Question

The internal angles of a convex polygon are inA.P. The smallest angle is $120^{\circ }$ and the common difference is $5^{\circ }$. The number to sides of the polygon is

• A. 8
• B. 9
• C. 10
• D. 16

Answer 1

Answer: (B)

From geometry, the sum of all internal angles $=(n-2)\times 180^{\circ }$
where $n$ is the number of sides of the polygon.
$\therefore \frac{n}{2}\left[2\times 120^{\circ }+(n-1)\times 5^{\circ }\right]=(n-2)\times 180^{\circ }$
$\Rightarrow n^2-25n+144=0$
$\Rightarrow (n-16)(n-9)=0$
If $n=16$ then the $16^{\text{th }}$ internal angle
$=120^{\circ }+(16-1)\times 5^{\circ }=195^{\circ }>180^{\circ }$
$\therefore n\ne 16$. Hence $n=9$