Problem 34 - Algebra for College Students

Question

Solve each problem. The Vietnam Veterans Memorial in Washington, \(D C\), is in the shape of two sides of an isosceles triangle. If the two walls of equal length were joined by a straight line of \(438 \mathrm{ft}\), the perimeter of the resulting triangle would be \(931.5 \mathrm{ft}\). Find the lengths of the two walls. (Data from pamphlet obtained at Vietnam Veterans Memorial.)

Step-by-Step Solution

Answer

The lengths of the two walls are 246.75 feet each.

Step 1: Define the Variables

Let the length of each of the two equal walls of the isosceles triangle be denoted as \(x\).

Step 2: Write the Perimeter Formula

The perimeter of the triangle is given by the sum of all three sides. In this case: \[2x + 438 = 931.5\].

Step 3: Solve for \(x\)

Rearrange the equation to isolate \(x\): \[2x = 931.5 - 438\]. Then, calculate the subtraction: \[2x = 493.5\]. Finally, divide by 2 to find \(x\): \[x = 246.75\].

Step 4: Verify the Solution

Replace \(x\) in the initial perimeter equation to confirm: \[2(246.75) + 438 = 931.5\]. Calculate: \[493.5 + 438 = 931.5\]. The calculation is correct.

Key Concepts

Perimeter of a Triangle

The perimeter of a triangle is the sum of all three sides. In geometric problems involving triangles, knowing how to calculate the perimeter is essential. If you know the lengths of all sides, you simply add them up. For an isosceles triangle, which has two equal sides, you can simplify the process by focusing on the unique sides. For the problem at hand, the equation for the perimeter is set up as the sum of the two equal sides and the base:

• Let the lengths of the two equal walls be represented by \(x\)
• The base is given as 438 feet.

By setting these terms into the perimeter formula, you get the equation \[2x + 438 = 931.5\]. This equation contains all the parameters needed to derive the unknown length of each equal side.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. They often contain one or more unknown variables that you need to solve for. In solving the isosceles triangle problem, we use algebra to find the lengths of the two equal sides. We start with the equation \[2x + 438 = 931.5\].After isolating the term with the variable, the equation becomes \[2x = 931.5 - 438\].Next, performing the subtraction, we have \[2x = 493.5\]. Finally, dividing by 2 to isolate \(x\), we get \[x = 246.75\]. By following these steps clearly and systematically, you can solve for the required lengths effortlessly.

Geometry Applications

Geometry has numerous real-world applications, from art and architecture to engineering and more. In this exercise, we're applying geometric principles to understand the structure of the Vietnam Veterans Memorial. The memorial forms an isosceles triangle. Using the given data, and understanding the relationships between the sides of the triangle, helps us to calculate unknown lengths.Here, knowing the perimeter helps us set up an equation that allows for solving basic algebraic expressions to find side lengths. The practical application of this knowledge not only helps in solving textbook exercises but also in understanding real-world structures and their properties. Always remember that breaking down shapes into simpler parts and using known formulas makes complex problems more approachable.