Problem 26 - Elementary Algebra

Question

The second side of a triangle is 3 meters shorter than twice the first side. The third side is 4 meters longer than the second side. If the perimeter is 58 meters, find the length of each side of the triangle.

Step-by-Step Solution

Answer

The sides of the triangle are 12, 21, and 25 meters long.

Step 1: Define Variables

Let's define the first side of the triangle as \( x \). This will help us express the other sides in terms of \( x \).

Step 2: Express Other Sides in Terms of x

According to the problem, the second side is 3 meters shorter than twice the first side. So, the second side is \( 2x - 3 \). The third side is 4 meters longer than the second side, which gives us \( (2x - 3) + 4 = 2x + 1 \).

Step 3: Write an Equation for the Perimeter

The problem states that the perimeter is 58 meters. The perimeter is the sum of all three sides. So, the equation for the perimeter is: \( x + (2x - 3) + (2x + 1) = 58 \).

Step 4: Simplify the Equation

Combine like terms in the equation: \( x + 2x - 3 + 2x + 1 = 58 \). This simplifies to \( 5x - 2 = 58 \).

Step 5: Solve for x

Add 2 to both sides to isolate the term with \( x \): \( 5x = 60 \). Then, divide by 5 to find \( x \): \( x = 12 \).

Step 6: Find the Length of Each Side

Now that we know \( x = 12 \), substitute it back into the expressions for the other sides. The first side is \( 12 \) meters, the second side is \( 2x - 3 = 2(12) - 3 = 21 \) meters, and the third side is \( 2x + 1 = 2(12) + 1 = 25 \) meters.

Key Concepts

Perimeter Calculation

Calculating the perimeter of a shape, especially triangles, is a key skill in geometry. The perimeter is simply the total measure of the boundary of the shape. For a triangle, this means adding all three sides together.
In this example, you're tasked with finding the perimeter of a triangle by considering unknown side lengths. Once these lengths are expressed relative to one another using a variable, it's possible to write an equation for their combined length.

• First side: typically labeled as your starting variable, let's say \( x \)
• Second side: given as a relationship to the first, \( 2x - 3 \)
• Third side: related to the second, \( 2x + 1 \)

Perimeter, therefore, becomes \( x + (2x - 3) + (2x + 1) \). Add these together to get a simple expression in terms of \( x \). Once simplified, you can solve for \( x \), revealing the lengths of the sides.

Equation Solving

Solving equations is all about finding the value of unknown variables that make the equation true. It's an essential part of algebra. In this problem, we needed to solve for \( x \) to find the side lengths of the triangle.
Let's breakdown the process:

• Combine like terms: Gather terms with \( x \) together, and constants together.
• Simplify: Turn complex equations into a simpler equation, \( 5x - 2 = 58 \).
• Isolate \( x \): Add 2 to both sides to start freeing \( x \), giving you \( 5x = 60 \).
• Solve for \( x \): Divide by 5 to finally get \( x = 12 \).

This sequential unraveling helps make equations clearer and shows how to handle expressions involving variables.

Geometry

Geometry isn't just about shapes; it's about how those shapes relate to space and each other. A triangle, being the simplest polygon, often serves as a fundamental problem in geometry. With this problem, understanding the spatial relationships between the sides is crucial.
Unlike the area or volume, perimeter only deals with the distance around. Knowing relationships between sides, like one side being longer or shorter by certain amounts, is a geometric insight that makes these calculations possible.

• Triangles have three sides which can be related algebraically.
• Equations can represent geometric properties such as length.
• Choose a side to represent with a variable, making relationships easier to express.

By translating geometric problems into algebraic expressions, we make complex relationships easier to manage and ultimately solve.