Problem 33 - Elementary Algebra

Question

The second side of a triangle is five times the length of the smallest side. The third is twice the length of the second side. The perimeter of the triangle is 48 inches. Find the length of each side.

Step-by-Step Solution

Answer

Answer: The lengths of the sides of the triangle are 3 inches, 15 inches, and 30 inches.

Step 1: Define the variables

Let x represent the length of the smallest side. Then, the second side is 5x, and the third side is 2(5x) = 10x.

Step 2: Set up an equation for the perimeter

We're given that the perimeter of the triangle is 48 inches. The perimeter of a triangle is the sum of the lengths of its sides. So, we can set up the equation: x + 5x + 10x = 48

Step 3: Solve for x

Combine like terms to simplify the equation: 16x = 48 Now, divide both sides by 16 to solve for x: x = 3

Step 4: Determine the length of each side

Now that we have the value of x, we can find the length of each side: - Smallest side: x = 3 inches - Second side: 5x = 5(3) = 15 inches - Third side: 10x = 10(3) = 30 inches So, the lengths of the triangle's sides are 3 inches, 15 inches, and 30 inches.

Key Concepts

Algebraic Expressions

In our triangle perimeter problem, we encountered algebraic expressions, which are combinations of variables, numbers, and operations like addition and multiplication. For instance, when we set the second side of the triangle to be five times the length of the smallest side, we expressed this relationship algebraically as 5x, where x is any number that represents the length of the smallest side. Similarly, for the third side being twice the length of the second side, we used the expression 10x.

Understanding algebraic expressions is critical because they are the backbone of forming equations that can be solved to discover unknown values. In our example, by establishing that the sum of all sides equals the perimeter, we could create a coherent mathematical statement that when simplified, led us to the value of x.

Solving Linear Equations

The next step in our journey to find the length of each side of the triangle involved solving a linear equation. A linear equation is an equation that can be written in the form ax + b = c, where a, b, and c are constants and x is the variable we seek to solve for.

Our equation from the triangle problem, x + 5x + 10x = 48, is a simple linear equation without a b term. To solve it, we combined like terms to get 16x = 48 and then performed a division by the coefficient 16 to isolate the variable x, yielding x = 3. This process illustrates an important problem-solving technique in algebra: condensing the equation step by step until the unknown variable is singled out. You can check your solution by substituting the value of x back into the original equation to ensure it satisfies the equation.

Triangle Properties

Our last concept revolves around the inherent triangle properties which played a significant role in formulating our initial equation. One fundamental property is that the perimeter of a triangle is the total length of all its sides. This understanding is key when setting up our equation for the perimeter, as we demonstrated by adding the lengths of all sides, represented by algebraic expressions, and setting them equal to 48 inches.

Another property that comes into play, albeit indirectly, is the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. While our problem doesn't explicitly use this property, it's a quick way to verify the plausibility of our found lengths. In the context of our problem, we didn't face any complications, but it's worth noting that such validations are a useful step in assessing the correctness of the solution.