Question

The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.

Options

• 8

• 9

• 10

• 16

Answer 1

The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is 9.

Explanation:

From geometry, the sum of all internal angles

= (n – 2) × 180°

where n is the number of sides of the polygon.

∴ `n/2[2 xx 120^circ + (n - 1) xx 5^circ]` = (n – 2) × 180°

`\implies` n² – 25n + 144 = 0

`\implies` (n – 16) (n – 9) = 0

If n = 16 then the 16^th internal

angle = 120° + (16 – 1) × 5°

= 195° > 180°

∴ n ≠ 16.

Hence n = 9