Question

Find the number of sides of a polygon having 35 diagonals.

Hint- The Number of diagonals are given here .So, we use the formula of finding the Number of diagonals of a polygon having n sides $= \dfrac{{n\left( {n - 3} \right)}}{2}$
As we know that the number of diagonals of polygon having n sides $= \dfrac{{n\left( {n - 3} \right)}}{2}$
Now it is given that polygons have 35 diagonals.
$\therefore 35 = \dfrac{{n\left( {n - 3} \right)}}{2}$
$\begin{gathered} \Rightarrow {n^2} - 3n = 70 \\ \Rightarrow {n^2} - 3n - 70 = 0 \\ \end{gathered}$
Now factorize the equation we have
$\begin{gathered} \Rightarrow {n^2} - 10n + 7n - 70 = 0 \\ \Rightarrow n\left( {n - 10} \right) + 7\left( {n - 10} \right) = 0 \\ \Rightarrow \left( {n - 10} \right)\left( {n + 7} \right) = 0 \\ \Rightarrow \left( {n - 10} \right) = 0{\text{ \& }}\left( {n + 7} \right) = 0 \\ \therefore n = 10,{\text{ - 7}} \\ \end{gathered}$
But the number of sides of a polygon cannot be negative.
So, the number of sides of a polygon having 35 diagonals is 10.

Note- In such types of questions the key concept we have to remember is that always recall the formula of number of diagonals of a polygon having n sides, then according to given condition substitute the value and simplify, we will get the required number of sides having 35 dia