If the number of diagonals of a regular polygon is 27, then the number of sides is :

Question

Accepted Answer

The correct option is 2:

The problem requires us to find the number of sides ($n$) of a regular polygon when the number of its diagonals ($D$) is given as 27. To solve this, we must recall the fundamental formula used to calculate the total number of diagonals in any polygon with $n$ sides:

$$D = \frac{n(n-3)}{2}$$

In this formula, $n$ represents the number of sides (and vertices) of the polygon. A diagonal is a line segment connecting two non-adjacent vertices. The formula works because from each of the $n$ vertices, we can draw a diagonal to every other vertex except itself and its two immediate neighbors, resulting in $(n-3)$ diagonals from each vertex. Since we count each diagonal twice (once from each end), we divide the product $n(n-3)$ by 2.

Substituting the given number of diagonals ($D=27$) into the formula, we get the equation:

$$27 = \frac{n(n-3)}{2}$$

To solve for $n$, we first multiply both sides by 2:

$$54 = n(n-3)$$
$$54 = n^2 - 3n$$

Rearranging this into a standard quadratic equation gives us:

$$n^2 - 3n - 54 = 0$$

We can solve this quadratic equation by factoring. We look for two numbers that multiply to -54 and add up to -3. These numbers are -9 and +6:

$$n^2 - 9n + 6n - 54 = 0$$
$$n(n - 9) + 6(n - 9) = 0$$
$$(n - 9)(n + 6) = 0$$

This gives us two possible values for $n$: $n=9$ or $n=-6$. Since the number of sides of a polygon must be a positive integer, we discard the negative solution. Therefore, the number of sides of the polygon is 9. A polygon with 9 sides is called a nonagon.

The incorrect option is 1:
If the number of sides ($n$) were 6 (a hexagon), the number of diagonals would be $D = \frac{6(6-3)}{2} = \frac{6 \times 3}{2} = 9$. Since the question specifies 27 diagonals, 6 is incorrect.

The incorrect option is 3:
If the number of sides ($n$) were 10 (a decagon), the number of diagonals would be $D = \frac{10(10-3)}{2} = \frac{10 \times 7}{2} = 35$. This value (35) is significantly higher than the given 27 diagonals, so 10 is incorrect.

The incorrect option is 4:
If the number of sides ($n$) were 12 (a dodecagon), the number of diagonals would be $D = \frac{12(12-3)}{2} = \frac{12 \times 9}{2} = 54$. This result is twice the given number of diagonals, confirming that 12 sides is incorrect for a polygon with 27 diagonals.