Problem 39 - Elementary Algebra

Question

Set up an algebraic equation and then solve. The perimeter of an equilateral triangle measures 63 centimeters. Find the length of each side.

Step-by-Step Solution

Answer

Each side of the triangle is 21 cm long.

Step 1: Understanding Perimeter of Equilateral Triangle

Since the triangle is equilateral, all three sides are equal. If each side is denoted as \( s \), the perimeter \( P \) can be expressed as \( P = s + s + s = 3s \). Given that the perimeter is 63 cm, we have the equation \( 3s = 63 \).

Step 2: Setting Up the Equation

Using the information that the perimeter is 63 cm, set up the equation: \( 3s = 63 \). This equation will help us find the length of each side (\( s \)).

Step 3: Solving the Equation for s

To find \( s \), divide both sides of the equation by 3: \( s = \frac{63}{3} \).

Step 4: Calculating the Length of Each Side

Perform the division to solve for \( s \): \( s = 21 \). Thus, each side of the triangle measures 21 cm.

Key Concepts

Perimeter

Understanding the concept of perimeter is fundamental in geometry. The perimeter of a shape is the total distance around its edges. In the case of an equilateral triangle, which is a triangle where all sides have the same length, calculating the perimeter becomes straightforward. You simply add up the length of all three sides.
For an equilateral triangle with side length denoted as \( s \), the perimeter \( P \) is calculated as:

• \( P = s + s + s = 3s \)

In this exercise, the perimeter is given as 63 cm, allowing us to set up an equation to find the side length, which is a concept we delve deeper into in the next section.

Algebraic Equation

Algebraic equations are powerful tools in problem-solving. They allow us to express relationships between quantities and solve for unknown values. In the context of this exercise, we need to find the side length of an equilateral triangle given its perimeter.

• We start by setting up the equation from the perimeter: \( 3s = 63 \).
• This equation represents the fact that three times the side length equals the given perimeter.

Solving this equation involves dividing both sides of the equation by 3, which will isolate \( s \) (the side length), making it easy to solve.

Problem-Solving

Problem-solving is an essential skill that applies to many aspects of life, including mathematics. Tackling problems like finding the length of a triangle's side involves a series of logical steps to reach a solution:

• First, understand the problem - recognize that you need to find the side length given a perimeter.
• Next, set up a relevant algebraic equation that accurately reflects the mathematical relationship.
• Then, solve the equation using algebraic techniques like dividing or factoring.
• Finally, interpret the solution in the context of the problem to ensure it makes sense.

By following these steps, you enhance not just your ability to solve math problems but also critical thinking skills applicable to real-world situations.

Geometry

Geometry is a branch of mathematics concerned with shapes, sizes, and the properties of space. An equilateral triangle, as featured in this exercise, is a fundamental geometric shape where all sides and angles are equal.

• This symmetry makes calculations like the perimeter straightforward, as each side length contributes equally to the total.
• Knowing geometry helps visualize and understand spatial relationships, which aids in setting up and solving equations.

In this case, recognizing an equilateral triangle's properties allows us to set up the problem correctly and reach a solution by using its inherent characteristics.