Problem 28 - Fundamentals of Mathematics

Question

The perimeter of a triangle is 75 meters. If each of two legs is exactly twice the length of the shortest leg, how long is the shortest leg?

Step-by-Step Solution

Answer

The shortest leg is 15 meters long.

Step 1: Understand the Problem

We are given a triangle with a perimeter of 75 meters. Two sides are each twice the length of the shortest leg. We need to find the length of the shortest leg.

Step 2: Assign Variables

Let the length of the shortest leg be denoted as \( x \). Then, each of the other two longer legs, being twice the shortest, can be represented as \( 2x \).

Step 3: Write the Perimeter Equation

The perimeter of the triangle can be expressed in terms of the side lengths as follows: \[ x + 2x + 2x = 75 \]

Step 4: Simplify the Equation

Combine like terms in the equation from the previous step: \[ 5x = 75 \]

Step 5: Solve for \( x \)

To find \( x \), divide both sides of the equation by 5: \[ x = \frac{75}{5} = 15 \]

Step 6: Confirm the Solution

The solution states the shortest leg is 15 meters. To confirm, check the total perimeter: \( 15 + 2(15) + 2(15) = 75 \). This confirms that the calculation is correct.

Key Concepts

Perimeter of a Triangle

The perimeter of a triangle is the total distance around the triangle. You calculate it by adding up the lengths of all three sides.

• If you know the lengths of each side, simply add them together.
• The formula for the perimeter, when sides are known, is expressed as: \[ P = a + b + c \]where \(a, b,\ and \ c\) are the side lengths of the triangle.

In problems involving the perimeter, you often need to work backward, using given total lengths to find unknown side lengths.
When one side is described in relation to others, as in the problem with one shortest leg and two longer legs double the length, this relationship forms the basis for setting up your equation.

Algebraic Equations

Algebraic equations are mathematical statements that show the relationship between different quantities. They usually involve variables and constants.
In our triangle problem, we used an algebraic equation to represent the perimeter in terms of a single variable, which helps in solving for unknowns. Here's how it works:

• Assign a variable, such as \(x\), to the unknown quantity, like the shortest leg length.
• Express the other sides in terms of \(x\). Here, the longer sides are \(2x\) each.
• Write an equation based on the perimeter: \[ x + 2x + 2x = 75 \].
• Simplify to connect the variable with the total perimeter, making it easier to solve.

Algebra allows converting real-life problems into mathematical expressions, making them easier to tackle step-by-step.

Problem Solving

Problem solving is the process of working through the details of a problem to reach a solution. When solving math problems like our triangle perimeter example, follow these steps:

• Read and understand the problem thoroughly. Identify what is given and what needs to be found.
• Translate the words into mathematical expressions using variables and equations.
• Simplify and solve the equations step-by-step.
• Check your work; make sure the solution fits all parts of the original problem.

Using a structured approach ensures each part of the problem is addressed and ultimately makes for effective problem solving in both lateral thinking and numerical calculations.