Question

A polygon has 135 diagonals. determine the number of side polygon​

Answer 1

Answer:

18 sides

Step-by-step explanation:

by using formula

d=n(n-3)/2 where d is no. of diagonals and n is no. of sides of a polygon.

Diagonal =135

Solution :–

135=n(n-3)/2

135(2)=2(n-3)

270=n(square) -3n

n (square)-3n-270=0

(n+15)(n-18)=0

n= -15;n=18 ==>Thus, we don't have any negative side.So, the number of sides are 18.

Check:–

n=18

d=n(n-3)/2

135=( (18) (18-3) ) /2

135=( (18) (15) ) /2

135=270/2

135=135

L. H. S. = R. H. S.

A regular polygon that has 18 number of sides is known as Octakaidecagon.

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Answer 2

Given:

A polygon has 135 diagonals

Solution:

Know that,

The number of diagonals of a polygon$\[ = \frac{{n\left( {n - 3} \right)}}{2}\]$

$\[\therefore \frac{{n\left( {n - 3} \right)}}{2} = 135\]$

$\[ \Rightarrow {n^2} - 3n = 270\]$

$\[ \Rightarrow {n^2} - 3n - 270 = 0\]$

Solve the quadratic equation,

$\[\begin{array}{l}{n^2} - 18n + 15n - 270 = 0\\ \Rightarrow n\left( {n - 18} \right) + 15\left( {n - 18} \right) = 0\\ \Rightarrow \left( {n - 18} \right)\left( {n + 15} \right) = 0\\ \Rightarrow n = 18, - 15\end{array}\]$

Neglect negative value.

$\[n = 18\]$

Hence the number of sides of the polygon is 18.