Problem 228 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. The perimeter of a triangle is 35 feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.

Step-by-Step Solution

Answer

The side lengths are 9 feet, 14 feet, and 12 feet.

Step 1: Define Variables

Let the length of the second side be denoted by the variable \(x\). Thus, the first side can be represented as \(x + 5\) and the third side can be represented as \(x + 3\).

Step 2: Set Up the Perimeter Equation

Since the perimeter of the triangle is the sum of all its sides, we can set up the equation: \( (x) + (x + 5) + (x + 3) = 35\).

Step 3: Combine Like Terms

Combine the terms in the equation to get: \( 3x + 8 = 35 \).

Step 4: Solve for \(x\)

Isolate \(x\) by subtracting 8 from both sides: \( 3x = 27 \). Then, divide both sides by 3: \( x = 9 \).

Step 5: Find All Side Lengths

Substitute \(x = 9\) back into the expressions for the side lengths. The second side is \( 9 \), the first side is \( 9 + 5 = 14 \), and the third side is \( 9 + 3 = 12 \).

Step 6: Verify the Perimeter

Check the sum of the side lengths to ensure they add up to the given perimeter: \( 9 + 14 + 12 = 35 \). This confirms that the calculations are correct.

Key Concepts

triangle properties

Triangles are fundamental geometric shapes, consisting of three sides and three angles. They can take various forms, such as equilateral, isosceles, or scalene, depending on the lengths of their sides and the measures of their angles. The given triangle problem is a scalene triangle because it has three sides of different lengths. Understanding the properties of triangles helps us use the correct methods to solve problems. For instance, knowing that the sum of the angles in any triangle is always 180° can be useful. However, in this specific problem, we focus more on the perimeter, which is the total length around the triangle by adding up all its side lengths.

perimeter of a triangle

The perimeter of a triangle is a simple yet vital concept in geometry. It represents the total distance around the triangle, which is achieved by summing the lengths of all three sides. In this exercise, the perimeter is given as 35 feet. By using the perimeter property, we are able to frame an equation that helps us determine the unknown side lengths. This is particularly useful when some sides are expressed in terms of another side, as it allows us to translate the geometric framework into an algebraic one. Remember, the formula for the perimeter of a triangle is: Perimeter = side1 + side2 + side3.

algebraic equations

Algebra is a powerful tool for solving geometric problems. By transforming the problem into an algebraic equation, we can find the unknown values. In this case, we start with the variable for one side ( x ), and then express the other sides in terms of this variable. The exercise provides transformations for the other two sides: (x + 5) and (x + 3). Adding these expressions forms our perimeter equation: x + x + 5 + x + 3 = 35. Combining like terms simplifies the equation further to: 3x + 8 = 35. Next, we isolate x to find its value by solving the equation step-by-step.

variable manipulation

Variable manipulation is essential for solving algebraic equations. In our problem, the variables represent the unknown side lengths. We start with the equation 3x + 8 = 35. Our goal is to isolate x. Here are the steps to do this effectively:

• Subtract 8 from both sides: 3x + 8 - 8 = 35 - 8 results in 3x = 27.
• Divide both sides by 3: x = 27 / 3 gives x = 9.

By calculating x this way, we find the second side length. Substituting x = 9 into our expressions for the other sides, we get: 14 feet for the first side ( 9 + 5 ) and 12 feet for the third side ( 9 + 3). Checking these values ensures they add up to the provided perimeter, confirming our solution.