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

Answer

Verified

272.1k+ views

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

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

Last updated date: 24th Sep 2023

•

Total views: 272.1k

•

Views today: 7.72k