Question

The difference between the exterior angles of two regular polygons, having the sides equal to (n – 1) and (n + 1) is 9°. Find the value of n.

Answer 1

We know that sum of exterior angles of a polynomial is 360°

(i) If sides of a regular polygon = n – 1

Then each angle = `360^circ/("n" - 1)`

and if sides are n + 1, then

each angle = `360^circ/("n" + 1)`

According to the condition,

`360^circ/("n" - 1) - 360^circ/("n" + 1)=9`

`=> 360 [1/("x" - 1) - 1/("x" + 1)] = 9`

`=> 360 [("n" + 1 - "n" + 1)/("n" - 1)("n" + 1)] = 9`

`=> (2 xx 360)/("n"^2 - 1) = 9`

 `=> "n"^2 - 1 = (2 xx 360)/9 = 80`

`=> n^2 - 1 = 80`

`=> n^2 = 1 - 80 = 0`

⇒ n² - 81 = 0

⇒ (n)² - (9)² = 0 

⇒ (n + 9)(n - 9) = 0

Either n + 9 = 0. then n = -9 which is not possible being negative,

or n - 9 = 0, then n = 9

∴ n = 9

∴ No. of. sides of a regular polygon = 9