The difference between any two consecutive interior angles of a polygon is 5 degree
Answers
9 Sides
Complete Question:
The difference between any two consecutive interior angles of a polygon is 5 degree
. If the smallest angle is 120, find the number of sides of the polygon.
First let us assume the number of sides of the polygon to be n.
Given:
First term of the Arithmetic Progression (a)
= 120
Common difference of the AP (d)
= 5
The Ap of the following question would be of the form:
120, 125, 130, 135, 140.......
Calculating:
The formula that is used to calculate the sum of n terms of an Arithmetic Progression:
Sn = n/2(2a + (n - 1) d)
Substituting all the values given to us into the following formula we get:
Sn = n/2(2(120) + (n - 1) (5)
Sn = n/2 (240 + 5n - 5)
Sn = n/2 (235 + 5n)
We are aware that the sum of interior angles of n sided polygon is
(n - 2) (180)
So putting the value of Sn as this we get:
(n - 2) (180) = n/2 (235 + 5n)
360 n - 720 = 235 n + 5n^2
5n^2 - 125 n + 720 = 0
Dividing this quadratic equation by 5 to simplify we get:
n^2 - 25n + 144 = 0
After solving this quadratic equation by splitting the middle term we get:
(n - 16) (n - 9) = 0
Number of sides can be 16 or 9
If the number of sides is 16 then,
a16 = 120 + 15 x 5 = 195 > 180 (It should be less than 180 not greater)
Which is not possible so we reject this value.
If the number of sides is 9 then,
a9 = 120 + 8 x 5 = 160
Which is possible
Therefore, the number of sides of the polygon is 9 Sides.