Question

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

• A. $(\frac{1}{3},\frac{1}{3})$
• B. $(\frac{2}{3},\frac{2}{3})$
• C. $(1,1)$
• D. $(1,3)$

Answer 1

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

For equilateral triangle,

Centroid, orthocentre, circumcentre and incentre all coincides, at one point

So, orthocentre(centroid) of $\Delta ABC=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$

$=(\frac{\frac{1}{1+\sqrt{3}}+\frac{\sqrt{3}}{1+\sqrt{3}}}{3},\frac{\frac{\sqrt{3}}{1+\sqrt{3}}+\frac{1}{1+\sqrt{3}}}{3})$

$=(\frac{1}{3},\frac{1}{3})$