Question

A polygon has 27 diagonals. The number of sides of the polygon is

• A. 7
• B. 9
• C. 8
• D. 10

Answer 1

Answer:

B. 9

• We are given that the polygon has 27 diagonals.

• Substitute this into the formula: 27 = n(n-3) / 2

• To solve for n (the number of sides), we can rearrange the equation:

1. Multiply both sides by 2: 54 = n(n-3)

2. Expand the right side: 54 = n² - 3n

3. Rearrange into a quadratic equation: n² - 3n - 54 = 0

• Now, we need to solve this quadratic equation for n . This can be done by factoring or using the quadratic formula.

• Factoring the quadratic equation: We look for two numbers that multiply to -54 and add up to -3. These numbers are -9 and 6.

• So, the equation becomes: (n - 9)(n + 6) = 0

• This gives two possible solutions for n :

• n - 9 = 0 => n = 9

• n + 6 = 0 => n = -6

• Since the number of sides of a polygon cannot be negative, we discard the solution n = -6.

• Therefore, the number of sides of the polygon is 9.

Exam-Oriented Tips

• Memorize the formula: The formula D = n(n-3) / 2 is crucial for solving problems related to diagonals of polygons in competitive exams.

• Practice with different values: Solve problems where the number of sides is given and you need to find the diagonals, and vice versa.

• Recognize quadratic equations: Be comfortable in solving quadratic equations that arise from these problems, as factoring is often the quickest method in an exam setting.

• Logical check: Always ensure your answer for the number of sides is a positive integer greater than or equal to 3 (as a polygon must have at least 3 sides).