If a vertex of an equilateral triangle is the origin and the side opposite to it has the equation x+y=1, then orthocentre of the triangle is :

Options

A. (1/3 , 1/3)
B. `(sqrt2//3 , sqrt2//3)`
C. (2/3 , 2/3)
D. none of these

Explanation

To find the orthocenter of the equilateral triangle with one vertex at the origin (0,0) and the side opposite to it given by the equation \(x + y = 1\), we can follow these steps:

### Step 1: Identify the vertices of the triangle
Let the vertices of the triangle be \(A(0, 0)\), \(B(x_1, y_1)\), and \(C(x_2, y_2)\). The line \(BC\) is given by the equation \(x + y = 1\).

### Step 2: Find the foot of the perpendicular from the origin to the line \(BC\)
The equation of the line can be rewritten in the form \(Ax + By + C = 0\):
- Here, \(A = 1\), \(B = 1\), and \(C = -1\).
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