Question

Express each statement as an equation involving the indicated variables. Area of an Equilateral Triangle The area \(A\) of an equilateral triangle is \(\frac{\sqrt{3}}{4}\) times the square of the length \(x\) of one side.

Short answer

A = \( \frac{\sqrt{3}}{4} x^2 \)

Step-by-step solution

Identify the Variables

Recognize the variables given in the problem. Here, the area is denoted by the variable \(A\) and the length of one side is denoted by the variable \(x\).

Understand the Relationship

The problem states that the area of an equilateral triangle is \(\frac{\sqrt{3}}{4}\) times the square of the length of one side. This tells us that the area is a function of \(x^2\).

Translate the Statement to an Equation

According to the description, the area \(A\) can be expressed as \(A = \frac{\sqrt{3}}{4} x^2\). This equation matches the verbal description provided in the problem.

Key concepts

equilateral triangle

An equilateral triangle is a special type of triangle where all three sides are of equal length. This results in all three internal angles also being equal, each measuring 60 degrees. The symmetry of equilateral triangles makes them unique and easy to work with in mathematics.
These properties allow for specific formulas, such as the area formula, that leverage the uniformity of the structure. Understanding these fundamental properties is crucial for solving problems related to equilateral triangles.

area formula

The formula to find the area of an equilateral triangle is derived from its symmetrical properties. The specific formula is:
\[ A = \frac{\sqrt{3}}{4} x^2 \]
Here, \(A\) represents the area of the triangle and \(x\) is the length of one side. This formula comes from dividing the triangle into two 30-60-90 right triangles and using trigonometric relationships. Knowing this formula allows you to quickly find the area without having to resort to more complex geometric calculations.
Using the formula effectively requires understanding why it works, which is why it's derived from basic geometric principles.

variable identification

In solving any equation involving geometry, identifying variables is a crucial step. Variables are symbols that represent unknown or changeable values. In our exercise, the area is denoted by the variable \(A\) and the length of one side is denoted by the variable \(x\).
Recognizing what each variable stands for helps in translating the verbal statement into a mathematical equation.
By substituting these variables into the area formula, we can solve for specific values or express relationships between the variables.

equation translation

Translating a verbal statement into a mathematical equation is an essential skill in algebra and geometry. In the given exercise, the statement 'The area of an equilateral triangle is \(\frac{\sqrt{3}}{4}\) times the square of the length \(x\) of one side' needs to be converted to an equation.
To do this, we recognize that \(A\) is the area and \(x\) is the side length. The phrase 'is \(\frac{\sqrt{3}}{4}\) times the square' translates to the multiplication and exponentiation operations in math.
Thus, the statement becomes:
\[ A = \frac{\sqrt{3}}{4} x^2 \]
This equation now quantitatively describes the relationship between the area and the side length of the equilateral triangle, enabling us to perform further calculations.