6.3 Q-13

Exercise 6.3 — Question 13

Problem

Find the number of diagonals of a polygon having $n$ sides. Also find the number of diagonals of a hexagon.

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Key Concept: Combinations and Polygon Diagonals

A polygon with $n$ sides has $n$ vertices. A diagonal is a line segment joining two non-adjacent vertices.

Total line segments joining any 2 vertices out of $n$: ${}^n{C}_2=\frac{n(n-1)}{2}$

Out of these, $n$ segments are the sides of the polygon (not diagonals).

Therefore, the number of diagonals is: $\text{Diagonals}={}^n{C}_2-n=\frac{n(n-1)}{2}-n=\frac{n(n-1)-2n}{2}=\frac{n^2-3n}{2}=\frac{n(n-3)}{2}$

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Worked Example: Hexagon ($n=6$)

$\text{Diagonals}=\frac{6(6-3)}{2}=\frac{6\times 3}{2}=\frac{18}{2}=9$

A hexagon has 9 diagonals.

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Symmetry Property of Combinations

For equations of the form ${}^n{C}_p={}^n{C}_q$:

${}^n{C}_p={}^n{C}_q⟹p=q\text{or}p+q=n$

Example: Solve ${}^n{C}_{15}={}^n{C}_7$.

Since $15\ne 7$, use $p+q=n$: $n=15+7=22$