Question

If a triangle has its orthocentre at $(1,1)$ and circumcentre at $(\frac{3}{2},\frac{3}{4}),$ then the coordinates of the centroid of the triangle are

• A. $(\frac{4}{3},-\frac{5}{6})$
• B. $(\frac{4}{3},\frac{5}{6})$
• C. $(-\frac{4}{3},\frac{5}{6})$
• D. $(-\frac{4}{3},-\frac{5}{6})$

Answer 1

The correct option is B $(\frac{4}{3},\frac{5}{6})$

Given, orthocentre $\equiv H(1,1)$ and circumcentre $\equiv S(\frac{3}{2},\frac{3}{4})$

Let the coordinates of the centroid be $(x,y).$
We know that, the centroid divides the line segment joining the orthocentre $\left(H\right)$ and circumcentre $\left(S\right)$ in the ratio $2:1$ internally.
So, $x=\frac{2\cdot \frac{3}{2}+1\cdot 1}{2+1}=\frac{4}{3}$
and $y=\frac{2\cdot \frac{3}{4}+1\cdot 1}{2+1}=\frac{5}{6}$

$\therefore$ Coordinates of the centroid are $(\frac{4}{3},\frac{5}{6})$