Chapter 7.3, Problem 21PSC

To determine

The importance of n , $n-3$ , and $2$ is to be explained in theorem $57$ .

Answer to Problem 21PSC

The importance is explained by using the formula of the number of diagonals

Explanation of Solution

Given information : Theorem $57$ states that the number of diagonals that can be drawn in a polygon of n sides is given by the formula

  $d=\frac{n(n-3)}{2}$

We can explain the theorem as given below

For a n convex sided polygon, we will have total n sides. Now, from each side we can draw $n-3$ diagonals. Thus, it gives us the value

  $n(n-3)$

Now, during this, we considered each diagonal twice. Once with both the vertices each. Hence, we divide the whole answer by $2$ . Hence, now we get

  $d=\frac{n(n-3)}{2}$