If a regular polygon has 20 diagonals, then the number of sides of this polygon is:

Question

Examoogle · Accepted Answer

The correct answer is Option C: 8. Substitute the given options into the diagonal formula: for n = 8, \frac{8(5)}{2} = 20, which matches the given number of diagonals.

Step-by-Step Solution & Concepts:
Shortcut Trick: Substitute the given options into the diagonal formula: for n = 8, \frac{8(5)}{2} = 20, which matches the given number of diagonals.

Formula Used: $$	ext{Number of diagonals} = \frac{n(n - 3)}{2}$$

Given: Number of diagonals of a regular polygon = 20

Steps: State the formula for diagonals of an n-sided polygon: The total number of diagonals in an $n$-sided polygon is given by the relation taking 2 vertices at a time minus the number of sides: $$	ext{Diagonals} = \frac{n(n - 3)}{2}$$. Set up the quadratic equation: Given that the polygon has $20$ diagonals, we equate the formula to $20$: $$\frac{n(n - 3)}{2} = 20$$. Solve the equation for n: Multiply both sides by $2$ to clear the denominator: $$n(n - 3) = 40$$ $$n^2 - 3n - 40 = 0$$ Factoring the quadratic equation: $$(n - 8)(n + 5) = 0$$ Since the number of sides must be positive ($n > 0$), we get $n = 8$.

Answer: C is correct because substituting n = 8 into the diagonal formula gives \frac{8(8-3)}{2} = 20 diagonals, which matches the given condition.

Exam Tip: Similar combinations formulas are used in graph theory to find the number of edges in a complete graph $K_n$, which is also given by $\frac{n(n-1)}{2}$.

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