If in a regular polygon, the number of diagonals are 54 , then the number of sides of the polygon are
If in a regular polygon, the number of diagonals are 54 , then the number of sides of the polygon are
10
12
9
6
Solution
Number of diagonals in a polygon of $\mathrm{n}$ sides is ${ }^{\mathrm{n}} \mathrm{C}_2-\mathrm{n}$.
$\begin{aligned}
\therefore \quad & { }^{\mathrm{n}} \mathrm{C}_2-\mathrm{n}=54 \\
& \Rightarrow \frac{\mathrm{n}(\mathrm{n}-1)}{2}-\mathrm{n}=54 \\
& \Rightarrow \mathrm{n}^2-3 \mathrm{n}-108=0 \\
& \Rightarrow(\mathrm{n}-12)(\mathrm{n}+9)=0
\end{aligned}$
But, number of sides cannot be negative.
$\therefore \quad \mathrm{n}=12$