Problem 42 - Elementary Algebra

Question

Set up an algebraic equation and then solve. A triangle has sides whose measures are consecutive even integers. If the perimeter is 42 inches, find the measure of each side.

Step-by-Step Solution

Answer

The sides of the triangle are 12 inches, 14 inches, and 16 inches.

Step 1: Define the variables

Let the three consecutive even integers be represented as \(x\), \(x + 2\), and \(x + 4\). These represent the lengths of the sides of the triangle.

Step 2: Set up the equation

The perimeter of a triangle is the sum of its sides. So, the equation for the perimeter is: \(x + (x + 2) + (x + 4) = 42\).

Step 3: Simplify the equation

Combine the like terms in the equation: \(3x + 6 = 42\).

Step 4: Solve for \(x\)

Subtract 6 from both sides to isolate the term with \(x\): \(3x = 36\).

Step 5: Divide by the coefficient of \(x\)

Divide each side by 3 to solve for \(x\): \(x = 12\).

Step 6: Find the measures of each side

Now that we know \(x = 12\), the sides of the triangle are: \(x = 12\), \(x + 2 = 14\), and \(x + 4 = 16\).

Key Concepts

Consecutive Even Integers

Consecutive even integers are numbers that follow each other in order without any gap between them, and each number is an even number.To better understand, let’s break down how to identify them with an example:

• If you start with an even number, say 10, the next consecutive even number would be 12.
• Continuing this sequence gives 14 next, because 12 plus 2 is 14.

In algebraic terms, if we denote the first even integer as \(x\), the next consecutive even integers can be expressed as \(x + 2\) and \(x + 4\). This pattern is because even numbers have a consistent difference of 2. This concept is very useful in setting up algebraic equations when dealing with problems involving sequences of even numbers.

Perimeter of a Triangle

The perimeter of a triangle is the total length around the triangle.It is calculated by adding up the lengths of all its sides.In our case of a triangle with sides as consecutive even integers:

• The sides we are considering are \(x\), \(x + 2\), and \(x + 4\).
• To find the perimeter, we add these values: \(x + (x + 2) + (x + 4)\).
• This sums up to a given perimeter, which in this problem, is 42 inches.

Understanding how to set up the equation correctly is crucial.You ensure that all side lengths are accounted for and any numerical value concerning the perimeter is correctly applied. This approach simplifies finding the individual lengths of each triangle’s side.

Variables in Algebra

Variables are symbols or letters, like \(x\), used to represent unknown numbers in algebra.They allow us to create equations and solve for unknown quantities. In algebraic problems, understanding the role of a variable is key:

• They can represent numbers in sequences, trends, or unknown values to be determined.
• In the triangle problem, \(x\) becomes the foundational variable representing the smallest of the consecutive even integers.

By manipulating these variables, we can derive relationships between different parts of an equation. For example, in solving \(x + (x + 2) + (x + 4) = 42\), \(x\) helps us encapsulate the idea of an initial even number in this context.This method of using and manipulating variables is crucial for forming an equation that gives a clear pathway to a solution.