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: \[ \boxed{25}. \]

Answer 2

Given the number of diagonals in an \( n \)-sided polygon is 275, find \( n \).

Step 1: Recall the formula for the number of diagonals in a polygon:
\[ \text{Number of diagonals} = \frac{n(n - 3)}{2} \]

Step 2: Set up the equation:
\[ \frac{n(n - 3)}{2} = 275 \]

Step 3: Multiply both sides by 2:
\[ n(n - 3) = 550 \]

Step 4: Expand:
\[ n^2 - 3n = 550 \]

Step 5: Rearrange to form a quadratic equation:
\[ n^2 - 3n - 550 = 0 \]

Step 6: Solve using quadratic formula:
\[ n = \frac{3 \pm \sqrt{(-3)^2 - 4 \times 1 \times (-550)}}{2} = \frac{3 \pm \sqrt{9 + 2200}}{2} = \frac{3 \pm \sqrt{2209}}{2} \]

Step 7: Calculate the square root:
\[ \sqrt{2209} = 47 \]

Step 8: Find the two possible roots:
\[ n = \frac{3 + 47}{2} = \frac{50}{2} = 25 \]
\[ n = \frac{3 - 47}{2} = \frac{-44}{2} = -22 \quad \text{(discard negative)} \]

Therefore, the number of sides \( n \) is:
\[ \boxed{25} \]