Problem 37 - Algebra A Combined Function

Question

A flower bed is in the shape of a triangle with one side twice the length of the shortest side, and the third side is 30 feet more than the length of the shortest side. Find the dimensions if the perimeter is 102 feet.

Step-by-Step Solution

Answer

The sides of the triangle are 18 feet, 36 feet, and 48 feet.

Step 1: Define Variables

Let the shortest side of the triangle be denoted as \( x \). Then the second side, which is twice the shortest side, is \( 2x \) and the third side, which is 30 feet more than the shortest side, is \( x + 30 \).

Step 2: Set Up Equation for Perimeter

The perimeter of the triangle is the sum of all its sides. Therefore, the equation is:\[ x + 2x + (x + 30) = 102 \]

Step 3: Simplify the Equation

Combine like terms in the equation:\[ 4x + 30 = 102 \]

Step 4: Solve for x

First, subtract 30 from both sides of the equation:\[ 4x = 72 \]Now, divide both sides by 4:\[ x = 18 \]

Step 5: Find the Lengths of All Sides

Now that we know \( x = 18 \), substitute back to find the lengths of all sides:- Shortest side: \( x = 18 \) feet- Second side: \( 2x = 36 \) feet- Third side: \( x + 30 = 48 \) feet.

Key Concepts

Understanding Triangle Dimensions

Triangles come in many shapes and sizes, each defined by the length of their sides and angles. In this problem, we're dealing with a triangle where the sides have specific relationships. We start by identifying the shortest side, which we'll call \( x \). This value influences the other sides:

• The second side is twice as long as the shortest, given by \( 2x \).
• The third side is the shortest side plus 30 feet, or \( x + 30 \).

Understanding these relationships is crucial, as they help us set up equations to find the unknowns. Defining these dimensions via variables transforms the problem from words into math, making it manageable.

Formulating the Perimeter Equation

The perimeter of a triangle is simply the sum of its three sides. In our problem, we know the perimeter equals 102 feet. Therefore, we need to add up all the side lengths:

• Shortest side: \( x \)
• Second side: \( 2x \)
• Third side: \( x + 30 \)

The perimeter equation is then:\[ x + 2x + (x + 30) = 102 \] This equation reflects how the total length around the triangle is made up of its individual sides. Setting this equation is a key step in moving from understanding dimensions to calculating actual lengths.

Solving Equations to Find Side Lengths

Solving equations is a critical skill in geometry, allowing us to find unknown values through algebraic manipulation. Start simplifying the perimeter equation by combining like terms:\[ 4x + 30 = 102 \]Next, solve for the variable \( x \):

• Subtract 30 from both sides to isolate terms with \( x \): \( 4x = 72 \)
• Divide both sides by 4 to solve for \( x \): \( x = 18 \)

Now we've found the shortest side to be 18 feet. This value acts as a foundation to find the lengths of the other sides, using the relationships established earlier.

Applying Geometry Solutions in Real Problems

Geometry often involves translating real-world problems into mathematical ones and then interpreting the results back to reality. We've found that \( x = 18 \), meaning:

• The shortest side is 18 feet.
• The second side, being twice as long, is \( 36 \) feet (i.e., \( 2x \)).
• The third side is \( 48 \) feet (i.e., \( x + 30 \)).

Translating the equation solution back to dimensions confirms the physical characteristics of our triangle. Being able to fluently convert between spatial reasoning and algebraic reasoning is essential for solving geometry problems efficiently.