Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of a polygon.) Decagon (10 sides)

Step-by-Step Solution

Answer

The decagon has 35 diagonals.

Step 1: Substitute the number of sides

The first step is to plug the count of sides into the formula. This gives us \(10(10-3)/2\).

Step 2: Simplify Mathematical Expression

Next, continue by subtracting 3 from 10 in the parentheses, resulting in \(10(7)/2\). Then continue to multiply 10 and 7 which results in \(70/2\).

Step 3: Final Calculation

Lastly, divide 70 by 2. This gives us the total amount of diagonals.

Key Concepts

Decagon Properties

Understanding the properties of a decagon is essential when learning about polygons. A decagon is a ten-sided figure, each angle in a regular decagon measures 144 degrees. One key characteristic of a decagon, or any polygon, relates to its diagonals. A diagonal is a line segment connecting two nonadjacent vertices. The number of diagonals in a polygon is given by the formula \( \frac{n(n-3)}{2} \) where \( n \) is the number of sides. For a decagon \( n = 10 \) therefore, it has \( \frac{10(10-3)}{2} \) diagonals. It is crucial to note that this formula only applies to simple polygons, where the sides do not intersect.

In a regular decagon, because of its symmetry, the length of all sides and diagonals are congruent. However, in irregular decagons, side and diagonal lengths can vary. Understanding these properties not only helps in calculating the number of diagonals but also aids in solving more complex geometric problems involving decagons.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operation symbols that represent a specific value or quantity. In the context of geometry and the calculation of diagonals in polygons, mathematical expressions are used to describe the relationship between the number of polygon sides and its diagonals through a formula.

For example, we used the expression \( \frac{10(10-3)}{2} \) to find the diagonals in a decagon. This mathematical expression involves substitution where we replaced \( n \) with 10 (the number of sides in a decagon), arithmetic operations of multiplication and subtraction to simplify the expression, and finally division to calculate the exact number of diagonals. It is important to follow the correct order of operations, which in this case is to perform the subtraction within the parentheses first, then the multiplication, and finally the division. Sure, calculators can help, but understanding how to manipulate these expressions is a fundamental skill in mathematics.

Geometric Formulas

Geometric formulas are tools that allow us to solve problems in geometry by providing mathematical relationships between different geometric quantities. For polygons, one of the most useful formulas is the one we used to calculate the number of diagonals: \( \frac{n(n-3)}{2} \) where \( n \) is the number of sides. This formula is derived from considering that each vertex can connect to every other vertex except for itself and its adjacent vertices, which are already connected by sides.

This formula is a powerful tool since by simply knowing the number of sides, we can immediately determine the number of diagonals, allowing for efficient problem-solving. Remember, geometric formulas can often be visualized to help understand the rationale behind them, which aids in better grasping the concepts and applying them to a variety of situations in geometry.