91Ó°ÊÓ

A function is a relationship between two variables, typically written as \( f(x) \), where \( f \) is the function and \( x \) is the input variable. In simpler terms, a function produces an output for each input. For this exercise, we express the area of an equilateral triangle as a function of the length of its side. This means we define the area, \( A(s) \), in terms of \( s \), the side length.

The domain of a function refers to the set of possible input values (\( s \) in our case) for which the function is defined. Here, since side lengths cannot be negative, \( s \) must be a non-negative real number. Thus, the domain is expressed as \( s \geq 0 \). In this case, any non-negative \( s \) could be plugged into the function to return an area, making it physically meaningful for describing lengths.

A 30-60-90 triangle is a special right triangle with angles measuring 30, 60, and 90 degrees. The sides of this triangle have a unique ratio which is always the same: 1: \( \sqrt{3} \): 2. This ratio indicates that the shortest side (opposite the 30-degree angle) has a length of 1, while the side opposite the 60-degree angle is \( \sqrt{3} \) times as long, and the hypotenuse (the longest side) is twice as long as the shortest side.

In the context of an equilateral triangle, when we draw a height from one vertex to the opposite side, creating two 30-60-90 triangles, we can use these properties to find the height. Since the height is opposite the 60-degree angle, it is \( \frac{\sqrt{3}}{2} \) times the hypotenuse (the side \( s \) of the equilateral triangle). This gives us a height \( h = \frac{\sqrt{3}}{2}s \). This property is crucial for calculating the area of the equilateral triangle.

Calculating the area of a triangle involves using the basic formula: \( A = \frac{1}{2} \times \text{base} \times \text{height} \). For an equilateral triangle, every side is equal to the base \( s \). We learned earlier that our height \( h \) is \( \frac{\sqrt{3}}{2} s \).

Substituting these values into the area formula gives us:

• Base is \( s \)
• Height is \( \frac{\sqrt{3}}{2}s \).

This results in the expression \( A = \frac{1}{2} \times s \times \frac{\sqrt{3}}{2} s \). Simplifying this expression, we find \( A = \frac{\sqrt{3}}{4} s^2 \). This formula allows us to express the area of an equilateral triangle purely in terms of its side length, \( s \), providing a simple and effective means to calculate triangle area.