Question

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

• A. 25
• B. 35
• C. 20
• D. 15

Hint:

To find the number of diagonals in a polygon, use the formula \( D = \frac{n(n-3)}{2} \), where \( n \) is the number of sides. Solve for \( n \) when the number of diagonals is given.

Answer 1

The Correct Option is A

The formula for the number of diagonals \( D \) in a polygon with \( n \) sides is given by: \[ D = \frac{n(n-3)}{2}. \] We are given that the polygon has 275 diagonals, so: \[ \frac{n(n-3)}{2} = 275. \] Multiply both sides of the equation by 2 to eliminate the fraction: \[ n(n-3) = 550. \] Expanding the equation: \[ n^2 - 3n = 550. \] Rearranging the terms: \[ n^2 - 3n - 550 = 0. \] Now, solve this quadratic equation using the quadratic formula: \[ n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-550)}}{2(1)} = \frac{3 \pm \sqrt{9 + 2200}}{2} = \frac{3 \pm \sqrt{2209}}{2}. \] Taking the square root of 2209: \[ n = \frac{3 \pm 47}{2}. \] Thus, the two possible values for \( n \) are: \[ n = \frac{3 + 47}{2} = 25 \quad \text{or} \quad n = \frac{3 - 47}{2} = -22. \] Since \( n \) must be a positive integer, we conclude that: \[ n = 25. \] Thus, the correct answer is: \[ \boxed{25}. \]