Question

Tick (✓) the correct answer:
A polygon has $27$ diagonals. How many sides does it have?
(a) $7$
(b) $8$
(c) $9$
(d) $12$

Answer 1

The correct option is (c): $9$

The number of the diagonals in the given polgon is $27.$
We know that the number of diagonals of a ${}^{\prime }{n}^{\prime }$ sided polygon is given by $\frac{n(n-3)}{2}$

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

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

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

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

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

$\Rightarrow n=-6,orn=9$

Number of sides cannot be negative.

$\therefore n=9$

Hence, the given polygon have $9$ sides.