Problem 46 - Intermediate Algebra

Question

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The perimeter of an equilateral triangle can be found by tripling the length of one of its sides.

Step-by-Step Solution

Answer

The mathematical model is \( P = 3s \), where \( s \) is the side length.

Step 1: Understanding the Problem

An equilateral triangle is a triangle where all three sides have equal length. The perimeter of any triangle is the sum of the lengths of all its sides. For an equilateral triangle, we can simply triple the length of one side because each side is equal.

Step 2: Choose a Variable

Let's designate a variable to represent the length of one side of the equilateral triangle. We will use the variable \( s \) to stand for the length of one side.

Step 3: Set Up the Mathematical Model

The perimeter \( P \) of the equilateral triangle is calculated by multiplying the length of one side \( s \) by 3, as there are three sides. This gives us the mathematical model: \[P = 3s\] This equation mathematically represents the verbal model given.

Key Concepts

Understanding Equilateral Triangles

An equilateral triangle is a special type of triangle where all three sides are exactly equal in length. This symmetry makes it unique and often simplifies calculations. In geometry, recognizing when a triangle is equilateral can help you use certain formulas that only apply to this specific shape.

• All angles in an equilateral triangle measure 60 degrees.
• The triangle's sides are congruent, meaning they are the same length.
• It is both equiangular (all angles are equal) and equilateral (all sides are equal).

Understanding these properties can help you identify an equilateral triangle and apply the correct perimeter formula, enriching your toolkit for solving geometric problems.

Using the Perimeter Formula

The concept of perimeter in geometry refers to the total length around a shape. For an equilateral triangle, it is particularly simple to calculate. The perimeter, denoted as \( P \), is the sum of all side lengths.
Since each side of an equilateral triangle is equal, you can compute the perimeter by simply tripling the length of one side. Mathematically, if the side of the triangle is represented as \( s \), then the perimeter \( P \) is given by:\[P = s + s + s = 3s\]

• This formula quickly gives the total boundary length of the triangle.
• You only need to know the length of one side to use it, making calculations straightforward.

Whether you're handling theoretical math problems or real-world geometry tasks, this formula will be your go-to for equilateral triangles.

Significance of Variable Representation

In mathematical modeling, variables play a crucial role. In the context of geometry, selecting a variable to represent a specific quantity simplifies the process of writing and solving equations.
For the problem of finding the perimeter of an equilateral triangle, choosing a variable is the first key step. Here, we've used \( s \) to denote the length of one side. This choice is arbitrary but serves as a convenient representation to create our mathematical equation:

• \( s \) stands for the side length of the triangle.
• Choosing a clear and consistent variable name aids understanding and communication.

Using the variable \( s \), we can express the perimeter in a simple algebraic form: \( P = 3s \). This approach shows how variable representation bridges verbal descriptions with mathematical expressions, making complex concepts more accessible.