The equation of an altitude of an equilateral triangle is `sqrt3x + y = 2sqrt3` and one of its vertices is `(3,sqrt3)` then the possible number of triangles is

Options

A. 0,0
B. `0, 2sqrt(3)`
C. `3, -sqrt(3)`
D. none of these

Explanation

To solve the problem, we need to find the possible number of equilateral triangles given the altitude and one vertex. Let's break down the steps:

### Step 1: Understand the given information
We have the equation of the altitude of an equilateral triangle given by:
\[ \sqrt{3}x + y = 2\sqrt{3} \]
and one of its vertices is at the point \( (3, \sqrt{3}) \).

### Step 2: Verify if the vertex lies on the altitude
We will check if the vertex \( (3, \sqrt{3}) \) satisfies the altitude equation:
\[ \sqrt{3}(3) + \sqrt{3} = 2\sqrt{3} \]
Calculating the left-hand side:
\[ 3\sqrt{3} + \sqrt{3} = 4\sqrt{3} \]
Since \( 4\sqrt{3} \neq 2\sqrt{3} \), the vertex does not lie on the altitude. This means the vertex can either be point B or point C of the triangle.

### Step 3: Determine the slope of the altitude
The altitude's equation can be rearranged to find its slope:
\[ y = -\sqrt{3}x + 2\sqrt{3} \]
The slope of the altitude is \( -\sqrt{3} \).

### Step 4: Find the slope of the sides of the triangle
Since the triangle is equilateral, the angles between the altitude and the sides are \( 60^\circ \). The slopes of the sides can be calculated as follows:
- The slope of one side (let's say BC) will be \( \tan(60^\circ) = \sqrt{3} \).
- The slope of the other side (let's say AB) will be \( \tan(-60^\circ) = -\sqrt{3} \).

### Step 5: Find the equations of the sides
Using the vertex \( (3, \sqrt{3}) \) and the slopes, we can find the equations of the two sides:
1. For side AB with slope \( \sqrt{3} \):
\[ y - \sqrt{3} = \sqrt{3}(x - 3) \]
Simplifying gives:
\[ y = \sqrt{3}x - 3\sqrt{3} + \sqrt{3} = \sqrt{3}x - 2\sqrt{3} \]

2. For side AC with slope \( -\sqrt{3} \):
\[ y - \sqrt{3} = -\sqrt{3}(x - 3) \]
Simplifying gives:
\[ y = -\sqrt{3}x + 3\sqrt{3} + \sqrt{3} = -\sqrt{3}x + 4\sqrt{3} \]

### Step 6: Find the intersection points
To find the other vertices (B and C), we need to find the intersection points of these lines with the altitude line:
1. Set \( \sqrt{3}x - 2\sqrt{3} = -\sqrt{3}x + 4\sqrt{3} \) and solve for \( x \).
2. Substitute \( x \) back into either line to find \( y \).

### Step 7: Determine the possible triangles
Since the altitude is fixed and the vertex can be either B or C, we can reflect the triangle across the altitude to find the second possible triangle. Therefore, the maximum number of distinct equilateral triangles that can be formed is 2.

### Conclusion
The possible number of equilateral triangles that can be formed is:
\[ \text{Possible number of triangles} = 2 \]