Question

If a polygon of \( n \) sides has 275 diagonals, then \( n \) is equal to:

• A. 25
• B. 35
• C. 20
• D. 15

Hint:

Use the formula for the number of diagonals in a polygon to find the number of sides, and then solve the resulting quadratic equation.

Answer 1

The Correct Option is A

We are given that the polygon has 275 diagonals. The formula for the number of diagonals in a polygon with \( n \) sides is: \[ \text{Number of diagonals} = \frac{n(n - 3)}{2} \] We are given that this is equal to 275: \[ \frac{n(n - 3)}{2} = 275 \] Multiply both sides by 2: \[ n(n - 3) = 550 \] Expanding the equation: \[ n^2 - 3n - 550 = 0 \] Solve this quadratic equation using the quadratic formula: \[ n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-550)}}{2(1)} \] \[ n = \frac{3 \pm \sqrt{9 + 2200}}{2} \] \[ n = \frac{3 \pm \sqrt{2209}}{2} \] \[ n = \frac{3 \pm 47}{2} \] Thus, \( n = 25 \) (since \( n \) must be positive). Thus, the correct answer is 25.