Problem 43 - Elementary Algebra

Question

Set up an algebraic equation and then solve. A triangle has sides whose measures are consecutive odd integers. If the perimeter is 21 inches, find the measure of each side.

Step-by-Step Solution

Answer

The sides are 5 inches, 7 inches, and 9 inches.

Step 1: Define the Variables

Let's denote the smallest side of the triangle as \( x \). Since the sides are consecutive odd integers, the three sides can be expressed as \( x \), \( x + 2 \), and \( x + 4 \).

Step 2: Write the Equation for the Perimeter

The perimeter of the triangle is the sum of its sides. We're given that the perimeter is 21 inches. So, we set up the equation: \[ x + (x + 2) + (x + 4) = 21. \]

Step 3: Simplify the Equation

Combine the like terms in the equation: \[ x + x + 2 + x + 4 = 21 \] simplifies to \( 3x + 6 = 21 \).

Step 4: Solve for x

To isolate \( x \), subtract 6 from both sides of the equation: \[ 3x = 15. \]Now, divide both sides by 3: \[ x = 5. \]

Step 5: Find the Measures of Each Side

Since \( x = 5 \), the measures of the sides are:- Smallest side: \( x = 5 \)- Second side: \( x + 2 = 7 \)- Largest side: \( x + 4 = 9 \).

Key Concepts

Consecutive Odd Integers

Consecutive odd integers are numbers that follow each other in order and are odd. They differ by 2 each time. Examples include 3, 5, and 7 or 11, 13, and 15. These sequences maintain their odd nature as they increase linearly. When describing three consecutive odd integers algebraically, start with a variable; let's use \( x \) for the smallest number. Then, the sequence can be expressed as:\[ x, \ x + 2, \ x + 4. \]
These mathematical expressions help form equations when solving problems that involve properties such as sums, products or, as in our exercise, a triangle's perimeter.

Perimeter of a Triangle

The perimeter of a triangle is the total distance around the outside of the shape, calculated by adding the lengths of its three sides. For our triangle with sides that are consecutive odd integers, we have the sides represented by \( x \), \( x + 2 \), and \( x + 4 \).
This means the equation for perimeter can be expressed simply as:

• \( x + (x + 2) + (x + 4) = \text{perimeter} \)

In this specific problem, you know that the perimeter is 21 inches. Solving for \( x \), you'll have everything you need to determine each side's length. Understanding perimeter is vital as it forms the basis for setting up equations when dealing with shapes in algebra.

Solving Equations

Solving equations involves finding values for variables that make the equation true. In our situation, the goal is to solve the equation we developed for the triangle's perimeter, which is \( 3x + 6 = 21 \). Here are simple steps to reach the solution:

• Start by simplifying the equation by combining like terms, then use arithmetic operations to isolate the variable.


• Subtract 6 from both sides to simplify further: \( 3x = 15 \).


• Finally, divide each side by 3, giving \( x = 5 \).

These steps help reveal the smallest side's length in our problem. Solving equations efficiently requires understanding each operation's purpose and outcome, helping clarify more complex algebraic problems.