[tex] \frac{n(n-3)}{2}=d \ \ \ \ \ \text{[n - number of sides, d - number of diagonals] } \\ \\ \frac{n(n-3)}{2}=27 \\ n(n-3)=27*2 \\ n^2-3n = 54 \\ n^2-3n-54=0 \\ discriminant=(-3)^2-4*1*(-54)=225 \\ n_1= \frac{3- \sqrt{225} }{2}= \frac{3-15}{2} = -6 \ \ \ \O \\ n_2= \frac{3+ \sqrt{225} }{2}= \frac{3+15}{2} = 9[/tex]
The polygon has 9 sides.
