Step 1 : Understanding the Question:
In this problem, we are tasked with determining the specific number of sides of a convex polygon based on the information that it contains 54 diagonals. Geometrically, a diagonal is a line segment that connects two vertices that are not adjacent to each other. The challenge lies in establishing a mathematical relationship between the vertices, which correspond to the sides in a closed polygon, and the segments that can be drawn between them. This problem falls under the category of elementary geometry and combinatorics.
Step 2 : Key Formulas and approach:
The derivation of the diagonal formula is rooted in combination theory. If a polygon has $n$ vertices, the total number of line segments that can be formed by joining any two vertices is given by the combination formula ${}^nC_2$, which is mathematically expressed as $n(n-1)/2$. Out of these total segments, $n$ segments represent the actual boundary sides of the polygon. Therefore, to isolate the count of diagonals, we subtract the number of sides from the total segments. The resulting formula is:
\[ D = \frac{n(n - 1)}{2} - n = \frac{n^2 - n - 2n}{2} = \frac{n(n - 3)}{2} \]
Our approach will be to substitute $D = 54$ into this formula and solve for the unknown variable $n$.
Step 3 : Detailed Explanation:
We start by setting up the equation based on the given data: $54 = \frac{n(n - 3)}{2}$.
To clear the fraction and simplify the algebraic expression, we multiply both sides of the equation by 2, yielding $108 = n(n - 3)$.
Next, we expand the right side of the equation to form a standard quadratic expression: $n^2 - 3n = 108$.
By moving all terms to one side, we obtain the quadratic equation in its standard form: $n^2 - 3n - 108 = 0$.
To solve this, we employ the splitting the middle term (factorization) method. We need two integers whose product is $-108$ and whose algebraic sum is $-3$. The integers $-12$ and $+9$ satisfy these conditions perfectly.
We rewrite the equation as $n^2 - 12n + 9n - 108 = 0$ and group the terms: $n(n - 12) + 9(n - 12) = 0$.
Factoring out the common binomial gives $(n - 12)(n + 9) = 0$.
This equation provides two potential solutions: $n = 12$ and $n = -9$.
In the physical context of geometry, the number of sides of a polygon must be a positive integer greater than or equal to 3. Therefore, we discard $n = -9$.
Step 4 : Final Answer:
The value of $n$ is 12, indicating that the polygon has 12 sides.