Problem 227 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. The perimeter of a triangle is 39 feet. One side of the triangle is one foot longer than the second side. The third side is two feet Ionger than the second side. Find the length of each side.

Step-by-Step Solution

Answer

The sides are 12 feet, 13 feet, and 14 feet.

Step 1 - Define Variables

Let the length of the second side be denoted by \(x\). The first side, being one foot longer than the second side, can be represented as \(x + 1\). The third side, being two feet longer than the second side, can be represented as \(x + 2\).

Step 2 - Set Up the Perimeter Equation

The perimeter of the triangle is given as 39 feet. The perimeter can be expressed in terms of the sides: \(x + (x + 1) + (x + 2) = 39\).

Step 3 - Simplify the Equation

Combine the like terms in the equation \(x + (x + 1) + (x + 2) = 39\): \(3x + 3 = 39\).

Step 4 - Solve for x

Isolate \(x\) by subtracting 3 from both sides: \(3x = 36\). Then divide by 3: \(x = 12\).

Step 5 - Find the Length of Each Side

Now that \(x\) is known, find the lengths of each side. The second side is \(x = 12\) feet. The first side is \(x + 1 = 13\) feet. The third side is \(x + 2 = 14\) feet.

Step 6 - Verify the Solution

Sum the lengths of the sides to verify the perimeter: \(12 + 13 + 14 = 39\) feet. The solution is correct.

Key Concepts

Perimeter of a Triangle

Understanding the perimeter of a triangle is crucial to solving various geometry problems. The perimeter is the total distance around the triangle. It's calculated by adding the lengths of all three sides together. In this exercise, we know that the perimeter is 39 feet. If you know the perimeter and the relationship between the sides, you can set up an equation to find the unknown lengths. By knowing the perimeter, you help simplify the problem and make it easier to solve.

Variable Definition

In mathematics, defining variables is a foundational step in solving problems. Variables represent unknown values that we aim to find. In this problem, we started by defining the length of the second side of the triangle as x. This made it easier to express the other sides: the first side was defined as x + 1 and the third side as x + 2. Defining variables lets you create equations that model the problem, which is essential for finding the solution. Always pick simple symbols like x, y, or z for your variables to keep things clear.

Linear Equations

Linear equations are vital in mathematics and appear frequently in various problems. A linear equation is an equation that makes a straight line when graphed. It usually looks like ax + b = c, where a, b, and c are constants. In our exercise, we combined the terms of our defined sides to get the linear equation: x + (x + 1) + (x + 2) = 39. Simplifying it, we found 3x + 3 = 39. Solving linear equations involves isolating the variable to find its value. Mastering this concept helps in solving many algebraic problems efficiently.

Equation Solving

Solving equations is a core skill in algebra. The steps involve simplification, isolation, and solution verification. In our problem, we simplified the linear equation 3x + 3 = 39 by combining like terms. Next, we isolated x by subtracting 3 from both sides to get 3x = 36. We then divided by 3 to find x = 12. Finally, we used this value to determine the lengths of the triangle’s sides: 12, 13, and 14 feet. Always check your solution by substituting the values back into the original equation. This helps verify your answer and ensures accuracy.