Let n be the number of vertices of the polygon. then there are n  sides. N = number of  all connections between all the vertices =   N = n(n-1)/2 Number of diagonals =  N - n = n (n - 3)/2    =  27  given            n²  - 3 n  - 54 = 0         (n - 9) (n + 6) = 0            n = 9 so there are  9 sides and 9 vertices

Understanding the Problem:

We are given that a polygon has 27 diagonals, and we need to determine the number of sides it has. To solve this problem, we will start by understanding the concepts of diagonals and their relationships with the number of sides in a polygon.

Diagonals in a Polygon:

A diagonal is a line segment that connects two non-adjacent vertices of a polygon. In any polygon, the number of diagonals can be calculated using the formula:

n(n-3)/2

Where "n" represents the number of sides in the polygon. This formula is derived by considering that each vertex can be connected to (n-3) other vertices to form diagonals, and dividing by 2 to avoid counting each diagonal twice.

Calculating the Number of Sides:

To find the number of sides in the polygon, we need to rearrange the formula for the number of diagonals:

n(n-3)/2 = 27

Multiplying both sides by 2 gives:

n(n-3) = 54

Expanding the equation gives:

n^2 - 3n = 54

Rearranging the equation to form a quadratic equation gives:

n^2 - 3n - 54 = 0

Solving the Quadratic Equation:

To find the value of "n" that satisfies the equation, we can either factorize the quadratic equation or use the quadratic formula. In this case, factoring the equation might not be straightforward, so we will use the quadratic formula:

n = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -3, and c = -54. Plugging in these values gives:

n = (-(-3) ± √((-3)^2 - 4(1)(-54))) / (2(1))

Simplifying the equation further:

n = (3 ± √(9 + 216)) / 2

n = (3 ± √225) / 2

n = (3 ± 15) / 2

This gives two possible solutions:

n1 = (3 + 15) / 2 = 18/2 = 9
n2 = (3 - 15) / 2 = -12/2 = -6

Interpreting the Solution:

Since a polygon cannot have a negative number of sides, we discard the second solution (n2 = -6). Therefore, the polygon has 9 sides.