Question

The interior angles of a polygon are in Arithmetic Progression. If the smallest angle be $120°$ and the common difference be $5°$, then the number of sides of the polygon is

• A. $8$
• B. $10$
• C. $9$
• D. $6$

Answer 1

The correct option is C

$9$

Step 1: Find the first term $a$ and common difference $d$

Let the number of sides of the polygon is $n$.

It is given that, the interior angles are in Arithmetic Progression with smallest angle $120°$ and the common difference $5°$.

Thus, the sequence of the interior angles can be given by, $120°,125°,130°,......$

So, the sequence of the exterior angles can be given by, $\left(180-120\right)°,\left(180-125\right)°,\left(180-130\right)°,.......$

$\Rightarrow 60°,55°,50°,......$

Thus, the exterior angles are also in Arithmetic Progression with the first term, $a=60°$ and the common difference, $d=\left(-5\right)°$.

It is known that the sum of all exterior angles of a polygon is $360°$.

Step 2: Find the number of sides of the polygon based on given information

So, sum of $n$ exterior angles can be given by, $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]=360$.

$$\begin{gathered} \Rightarrow \frac{n}{2}\left[\left(2\times 60\right)+\left(n-1\right)\times \left(-5\right)\right]=360 \\ \Rightarrow n\left[120+\left(-5\right)n+5\right]=360\times 2 \\ \Rightarrow n\left[125-5n\right]=720 \\ \Rightarrow 125n-5n^2=720 \\ \Rightarrow 5n^2-125n+720=0 \\ \Rightarrow 5\left[n^2-25n+144\right]=0 \\ \Rightarrow n^2-16n-9n+144=0 \\ \Rightarrow n\left(n-16\right)-9\left(n-16\right)=0 \\ \Rightarrow \left(n-16\right)\left(n-9\right)=0 \end{gathered}$$

So, the roots of the quadratic equation are $n=16$ and $n=9$.

For $n=16$, the value of exterior angle is negative, so $n\ne 16$.

Hence, the number of sides of the polygon is $9$, so, the correct answer is option (C).