Question

If a polygon of $n$ sides has 560 diagonals, then $n$ = ?

• A. 35
• B. 36
• C. 37
• D. 38

Hint:

Diagonals in a polygon: $\binom{n}{2} - n = \frac{n(n - 3)}{2}$

Answer 1

The Correct Option is A

The number of diagonals in an $n$-sided polygon is given by: \[ \frac{n(n - 3)}{2} = 560 \Rightarrow n(n - 3) = 1120 \Rightarrow n^2 - 3n - 1120 = 0 \] Solve the quadratic: \[ n = \frac{3 \pm \sqrt{9 + 4 \cdot 1120}}{2} = \frac{3 \pm \sqrt{4489}}{2} = \frac{3 \pm 67}{2} \Rightarrow n = \frac{70}{2} = 35 \]