Question

A polygon has $54$ diagonals. Then, the number of its sides is

• A. 7
• B. 9
• C. 10
• D. 12

Answer 1

Answer: (D)

Number of diagonals in polygon $=54$
Let the number of sides in polygon $=n$
Then,
The number of diagonal in any polygon
$=\frac{n(n-3)}{2}$
$54=\frac{n(n-3)}{2}$
$n^{2}-3 n=108$
$n^{2}-3 n-108=0$
$n^{2}-12 n+9 n-108=0$
$n(n-12)+9(n-12)=0$
$(n+9)(n-12)=0$
$n=-9$ or $n=12$
(not possible) Hence, the number of sides in polygon is $n=12$