Question

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

• A. 12
• B. 11
• C. 10
• D. 9
• E. 8

Hint:

Always use formula: Diagonals in polygon = \(\binom{n}{2} - n = \frac{n(n-3)}{2}\).

Answer 1

Answer: (E)

Step 1: Formula for diagonals.
\[ \text{Diagonals} = \frac{n(n-3)}{2} \]
Step 2: Apply condition.
\[ \frac{n(n-3)}{2} = 20 \] \[ n(n-3) = 40 \] \[ n^2 - 3n - 40 = 0 \]
Step 3: Solve quadratic.
Discriminant = \(9 + 160 = 169\), root = 13.
\[ n = \frac{3 \pm 13}{2} = \frac{16}{2}, \frac{-10}{2} \] \[ n = 8 \quad \text{or} \quad n = -5 \] Reject negative.
Final Answer:
\[ \boxed{8} \]