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 a. 1 b. 2 c. 3 4. 4

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to determine how many possible equilateral triangles can be formed given the altitude and one vertex.

### Step-by-Step Solution:

1. **Understand the Given Information**:
We have the equation of the altitude of an equilateral triangle:
\[ \sqrt{3}x + y = 2\sqrt{3} \]
and one vertex of the triangle at the point \( (3, \sqrt{3}) \).

2. **Check if the Vertex Lies on the Altitude**:
To check if the vertex \( (3, \sqrt{3}) \) lies on the altitude, we substitute \( x = 3 \) and \( y = \sqrt{3} \) into the altitude equation.
\[ \sqrt{3}(3) + \sqrt{3} = 3\sqrt{3} + \sqrt{3} = 4\sqrt{3} \]
We compare this with \( 2\sqrt{3} \):
\[ 4\sqrt{3} \neq 2\sqrt{3} \]
Therefore, the vertex \( (3, \sqrt{3}) \) does not lie on the altitude.

3. **Determine the Position of the Vertex**:
Since the vertex does not lie on the altitude, it can be either above or below the line represented by the altitude.

4. **Visualizing the Triangle**:
The altitude divides the triangle into two equal halves. The vertex can be positioned on either side of the altitude.

5. **Finding Possible Triangles**:
Since the vertex can be positioned above or below the altitude, we can form two distinct equilateral triangles:
- One triangle with the vertex above the altitude.
- Another triangle with the vertex below the altitude.

6. **Conclusion**:
The maximum number of distinct equilateral triangles that can be formed with the given conditions is **2**.

### Final Answer:
The possible number of triangles is **2** (Option b).