Problem 27 - Elementary Algebra

Question

The perimeter of an equilateral triangle is 4 centimeters more than the perimeter of a square, and the length of a side of the triangle is 4 centimeters more than the length of a side of the square. Find the length of a side of the equilateral triangle. (An equilateral triangle has three sides of the same length.)

Step-by-Step Solution

Answer

The side of the equilateral triangle is 12 centimeters.

Step 1: Define the Variables

Let the length of a side of the square be \( x \) centimeters. Therefore, the perimeter of the square is \( 4x \) centimeters. Similarly, let the length of a side of the equilateral triangle be \( y \) centimeters. The perimeter of the triangle is \( 3y \) centimeters.

Step 2: Set Up the Equations

According to the problem, the perimeter of the equilateral triangle is 4 centimeters more than the perimeter of the square. This gives us the equation: \( 3y = 4x + 4 \). Also, it is given that the length of a side of the triangle is 4 centimeters more than the length of a side of the square, which gives us: \( y = x + 4 \).

Step 3: Substitute and Solve for the Side of the Square

Substitute \( y = x + 4 \) into the equation for the perimeter of the triangle: \( 3(x + 4) = 4x + 4 \). Simplifying this equation gives: \( 3x + 12 = 4x + 4 \). By rearranging, we get: \( x = 8 \). So, the length of a side of the square is 8 centimeters.

Step 4: Calculate the Side of the Triangle

Using \( y = x + 4 \), substitute \( x = 8 \) into the equation to find \( y \): \( y = 8 + 4 = 12 \). Therefore, the length of a side of the equilateral triangle is 12 centimeters.

Key Concepts

Understanding an Equilateral Triangle

An equilateral triangle is a type of polygon characterized by having three sides of equal length. This means each side of the triangle is the same, giving it a symmetric and uniform appearance. Because all sides are equal, all internal angles are also equal, and they each measure 60 degrees.

This symmetry is not just aesthetic; it has mathematical implications as well. For example, knowing the length of one side of an equilateral triangle allows us to easily determine both its perimeter and also to consider other properties, such as altitude and area.

• An equilateral triangle simplifies calculations because of its uniformity.
• All internal angles are 60 degrees, always.
• It embodies both equality in sides and angles, making it unique among triangles.

In terms of practical applications, recognizing an equilateral triangle helps in solving problems involving uniformity and symmetry.

Calculating the Perimeter

The perimeter is often a central concept in geometry problems, representing the total distance around a figure. For an equilateral triangle, calculating the perimeter is straightforward: simply multiply the length of one side by three.

Given an equilateral triangle with side length denoted as \( y \), the perimeter \( P \) is expressed as \( P = 3y \).

• The formula involves a simple multiplication, reflecting the triangle's uniform side lengths.
• The perimeter provides a measure of the triangle's total boundary.

In algebra word problems, comparing the perimeter of different shapes, like a triangle and a square, helps set up equations that we can solve to find unknown measurements, as seen in this exercise.

Utilizing Variable Substitution

Variable substitution is an essential technique in algebra used to simplify and solve equations. In the original exercise, we define the side length of the square as \( x \), and the side length of the triangle as \( y \). This allows us to express their perimeters and relationships in algebraic form.

The equations from the problem are \( 3y = 4x + 4 \) and \( y = x + 4 \). By substituting \( y = x + 4 \) into the first equation, we eliminate one variable, making it easier to solve for \( x \).

• Variable substitution simplifies complex problems by reducing the number of variables.
• It involves replacing one variable with another equivalent expression.
• This method is particularly helpful in solving simultaneous equations.

Knowing how to effectively perform variable substitution is a fundamental skill in algebra, enabling students to tackle a wide array of word problems with confidence.