Question

a polygon has 324 diagonals determine the number of sides of this polygon​

Answer 1

Answer:

27

Step-by-step explanation:

Sides Diagonals

27 324

28 350

29 377

30 405

Answer 2

Given:-

$\red{➤}\:$$\sf Number \:of\: diagonals\;in\;polygon = 324$

To Find:-

$\orange{☛}\:$$\sf Number \:of \:Sides\: of \:polygon$

Solution:-

$\underline{\tt{Formula\:Applied}}$

$\green{ \underline { \boxed{ \sf{Number\:of\:diagonals=\dfrac{n(n-3)}{2}}}}}$

$\tt\pink{where\; "n"\:is\:number \:of\:sides\:of\:polygon}$

$\underline{\bf{Putting\;Values-}}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf \dfrac{n(n-3)}{2} = 324 \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf n(n-3)= 324 \times 2 \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf n^2-3n= 648\\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf n^2-3n-648 =0 \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf n^2-27n+24n-648 =0 \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf n(n-27)+24(n-27) =0 \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf (n-27)(n+24) =0 \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf n = 27 , n= -24\\\end{gathered}$

We will neglect negetive value

$\begin{gathered}\\\quad\therefore \quad\sf n= 27 \\\end{gathered}$

$\begin{gathered}\\\quad \maltese\quad\boxed{\sf {Number \:of \:Sides\: of \:polygon=27 }}\\\end{gathered}$

Note-

If factorisation seems difficult by splitting method ,do it by quadratic formula method-

$\sf {Supposing\; quadratic\; equation\; \red{ax^2+bx+c=0}}$

Quadratic formula-

$\orange{ \underline { \boxed{ \sf{x=\dfrac{-b±\sqrt{b^2-4ac}}{2a}}}}}$