              ...(ii) 

From (i) and (ii), 

(n – 2) × 180° = \(\frac{n}{2}\) [240° +  5n – 5°]

⇒ (n – 2) × 360 = 5n² + 235n 

⇒ 5n² + 235n – 360n + 720 = 0 

⇒ 5n² – 125n + 720 = 0 ⇒ n² – 25n + 144 = 0 

⇒ (n – 16) (n – 9) = 0 ⇒ n = 16 or 9. 

when n = 16, the last angle a_n = a + (n – 1) d 

= 120° + (16 – 1) × 5° = 195° 

which is not possible. 

Hence, n = 9.