Question

The interior angles of a convex polygon are in A.P. The smallest angle is \[{120^ \circ }\] and the common difference is \[{5^ \circ }\] . The number of sides is
A. \[7\]
B. \[16\]
C. \[9\]
D. \[9\] or \[16\] both values being permissible

Answer 1

Hint: Using the formula of sum of all the angles of a polygon form an equation. Recall the properties of an arithmetic progression. Check how the numbers of an arithmetic progression related to each other and what the sum of all the numbers of the series is. Use these facts to find the number of sides of the given convex polygon.

Complete step-by-step answer:
Given, smallest angle is \[{120^ \circ }\]
Common difference between the angles is \[{5^ \circ }\]
Let the number of sides of the convex polygon be \[n\] .
For any polygon sum of its interior angles is
\[S = (n - 2){180^ \circ }\] (i)
It is said that interior angles of a convex polygon are in A.P that is arithmetic progression. By arithmetic progression we mean a series of numbers in which the difference between any two consecutive numbers is the same.
The smallest angle here is \[{120^ \circ }\] and the common difference is \[{5^ \circ }\] . So, the angles of the convex polygon will be \[{120^ \circ },{125^ \circ },{130^ \circ },{135^ \circ }\] and so on.
Sum of all the numbers of an arithmetic progression is given by,
 \[{\text{Sum}} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right] \] , where \[n\] is the number of terms, \[a\] is the first term of the series and \[d\] is the common difference.
Here, \[a = {120^ \circ }\] , \[d = {5^ \circ }\]
Using these values in the general equation we get,
 \[{\text{S}} = \dfrac{n}{2}\left[ {2 \times 120 + (n - 1)5} \right] \]
 \[ \Rightarrow {\text{S}} = \dfrac{n}{2}\left[ {240 + (n - 1)5} \right] \] (ii)
Equating (i) and (ii) we get,
 \[\dfrac{n}{2}\left[ {240 + (n - 1)5} \right] = (n - 2)180\]
 \[\Rightarrow 240n + 5{n^2} - 5n = 360n - 720 \\
   \Rightarrow 5{n^2} - 125n + 720 = 0 \\
   \Rightarrow {n^2} - 25n + 144 = 0 \]
 \[\Rightarrow {n^2} - 16n - 9n + 16 \times 9 = 0 \\
   \Rightarrow (n - 16)(n - 9) = 0 \\
   \Rightarrow n = 16,9 \]
Therefore, the number of sides can be \[9\] or \[16\] .
Hence, the correct answer is option (D) \[9\] or \[16\] both values being permissible
So, the correct answer is “Option D”.

Note: A polygon is a closed plane figure with finite number of line segments or we can say a closed plane figure having more than two sides. Here, convex polygon was mentioned, convex polygon is a polygon having all its angles less than \[{180^ \circ }\] and a line segment joining any two vertices of the polygon lies inside the polygon.