Problem 19 - Algebra A Combined Function

Question

Find the number of diagonals for a polygon that has 15 sides.Find the number of sides \(n\) for a polygon that has 35 diagonals.

Step-by-Step Solution

Answer

The polygon has 10 sides.

Step 1: Understanding the Diagonal Formula

The formula for calculating diagonals in a polygon with \( n \) sides is given by \( \frac{n(n - 3)}{2} \). This formula arises because each vertex can connect to \( n - 3 \) other vertices to form a diagonal, excluding itself and its two adjacent vertices.

Step 2: Establish Equation for 35 Diagonals

We are given that the polygon has 35 diagonals. Using the formula from Step 1, set up the equation: \( \frac{n(n - 3)}{2} = 35 \).

Step 3: Clear the Fraction by Multiplying

Multiply both sides of the equation by 2 to eliminate the fraction: \( n(n - 3) = 70 \).

Step 4: Simplify and Solve the Quadratic Equation

Expand and rearrange the equation to form a standard quadratic equation: \( n^2 - 3n - 70 = 0 \).

Step 5: Factor the Quadratic Equation

Factor the equation \( n^2 - 3n - 70 = 0 \) to \( (n - 10)(n + 7) = 0 \).

Step 6: Solve for Possible Values of n

Set each factor equal to zero: \( n - 10 = 0 \) and \( n + 7 = 0 \). Solving these gives \( n = 10 \) or \( n = -7 \). Since the number of sides, \( n \), cannot be negative, we choose \( n = 10 \).

Key Concepts

Formula for Diagonals

When dealing with polygons, understanding how to find diagonals is crucial. A diagonal refers to a line segment that connects two non-adjacent vertices (corners) of a polygon. To find the number of diagonals in a polygon with \( n \) sides, we use the formula:\[ \frac{n(n - 3)}{2} \]This formula works because each vertex can be connected to \( n - 3 \) other vertices to make a diagonal. The \( n - 3 \) term represents the total number of vertices a vertex can connect to, excluding itself and its two immediate neighbors.

• The formula accounts for all possible line segments a vertex can form with other vertices.
• We divide by 2 because each diagonal is counted twice when considering both endpoints.

Keep in mind that this formula only applies to simple polygons, meaning polygons that do not intersect themselves.

Quadratic Equations

Quadratic equations are used to solve problems where relationships involve squared terms. The general form of a quadratic equation is:\[ ax^2 + bx + c = 0 \]In our case, we derived the quadratic equation \( n^2 - 3n - 70 = 0 \) from the diagonal formula by setting it equal to 70, which is twice the number of given diagonals. Solving quadratic equations can typically be done in several ways:

• Factoring: Break the equation into simpler binomial expressions. For example, \( (n - 10)(n + 7) = 0 \).
• Quadratic Formula: Use \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) when factoring is complex.
• Completing the Square: Adjust the equation to make the squared term easier to solve.

In solving \( n^2 - 3n - 70 = 0 \), factoring was efficient, providing solutions \( n = 10 \) and \( n = -7 \). Because a polygon cannot have a negative number of sides, \( n = 10 \) is the valid solution.

Polygon Sides

The number of sides in a polygon is a fundamental property that determines many aspects of geometry and symmetry within the shape. A polygon with \( n \) sides is called an n-gon.

• When \( n = 3 \), it is a triangle.
• When \( n = 4 \), it is a quadrilateral.
• With \( n = 5 \), it becomes a pentagon, and so on.

For instance, in our problem of finding a polygon with 35 diagonals, we discovered that the number of sides is actually 10, making it a decagon. Simply put, understanding how sides determine shape assists in recognizing various polygon properties and is key in using formulas effectively. From angles to diagonals, the number of sides plays an essential role in defining the polygon’s characteristics.