Problem 24 - Elementary Algebra

Question

A triangular plot of ground has a perimeter of 54 yards. The longest side is twice the shortest side, and the third side is 2 yards longer than the shortest side. Find the lengths of the sides of the triangle.

Step-by-Step Solution

Answer

Sides are 13, 26, and 15 yards.

Step 1: Identify Variables

Let the shortest side of the triangle be represented by the variable \( x \). According to the problem, the longest side will be twice as long, or \( 2x \). The third side is 2 yards longer than the shortest side, which can be expressed as \( x + 2 \).

Step 2: Set Up the Equation for Perimeter

The perimeter of the triangle, which is the sum of all three sides, is given as 54 yards. We write the equation as follows: \( x + 2x + (x + 2) = 54 \).

Step 3: Simplify and Solve the Equation

Simplify the equation: \( x + 2x + x + 2 = 54 \) to get \( 4x + 2 = 54 \). Subtract 2 from both sides to get \( 4x = 52 \). Divide both sides by 4 to find \( x \): \( x = 13 \).

Step 4: Calculate the Length of Each Side

Now that we have \( x = 13 \), we can find the lengths of the sides: - Shortest side: \( x = 13 \) yards, - Longest side: \( 2x = 26 \) yards, - Third side: \( x + 2 = 15 \) yards.

Key Concepts

Perimeter of Triangle

The perimeter of a triangle is the total distance around the outside of the triangle. In simpler terms, it's the sum of the lengths of all three sides of the triangle. Establishing the perimeter as a starting point is essential in solving algebra word problems involving triangles. Knowing the perimeter helps us set up equations to find unknown side lengths. In our exercise, we are given a perimeter of 54 yards. This information allows us to form an equation that includes the triangle's sides and solve for unknown values. Understanding perimeter helps translate word problems into a mathematical expression. Often, this step is crucial for finding the solution.

Variable Identification

Identifying variables is the foundation of solving algebraic word problems. This step involves translating descriptive statements into mathematical symbols. In our problem, we need to discover the lengths of the triangle's sides. Let's break it down:

• Shortest side: Represented as the variable \( x \).
• Longest side: Stated to be twice the shortest side, so we use \( 2x \).
• Third side: Is 2 yards longer than the shortest side, hence \( x + 2 \).

Clearly defining these variables is vital because they form the basis of the equation we'll solve. Remember, choosing the simplest variable (\( x \) for the shortest side) will simplify the problem-solving process and reduce complexity.

Equation Solving Steps

Once we have identified the variables, we need to configure them into a workable equation. This process begins with setting up an equation that accurately represents the problem statement. In our case, the problem provides us with the triangle's perimeter (54 yards), allowing us to establish the following equation: \[ x + 2x + (x + 2) = 54 \] The next step involves simplifying the equation to combine like terms: \[ 4x + 2 = 54 \] From there, we solve for \( x \) using standard algebraic techniques:

• Subtract 2 from both sides: \( 4x = 52 \)
• Divide both sides by 4: \( x = 13 \)

By following these equation-solving steps, we pinpoint the value of \( x \), laying the groundwork for determining the actual triangle side lengths.

Triangle Side Lengths

Once \( x \) is discovered, we can calculate the specific lengths of each triangle side. Using the variable definitions we set earlier:

• Shortest side: Simply \( x = 13 \) yards.
• Longest side: Calculated as \( 2x = 26 \) yards.
• Third side: By adding 2 to the shortest side \( x + 2 = 15 \) yards.

These calculations verify that each side length satisfies the original condition of the triangle's perimeter, which should sum to 54 yards: \( 13 + 26 + 15 = 54 \). Understanding how each side contributes to the triangle's structure allows us to ensure our solution is mathematically consistent with all given constraints.