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 \( A(0, 0) \), \( B(\sqrt{3}, 0) \), and \( C(0, \sqrt{6}) \), 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
We observe that:
- Point \( A \) is at the origin.
- Point \( B \) lies on the x-axis.
- Point \( C \) lies on the y-axis.

Since \( A \) is at \( (0, 0) \), \( B \) is at \( (\sqrt{3}, 0) \), and \( C \) is at \( (0, \sqrt{6}) \), we can see that the triangle \( ABC \) is a right triangle with the right angle at point \( A \).

### Step 3: Use the property of the orthocenter in a right triangle
In a right triangle, the orthocenter is located at the vertex where the right angle is formed. Since \( A(0, 0) \) is the vertex with the right angle, the orthocenter of triangle \( ABC \) is at point \( A \).

### Step 4: State the orthocenter coordinates
Thus, the orthocenter of triangle \( ABC \) is:
\[ \text{Orthocenter} = A(0, 0) \]

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

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