Problem 81 - Elementary Algebra

Question

A triangle has sides whose measures are consecutive even integers. If the perimeter is 24 meters, then find the measure of each side.

Step-by-Step Solution

Answer

The sides are 6 m, 8 m, and 10 m.

Step 1: Define Variables for the Sides

Let's denote the sides of the triangle as \( x \), \( x+2 \), and \( x+4 \) since they are consecutive even integers.

Step 2: Set Up the Perimeter Equation

The perimeter of the triangle is given as 24 meters. Therefore, you can write the equation for the perimeter as: \( x + (x + 2) + (x + 4) = 24 \).

Step 3: Simplify and Solve for x

Combine the terms in the equation: \( x + x + 2 + x + 4 = 24 \). Simplifying this gives \( 3x + 6 = 24 \). Subtract 6 from both sides to get \( 3x = 18 \). Finally, divide both sides by 3 to solve for \( x \): \( x = 6 \).

Step 4: Find the Measures of Each Side

Now that we know \( x = 6 \), we can find each side. The sides are: \( x = 6 \), \( x+2 = 8 \), and \( x+4 = 10 \).

Key Concepts

Perimeter of a Triangle

The perimeter of a triangle is the total distance around the triangle. It is the sum of the lengths of all its sides. For any triangle with sides of lengths \( a \), \( b \), and \( c \), the formula for the perimeter is \( a + b + c \). The perimeter gives us insight into how large a triangle is by telling us the total length around its boundary.
In this exercise, the problem mentions that the perimeter is 24 meters. This means that the sum of the sides of the triangle adds up to 24 meters. Perimeter is a crucial concept in geometry, as it provides a way to measure the boundary of two-dimensional shapes. It's also one of the first steps in understanding more complex geometrical concepts.

Equation Solving

Equation solving is the process of finding the value of the unknown variable that makes the equation true. In this problem, equation solving is used to determine the lengths of the sides of the triangle that add up to a known perimeter.
The equation given in the problem is \( x + (x + 2) + (x + 4) = 24 \). The purpose of this equation is to solve for \( x \) so that the sum equals 24. To solve it, you combine like terms: \( x + x + 2 + x + 4 \). This simplifies to \( 3x + 6 = 24 \).

• First, we remove the constant by subtracting 6 from both sides, yielding \( 3x = 18 \).
• Next, we divide each side by 3 to isolate \( x \), resulting in \( x = 6 \).

Solving equations often involves these kinds of steps: simplifying expressions, isolating variables, and finally calculating their values. This approach is foundational in algebra.

Problem Solving Steps

Problem solving, in mathematics, involves a systematic approach to finding a solution. It's about breaking down a problem into manageable parts and solving each part step by step. Let’s break down the steps used in this exercise.

**1. Define Variables:**
The problem begins by defining the sides of the triangle in terms of a single variable, \( x \). We express the sides as \( x \), \( x + 2 \), and \( x + 4 \) to ensure that they are consecutive even numbers.

**2. Formulate the Equation:**
Using these variables, we set up an equation based on the perimeter: \( x + (x + 2) + (x + 4) = 24 \). This equation translates the word problem into a mathematical expression that we can solve.

**3. Simplify and Solve:**
We simplify the equation into \( 3x + 6 = 24 \), then solve it to find \( x = 6 \). This gives us the baseline from which we can calculate the individual side lengths.

**4. Interpret the Solution:**
Finally, we use the solution \( x = 6 \) to find the lengths of the sides: \( 6 \), \( 8 \), and \( 10 \). Each step builds on the last, moving from identifying the problem to finding a conclusive answer. These systematic steps are important in all areas of math, illustrating clear logic pathways to solutions.