Question

If a polygon has $90$ diagonals, the number of its sides is given by?

• A. $12$
• B. $11$
• C. $10$
• D. $15$

Answer 1

The correct option is D

$15$

Explanation for the correct option:

Find the number of sides of the polygon :

Let, number of sides $=n$, number of vertex $=n$, number of lines $={}^{n}C_2$

Number of diagonals $={}^{\mathbf{n}}\mathit{C}_{\mathbf{2}}\mathbf{–}\mathbf{}\mathit{n}=\frac{\mathbf{}\mathbf{n}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{3}\mathbf{)}}{\mathbf{2}}$

$$\begin{gathered} \Rightarrow 90=\frac{n(n-3)}{2} \\ \Rightarrow n^2-3n=180 \\ \Rightarrow n^2–3n–180=0 \end{gathered}$$

Solving for $n$, we get;

$$\begin{gathered} n^2-15n+12n-180=0 \\ \Rightarrow (n-15)(n+12)=0 \end{gathered}$$

$n=15,-12$

The number of sides cannot be negative. Therefore, $n=15$.

Hence the correct option is D.