Question

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

• A. $\left(\frac{1}{3},\frac{1}{3}\right)$
• B. $\left(\frac{\sqrt{2}}{3},\frac{\sqrt{2}}{3}\right)$
• C. $\left(\frac{2}{3},\frac{2}{3}\right)$
• D. None of these

Answer 1

The correct option is A $\left(\frac{1}{3},\frac{1}{3}\right)$

Let $△OAB$ be the equilateral triangle.
The orthocentre and centroid of an equilateral triangle are same.
The given equation of side is $x+y=1$.
The foot of the perpendicular from the origin on $AB$ is $(\frac{1}{2},\frac{1}{2})$.
The centroid divides the median (altitude in this case as it is an equilateral triangle) in the ratio $2:1$. Hence, the centroid has the coordinates $(\frac{1}{3},\frac{1}{3})$