The orthocentre of a trianlge whose vertices are `(0,0) , (sqrt3,0) and (0,sqrt6)` is

Options

A. (2,1)
B. (3,2)
C. (4,1)
D. none of these

Answer

To find the orthocenter of the triangle with vertices at \( (0,0) \), \( (\sqrt{3}, 0) \), and \( (0, \sqrt{6}) \), we can follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are:
- \( A(0, 0) \)
- \( B(\sqrt{3}, 0) \)
- \( C(0, \sqrt{6}) \)

### Step 2: Determine the type of triangle
To determine the type of triangle, we can check the angles. The triangle formed by these points has:
- \( AB \) along the x-axis
- \( AC \) along the y-axis

Since \( AB \) and \( AC \) are perpendicular to each other, triangle \( ABC \) is a right triangle with the right angle at vertex \( A(0, 0) \).

### Step 3: Find the orthocenter of the triangle
For a right triangle, the orthocenter is located at the vertex where the right angle is formed. In this case, since the right angle is at vertex \( A(0, 0) \), the orthocenter \( H \) is also at point \( A \).

Thus, the orthocenter of the triangle is:
\[ H(0, 0) \]

### Conclusion
The orthocenter of the triangle with vertices \( (0,0) \), \( (\sqrt{3}, 0) \), and \( (0, \sqrt{6}) \) is \( (0, 0) \).

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