Question

If a triangle has its orthocenter at $(1,1)$ and circumcenter 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 $(\frac{4}{3},\frac{5}{6})$

Since the centroid divides the line joining the orthocenter and circumcenter in the ratio $2:1$ internally, therefore, if the centroid is $(x,y)$ then

$x=\frac{2\times \frac{3}{2}+1\times 1}{2+1}=\frac{4}{3}$ and $y=\frac{2\times \frac{3}{2}+1\times 1}{2+1}=\frac{5}{6}$....(using the ratio formula to find the cordinates)

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