Problem 17 - Elementary Algebra

Question

The perimeter of a triangle is \(105 \mathrm{~cm}\). If each of the two legs is exactly twice the length of the base, how long is each leg?

Step-by-Step Solution

Answer

Answer: Each leg of the triangle is 42 cm long.

Step 1: Define the variables

Let \(x\) represent the length of the base, then the length of the legs would be \(2x\).

Step 2: Set up the equation for the perimeter

Since the perimeter of a triangle is the sum of the lengths of its sides, we can write it as: $$x + 2x + 2x = 105$$

Step 3: Simplify and solve the equation

We can simplify the equation by adding the terms on the left-hand side: $$5x = 105$$ Then, divide both sides by 5 to solve for \(x\): $$x=\frac{105}{5}$$ $$x = 21$$

Step 4: Find the length of each leg

Now that we know \(x=21\), we can find the length of each leg by multiplying the base length by 2: $$\text{Length of each leg} = 2x = 2\times 21 = 42$$ Each leg of the triangle is 42 cm long.

Key Concepts

Perimeter of a Triangle

The perimeter of a triangle is the total distance around the triangle, which is calculated by adding together the lengths of all three sides. Calculating perimeters is an important concept in geometry. For any triangle, whether it's scalene, isosceles, or equilateral, the perimeter is consistently found using this method. If a triangle has sides of lengths \(a\), \(b\), and \(c\), the perimeter \(P\) is represented as: \[ P = a + b + c \]
In our exercise, we have a specific type of triangle where two legs are equal and each leg is twice the length of the base. By representing the base length as \(x\), the legs become \(2x\), giving the equation for the perimeter as \(x + 2x + 2x = 105 \, \text{cm}\). This allows us to easily manipulate the equation to find individual side lengths.

Equation Solving

Equation solving is a fundamental skill in algebra that involves finding the value of variables that satisfy a given equation. In the context of our problem, once we've defined the sides of the triangle (base and legs) in terms of the variable \(x\), we set up an equation using those expressions.
The equation for the perimeter of the triangle was: \[ x + 2x + 2x = 105 \]
By combining like terms, we simplify this to \(5x = 105\). To solve for \(x\), we perform simple arithmetic by dividing both sides by 5. This yields \(x = 21\). This value of \(x\) represents the length of the base of the triangle, which is an essential step to find the lengths of the other sides.

Variable Definition

Defining variables is one of the first critical steps when solving algebraic problems. It means choosing symbols to represent unknown values, allowing us to create equations and solve problems systematically.
In this exercise, we started by representing the base of the triangle with the variable \(x\). The problem states that the two legs of the triangle are each twice the length of the base. Thus, we can represent each leg as \(2x\). Using variables simplifies complex problems by creating a way to express relationships mathematically. This not only helps in forming equations that can be manipulated but also in understanding the problem structure.

• Base: Represented by \(x\).
• Legs: Each represented by \(2x\).

This logical structuring through variables enables equation setup and ultimately leads to finding the specific measurements of triangle sides.