Question

A regular polygon of \( n \) sides has 170 diagonals. Then \( n \) is equal to:

• A. -20
• B. -17
• C. 24
• D. 20

Hint:

Use the formula for diagonals: \( \frac{n(n-3)}{2} \), and solve the quadratic for \( n \).

Answer 1

The Correct Option is D

The number of diagonals in an \( n \)-sided polygon is given by:
\[ \frac{n(n - 3)}{2} = 170 \]
Multiply both sides by 2:
\[ n(n - 3) = 340 \Rightarrow n^2 - 3n - 340 = 0 \]
Solve the quadratic:
\[ n = \frac{3 \pm \sqrt{9 + 4 \cdot 340}}{2} = \frac{3 \pm \sqrt{1369}}{2} = \frac{3 \pm 37}{2} \]
\[ n = 20 \text{ (only positive value taken)} \]