Question

A regular polygon of n sides has $170$ diagonal . Then n =

Answer 1

Here, we have to apply combination.

$\therefore$ ${}^n{C}_2-n=120$

$\Rightarrow$ $\frac{n(n-1)}{2}-n=170$

$\Rightarrow$ $n^2-n-2n=340$

$\Rightarrow$ $n^2-3n-340=0$

$\Rightarrow$ $n^2-20n+17n-340=0$

$\Rightarrow$ $n(n-20)+17(n-20)=0$

$\Rightarrow$ $(n-20)(n+17)=0$

$\Rightarrow$ $n=20$ or $n=-17$

Number of sides cannot be negative.

$\therefore$ $n=20$

Hence, there are $20$ sides in polygon.