Chapter 8.1, Problem 23PSC

To determine

To show:There is no polygon exists if the ratio of no. of diagonals to the sum of the measures of the polygon’s angle is 1 to 18.

Explanation of Solution

Given information

The ratio of no. of diagonals to the sum of the measures of the polygon’s angle is 1 to 18

Formula used

No. of diagonals in a polygon of n − sides is given by:

n (n − 3 ) and

Sum of the measures of the polygon’s angles is given by:

180 (n − 2)

Proof:

Since, given that

The ratio of no. of diagonals to the sum of the measures of the polygon’s angle is :

1 : 18, i.e.

For polygon of n − sides

No. of diagonals: sum of measures of angles in polygon is 1:18

n (n − 3) : 180 (n − 2)::1:18

  $\begin{array}{l}\text{n }\left(\text{n}-\text{3}\right)\times 18=180\left(\text{n}-2\right)\\ \Rightarrow {\text{n}}^2-3\text{n}=10\text{n}-20\\ \Rightarrow {\text{n}}^2-13\text{n+20=0}\end{array}$

Since, equation: ${\text{n}}^2-13\text{n+20=0}$

has no rational solution,

Thus, no such polygon will exist

Hence proved