Question

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Answer 1

HERE IS YOURS SOLUTION;

◆ The angles of the polygon should form an AP.

So,

d= 5 or 5°

a= 120 or 120°

◆ We are using the formula for polygon to reference of angles & taking the number of sides as n.

So, the Formula comes out to be 180 × (n-2)

Sn = 180 × (n-2) = n/2 [ 2a +(n-1)d ]

=> n/2 [ 2 × 120 + (n-1) 5 ]

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

=> n [ 240 + (n-1)5] = 180 ×2 (n-2)

=> n [ 240 + (n-1)5] = 360 (n-2)

=> 240n +5n^2 -5n =360 (n-2)

=> 235n +5n^2 = 360n - 720

=> 235n -360n +5n^2 +720 =0

=> -125n +5n^2 +720 =0

=> 5n^2 -125n +720 =0

=> n^2 - 25 n + 144 =0 【 Divided By 5 】

=> n^2 - 16n -9n +144 =0 【 Splitting The Middle Term 】

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

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

=> n = 9 & 16

◆ So, the number of sides of the polygon is 9 or 16 ◆

HOPE IT HELPS

Answer 2

Hey!!!!

A difficult question

______________

ATQ,

=> Smallest angle = 120°

Difference = 5°

Next angle = 120 + 5 = 125°

let the number of sides be n

Then sum of all angles => (n - 2)180 (formula)

=> Sum of all interior angle = (n - 2)180 ------(1)

Thus can we say that the angles are in AP

=> 120° , 125° , 130° , ...

Here, a = 120
d = 5

Now we have the main calculation part

We know Sn =

$\frac{n}{2} (2a + (n - 1)d)$

Let's replace values

$= > 180(n - 2) = \frac{n}{2} (240 - (n - 1)5)$

=> 360(n - 2) = n(240 - 5n + 5)

=> 360n - 720 = n(245 - 5n)

=> 360n - 720 = 235n - 5n²

=> 5n² - 125n + 720 = 0

=> n² - 25n + 144 = 0

By middle term splitting method

=> 144 = 16 x 9

Thus

=> n² - 16n - 9n + 144 = 0

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

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

Thus n = 9 or n = 16

Thus the polygon may be nonagon or with 16 sides.

__________

The question was tricky

Hope this helps ✌️

Legally Good Morning :-)