Question

If in a regular polygon the numbers of diagonal is \[54\], then the number of sides of this polygon is
A) \[12\]
B) \[10\]
C) \[6\]
D) \[9\]

Answer 1

Hint: As we know that the formula to calculate the number of diagonals in n- sided polygon is \[\dfrac{{{\text{n(n - 3)}}}}{{\text{2}}}\] and hence here n is the number of sides of the polygon. As the number of diagonals is known then we can calculate the value of n and so the number of sides of the polygon can be calculated.

Complete step by step solution: As given the number of diagonals is \[54\].
And formula to calculate the number of diagonals is \[\dfrac{{{\text{n(n - 3)}}}}{{\text{2}}}\]. Hence,
\[ \Rightarrow \dfrac{{{\text{n(n - 3)}}}}{{\text{2}}} = 54\]
On multiplying with 2 we get,
\[ \Rightarrow {\text{n(n - 3)}} = 108\]
On opening the bracket we get,
\[ \Rightarrow {{\text{n}}^{\text{2}}}{\text{ - 3n - 108 = 0}}\]
On factorising we get,
\[ \Rightarrow {{\text{n}}^{\text{2}}}{\text{ - 12n + 9n - 108 = 0}}\]
On taking factor common we get,
\[ \Rightarrow {\text{n(n - 12) + 9(n - 12) = 0}}\]
On simplification we get,
\[ \Rightarrow ({\text{n + 9) = 0 and (n - 12) = 0}}\]
Hence, we get,
\[ \Rightarrow {\text{n = 12,( - 9)}}\]
As the number of sides of polygons cannot be negative,
Hence the number of sides of the polygon is 12.

Hence, option (A) is the correct answer.

Note: The number of diagonals in a polygon is equals to \[\dfrac{{{\text{n(n - 3)}}}}{{\text{2}}}\], where n is the number of polygon sides. For a convex n-sided polygon, there are n vertices, and from each vertex you can draw \[{\text{n - 3}}\] diagonals, so the total number of diagonals that can be drawn is \[{\text{n(n - 3)}}\].
Calculate the value of n and also solve the quadratic equation without any error as roots of the quadratic equation are calculated by factorization method in the above solution.