Problem 28 - Elementary Algebra

Question

The perimeter of a triangle is 40 centimeters. The longest side is 1 centimeter longer than twice the shortest side. The other side is 2 centimeters shorter than the longest side. Find the lengths of the three sides.

Step-by-Step Solution

Answer

The sides are 8 cm, 15 cm, and 17 cm.

Step 1: Define Variables

Let's define the variables for the sides of the triangle. Let the shortest side be \( x \) cm. Then, the longest side is \( 2x + 1 \) cm, and the other side is \( 2x + 1 - 2 = 2x - 1 \) cm.

Step 2: Write the Perimeter Equation

The perimeter of a triangle is the sum of its sides. The given perimeter is 40 cm. So, the equation is: \[ x + (2x + 1) + (2x - 1) = 40 \]

Step 3: Simplify the Equation

Simplify the equation from Step 2: \[ x + 2x + 1 + 2x - 1 = 40 \]Combine like terms: \[ 5x = 40 \]

Step 4: Solve for the Shortest Side

Solve the simplified equation for \( x \): \[ 5x = 40 \]Divide both sides by 5: \[ x = 8 \] So, the shortest side is 8 cm.

Step 5: Calculate the Other Sides

Use \( x = 8 \) from Step 4 to find the other sides. Longest side = \( 2x + 1 = 2(8) + 1 = 17 \) cm. Other side = \( 2x - 1 = 2(8) - 1 = 15 \) cm.

Key Concepts

Perimeter of a Triangle

Understanding the concept of a triangle's perimeter is fundamental in elementary algebra. The perimeter of a triangle is simply the sum of the lengths of its three sides. This concept can be compared to walking around the triangle's boundary and measuring the distance you cover.
For instance, if you know the lengths of all sides, you can easily calculate the perimeter by adding them together. In the given exercise, you need to find the perimeter, which is already provided as 40 cm.
However, what makes this problem interesting is that you have to find the side lengths without initially knowing them. This involves setting up an equation using the known perimeter formula:

• Perimeter = Side 1 + Side 2 + Side 3.

This nudges us into using algebraic methods to express all side lengths in terms of a single variable and then solve for that variable.

Linear Equations

A significant portion of solving the problem involves understanding linear equations. A linear equation represents a straight line when graphed, and it can be expressed in the standard form of \( ax + b = c \). Such equations are fundamental in expressing relationships between variables.
In the exercise solution, the variables were introduced as follows:

• The shortest side of the triangle is represented by \( x \).
• The longest side, in terms of \( x \), is expressed as \( 2x + 1 \).
• The other side is written as \( 2x - 1 \).

By writing the perimeter equation, the exercise uses these expressions to form a linear equation:

• \( x + (2x + 1) + (2x - 1) = 40 \)

This gives us a linear equation with a single variable that can be solved using standard algebraic techniques by combining like terms.

Solving Equations

Solving equations is crucial in finding unknown values, especially in algebra. To solve an equation, you need to isolate the unknown variable on one side of the equation.
In the given problem, we start with the equation \( 5x = 40 \) after combining like terms. Solving such an equation involves reversing operations applied to \( x \). In this case:

• Since 5 is multiplied by \( x \), you divide both sides by 5.
• This results in \( x = 8 \).

Now, with \( x \) determined, you can use it to find the other sides:

• Longest side: \( 2x + 1 = 2 \times 8 + 1 = 17 \) cm.
• Other side: \( 2x - 1 = 2 \times 8 - 1 = 15 \) cm.

Solving these equations reveals the actual side lengths that sum up to the given perimeter.