Question

Express each statement as an equation involving the indicated variables. Perimeter of an Equilateral Triangle The perimeter \(P\) of an equilateral triangle is 3 times the length \(x\) of one side.

Short answer

P = 3x

Step-by-step solution

Understand the Problem

An equilateral triangle has three equal sides. The given exercise involves expressing the perimeter (\(P\)) of an equilateral triangle as an equation involving the length of one side (\(x\)).

Recall the Formula for Perimeter

The perimeter (\(P\)) of any triangle is the sum of the lengths of its sides. For an equilateral triangle, the perimeter is three times the length of one side.

Write the Equation

Using the information from Step 2, you can write the equation for the perimeter of the equilateral triangle. Since each side has a length of \(x\), the perimeter is \[ P = 3x \].

Key concepts

Perimeter of a Triangle

To understand the concept of the perimeter of a triangle, let's start with what a perimeter means. The perimeter is the total distance around a shape. For any triangle, the perimeter is the sum of the lengths of its three sides.
Imagine you have a triangle with sides of lengths denoted by a, b, and c. The formula for the perimeter (\text{P}) would be:
\[ P = a + b + c \]

• Example: If a triangle has sides of 4 cm, 5 cm, and 6 cm, then the perimeter is 4 + 5 + 6 = 15 cm.

Understanding this is crucial as it forms the basis for calculating the perimeter of an equilateral triangle, which is a special type of triangle.

Equilateral Triangle

An equilateral triangle is a unique type of triangle where all three sides are equal in length. This equality simplifies many of the calculations we need to perform.
Since all three sides are equal, we can denote the length of each side as \text{x}. Because the perimeter of a triangle is the sum of all its sides, for an equilateral triangle, this becomes:
\[ P = x + x + x \]
Simplified, it becomes:
\[ P = 3x \]
This equation tells us that the perimeter (\text{P}) of an equilateral triangle is simply three times the length of one side (\text{x}).

• If each side is 5 cm, then the perimeter is 3 * 5 = 15 cm.

Knowing these characteristics helps in quickly deriving the perimeter equation.

Algebraic Equations

Algebraic equations are a way to represent relationships between variables and constants. When dealing with the perimeter of an equilateral triangle, the equation used involves algebraic expressions.
In our exercise, we defined the perimeter (\text{P}) in terms of the length of one side (\text{x}). The equation for the perimeter is:
\[ P = 3x \]
This is a simple linear equation in one variable (\text{x}). Solving or using this equation involves plugging in the value for \text{x} to find \text{P}.

• Example: Let's say the side length \text{x} is 7 cm. Plugging it into the equation gives \text{P} = 3 * 7 = 21 cm.

Understanding this equation allows us to calculate the perimeter of any equilateral triangle, as long as we know the side length. This is a fundamental concept in algebra and is applicable in various geometric contexts.