Let f(x) be a real valued function such that the area of an equilateral triangle with two of its vertices at (0, 0) and (x, f(x)) is `(sqrt(3))/(4)` square units. Then
Perimeter of the equilateral triangle is

Options

A. 1
B. 3
C. 6
D. `3sqrt(3)`

Answer

To solve the problem, we need to find the perimeter of the equilateral triangle given that the area is \(\frac{\sqrt{3}}{4}\) square units and two of its vertices are at (0, 0) and (x, f(x)).

### Step-by-Step Solution:

1. **Understanding the Area of the Triangle**:
The area \(A\) of an equilateral triangle with side length \(A\) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
where \(s\) is the length of a side of the triangle.

2. **Setting the Area Equal to Given Value**:
We know that the area of the triangle is \(\frac{\sqrt{3}}{4}\). Thus, we can set up the equation:
\[ \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \]

3. **Solving for Side Length \(s\)**:
To solve for \(s\), we can multiply both sides of the equation by 4:
\[ \sqrt{3} s^2 = \sqrt{3} \]
Now, divide both sides by \(\sqrt{3}\):
\[ s^2 = 1 \]
Taking the square root of both sides gives:
\[ s = 1 \]

4. **Finding the Perimeter of the Triangle**:
The perimeter \(P\) of an equilateral triangle is given by:
\[ P = 3s \]
Substituting the value of \(s\):
\[ P = 3 \times 1 = 3 \]

### Final Answer:
The perimeter of the equilateral triangle is \(3\) units.

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