Final answer:
The area of an equilateral triangle with side length s is \( \frac{s^2 \sqrt{3}}{4} \), derived by using the formula 1/2 × base × height with the height determined from the properties of 30-60-90 right triangles within the equilateral triangle.
Step-by-step explanation:
The area of an equilateral triangle with side length s is given by the formula \( \frac{s^2 \sqrt{3}}{4} \). To show this, we first need to recall the general formula for the area of a triangle, which is 1/2 × base × height.
For an equilateral triangle, all sides are equal in length, so the base is s. To find the height (h), we can draw an altitude from any vertex to the base, which bisects the base and creates two 30-60-90 right triangles. The altitude is the longer leg in these right triangles, and its length will be s \( \frac{\sqrt{3}}{2} \) because, in a 30-60-90 triangle, the length of the longer leg is \( \sqrt{3} \) times that of the shorter leg, which in this case is half the base \( \frac{s}{2} \).
Substitute these values into the formula for the area of a triangle:
\( \frac{1}{2} × s × \left( s \frac{\sqrt{3}}{2} \right) \)
This simplifies to:
\( \frac{1}{2} × s × s \frac{\sqrt{3}}{2} = \frac{s^2 \sqrt{3}}{4} \)
Thus, option 3 (s²*sqrt(3)) is correct after considering that the area must be divided by 4, not just multiplied by the square root of 3.