Question

A line segment can be drawn from each vertex of a polygon to every other vertex, forming the sides and

diagonals of the polygon. A square, for example, has 4 sides and 2 diagonals. Using combinations, write
an expression that would give you the total number of sides and diagonals for a polygon with
sides.

Answer 1

n(n - 3)/2

Explanation:

We are told that;

A square has 4 sides and 2 diagonals.

Now, a pentagon will have 5 sides and 5 diagonals

A hexagonal when drawn will have 6 sides and 9 diagonals

Thus, we see that;

When n = 4, d = 2

When n = 5, d = 5

When n = 6, d = 9

Using combination, we will find number of pairs of 2 vertices that can possibly be formed from n-vertices. Thereafter we will subtract n from that since it's neighboring vertices can't combine with the current one.

Thus, we have;

C(n, 2) - n

This gives s;

(n(n-1)/2) - n

Simplifying this gives;

(n(n - 1) - 2n)/2

>> (n² - n - 2n)/2

>> (n² - 3n)/2

>> n(n - 3)/2