Question

A polygon has 20 diagonals. How many sides does it have?

• A. 12
• B. 11
• C. 10
• D. 9
• E. 8

Hint:

Polygon diagonals = \(\binom{n}{2} - n\). Solve quadratic when diagonals are given.

Answer 1

Answer: (E)

Step 1: Formula.
\[ \text{Diagonals} = \frac{n(n-3)}{2} \]
Step 2: Apply condition.
\[ \frac{n(n-3)}{2} = 20 \] \[ n^2 - 3n - 40 = 0 \]
Step 3: Solve.
Discriminant = 169, roots = \(\frac{3 \pm 13}{2}\). So \(n = 8\) (valid).
Final Answer:
\[ \boxed{8} \]