Problem 23 - Elementary Algebra

Question

The perimeter of a triangle is 100 feet. The longest side is 3 feet less than twice the shortest side, and the third side is 7 feet longer than the shortest side. Find the lengths of the sides of the triangle.

Step-by-Step Solution

Answer

The sides of the triangle are 24 feet, 31 feet, and 45 feet.

Step 1: Define Variables for the Sides

Let \( x \) be the length of the shortest side of the triangle. Then, the longest side can be expressed as \( 2x - 3 \), and the third side as \( x + 7 \).

Step 2: Set Up the Perimeter Equation

The perimeter of a triangle is the sum of all its sides. Therefore, we can write the equation for the perimeter as:\[x + (2x - 3) + (x + 7) = 100\]

Step 3: Simplify the Equation

Combine like terms in the equation:\[x + 2x - 3 + x + 7 = 100 \rightarrow 4x + 4 = 100\]

Step 4: Solve for x

Subtract 4 from both sides of the equation:\[4x = 96\]Now divide both sides by 4:\[x = 24\]

Step 5: Find the Lengths of All Sides

Now that we have \( x = 24 \), we can find the lengths of the other sides:- Shortest side: \( x = 24 \) feet- Longest side: \( 2x - 3 = 2 \times 24 - 3 = 45 \) feet- Third side: \( x + 7 = 24 + 7 = 31 \) feet.

Key Concepts

Triangle Sides

Understanding the sides of a triangle is fundamental when solving problems involving its perimeter. A triangle has three sides, each of which has a different length. In our specific problem, the triangle's sides are uniquely related to each other.

Let's break it down:

• The shortest side is the basis for defining the other two sides. This means that the problem hinges on finding this side first.
• The longest side is defined as 3 feet less than twice the shortest side. This can be expressed algebraically as \(2x - 3\), assuming \(x\) is the length of the shortest side.
• The third side is 7 feet longer than the shortest side, which means it can be written as \(x + 7\).

Knowing how to define each side using algebra is crucial for setting up the equation that determines the perimeter.

Perimeter Equation

The perimeter of a triangle is simply the total length around the triangle. To find it, add up the lengths of all its sides.

What we'll do in the problem is write an equation for the perimeter. Given that the perimeter is 100 feet, we can express it as:

• First, we add the shortest side \(x\), the longest side \(2x - 3\), and the third side \(x + 7\).
• The expression becomes \(x + (2x - 3) + (x + 7)\).
• By simplifying, we get \(4x + 4 = 100\).

This equation represents the relationship of the triangle's sides to its perimeter, setting the stage for solving for \(x\).

Algebraic Equation Solving

Solving algebraic equations is a powerful tool for finding unknown values like the sides of a triangle. Here's how we approach it step by step:

We start with our equation \(4x + 4 = 100\). The goal is to isolate \(x\) on one side of the equation:

• First, subtract 4 from both sides to simplify the equation to \(4x = 96\).
• Next, divide each side by 4 to solve for \(x\), giving us \(x = 24\).

Once we have the value of \(x\), we substitute it back into expressions for the other sides: the shortest side is 24 feet, the longest side is \(2 \times 24 - 3 = 45\) feet, and the third side is \(24 + 7 = 31\) feet. Each step is important to ensure we're solving correctly while using basic algebraic principles.