Question

In a polygon number of diagonals are 54. The number of sides of polygon is
A.10
B.12
C.9
D.None of these

Answer 1

Hint: A two dimensional closed plane which has at least three straight lines called polygon. Numbers of sides in a polygon are finite. The number of diagonals in a polygon =\[\dfrac{n(n-3)}{2}\], where n is the polygon's sides.

Complete step-by-step answer:
We are given a number of diagonals which is 54.
Apply, above formula in order to find number of side,
The number of diagonal in a polygon =\[\dfrac{n(n-3)}{2}\]
Substituting 54 on the left hand side,
  $ \begin{align}
  & 54=\dfrac{{{n}^{2}}-3n}{2} \\
 & 108={{n}^{2}}-3n \\
\end{align} $
Move all quantity to one side,
 $ {{n}^{2}}-3n-108=0 $
Now we have a quadratic equation. In order to find the value of n split the middle part into two parts to factorize it.
 $ {{n}^{2}}-12n+9n-108=0 $
Take n common from first two terms and nine from last terms.
 $ n(n-12)+9(n-12)=0 $
Take (n-12) common from both terms.
 $ (n-12)(n+9)=0 $
If $ n-12=0 $
Then, $ n=12 $
If $ n+9=0 $
Then, $ n=-9 $

Neglect the last value of n because the number of sides can’t be a negative number.
So, the number of sides of a polygon which have 54 diagonals is 12.

Option (B) 12 is correct.

Note:Triangle, Square, Quadrilateral all are a kind of Polygon. We can withdraw the relation between diagonal and sides of a polygon by using the combination of diagonals that each vertex sends to another vertex then subtracting total sides.