Question:

A polygon has 27 diagonals. How many sides does it have?

(a) 7

(b) 8

(c) 9

(d) 12

Solution:

(c) 9

$\frac{n(n-3)}{2}=27$

$\Rightarrow n(n-3)=54$

$\Rightarrow n^{2}-3 n-54=0$

$\Rightarrow n^{2}-9 n+6 n-54=0$

$\Rightarrow n(n-9)+6(n-9)=0$

$\Rightarrow n=-6$ or $n=9$

Number of sides cannot be negative.

$\therefore \mathrm{n}=9$