Problem 25 - Elementary Algebra

Question

The second side of a triangle is 1 centimeter longer than three times the first side. The third side is 2 centimeters longer than the second side. If the perimeter is 46 centimeters, find the length of each side of the triangle.

Step-by-Step Solution

Answer

The sides are 6 cm, 19 cm, and 21 cm.

Step 1: Define Variables

Let the length of the first side of the triangle be \( x \) cm. Then the second side, which is 1 cm longer than three times the first side, is \( 3x + 1 \) cm. The third side, which is 2 cm longer than the second side, is \( 3x + 1 + 2 = 3x + 3 \) cm.

Step 2: Set Up the Equation

The perimeter of the triangle is the sum of the lengths of all three sides. The problem states that this sum is 46 cm. So, we can write the equation: \[ x + (3x + 1) + (3x + 3) = 46 \]

Step 3: Simplify the Equation

Combine like terms in the equation to simplify it: \[ x + 3x + 1 + 3x + 3 = 46 \] This simplifies to \[ 7x + 4 = 46 \]

Step 4: Solve for x

Subtract 4 from both sides to isolate the variable term \( 7x \): \[ 7x = 42 \] Then, divide both sides by 7 to solve for \( x \): \[ x = 6 \]

Step 5: Calculate Each Side's Length

Now that we have \( x = 6 \), calculate each side's length: The first side is \( x = 6 \) cm. The second side is \( 3x + 1 = 3(6) + 1 = 19 \) cm. The third side is \( 3x + 3 = 3(6) + 3 = 21 \) cm.

Key Concepts

Triangles

Triangles are fundamental shapes in geometry, made up of three sides and three angles. They are versatile and appear in various applications, from architecture to art. In this context, we are specifically focusing on a type of triangle with sides of varying lengths.

To describe a triangle, you can classify it by its sides:

• Equilateral: All sides are equal.
• Isosceles: Two sides are of equal length.
• Scalene: All sides have different lengths.

In our problem, we are dealing with a scalene triangle, where each side varies in length. Recognizing the type of triangle can help in understanding and applying the right formulas.

Perimeter

Perimeter can be thought of as the distance around a shape. For a triangle, the perimeter is simply the sum of its three sides.

The formula for the perimeter of a triangle is:

• Perimeter = Side 1 + Side 2 + Side 3

Understanding this concept is crucial when determining unknown side lengths, as in the given problem where the overall perimeter was given as 46 cm. From this information, along with relationships between the sides, you can solve for each individual side length.

Linear Equations

Linear equations are mathematical statements of equality involving linear expressions. These expressions consist of constants and variables raised only to the power of one.

In the context of the problem, you established an equation based on the perimeter's expressions for each side of the triangle:

• The first side: \( x \)
• The second side: \( 3x + 1 \)
• The third side: \( 3x + 3 \)

Combining these, the equation \( x + 3x + 1 + 3x + 3 = 46 \) was formed and then simplified to \( 7x + 4 = 46 \). Solving linear equations involves isolating the variable to find its value, which was accomplished by performing algebraic operations.

Problem Solving

Problem solving in mathematics involves understanding the problem, devising a plan, carrying out that plan, and then reviewing the solution and process.

In this exercise, you went through these steps:

• Understanding the problem: Recognizing the relationships among the triangle's sides and establishing what you need to find.
• Devising a plan: Setting up an appropriate equation using the perimeter and side relationships.
• Carrying out the plan: Simplifying and solving the linear equation for the variable.
• Reviewing: Checking that the calculated side lengths add up to the given perimeter.

Each step is vital to ensure that the solution is correct and complete.