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The problem asks you to find the analytical expression that relates the number of diagonals in a polygon to the number of its sides, based on the table provided.

### Step A: Finding the Expression

The number of diagonals \( d \) in a polygon with \( n \) sides can be calculated using the formula:

\[
d = \frac{n(n-3)}{2}
\]

Where:
- \( n \) is the number of sides of the polygon.
- \( d \) is the number of diagonals.

Let's verify this with the table:

- **Triángulo (3 sides):**
  \[
  d = \frac{3(3-3)}{2} = \frac{3 \times 0}{2} = 0
  \]
- **Cuadrilátero (4 sides):**
  \[
  d = \frac{4(4-3)}{2} = \frac{4 \times 1}{2} = 2
  \]
- **Pentágono (5 sides):**
  \[
  d = \frac{5(5-3)}{2} = \frac{5 \times 2}{2} = 5
  \]
- **Hexágono (6 sides):**
  \[
  d = \frac{6(6-3)}{2} = \frac{6 \times 3}{2} = 9
  \]
- **Heptágono (7 sides):**
  \[
  d = \frac{7(7-3)}{2} = \frac{7 \times 4}{2} = 14
  \]
- **8-sided polygon (?):**
  Using \( n = 8 \), calculate \( d \):
  \[
  d = \frac{8(8-3)}{2} = \frac{8 \times 5}{2} = 20
  \]
- **Heptadecágono (17 sides):**
  \[
  d = \frac{17(17-3)}{2} = \frac{17 \times 14}{2} = 119
  \]

### Step B: Use the Formula to Determine the Polygon

Given the diagonals \( d = 65 \), we use the formula and solve for \( n \):

\[
65 = \frac{n(n-3)}{2}
\]

Multiply both sides by 2:

\[
130 = n(n-3)
\]

Solve the quadratic equation:

\[
n^2 - 3n - 130 = 0
\]

Using the quadratic formula:

\[
n = \frac{-(-3) \pm \sqrt{(-3)^2 + 4(1)(130)}}{2(1)}
\]

This yields \( n = 13 \). Therefore, the polygon with 65 diagonals is a 13-sided polygon (Tridecagon).

Would you like further details or have any other questions?

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Here are five related questions:
1. How would you derive the formula for the number of diagonals?
2. How does the number of sides of a polygon affect the number of diagonals?
3. What happens to the number of diagonals if you double the number of sides?
4. Can a polygon with less than three sides have diagonals?
5. How does this formula apply to regular vs. irregular polygons?

**Tip:** When solving quadratic equations, always check both roots to ensure the correct number of sides is identified.

Find the analytical expression for the number of diagonals in a polygon based on the given table of polygons and their corresponding number of diagonals.

Learn how to find the number of diagonals in various polygons using a mathematical formula. This guide explains the expression d = n(n - 3) / 2 and applies it to polygons like triangles, quadrilaterals, pentagons, and more. Suitable for high school students.

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