Question

If the centroid and circumcentre of a triangle are $(3,3)$ and $(6,2)$, respectively, then the orthocentre is

• A. $(-3,5)$
• B. $(-3,1)$
• C. $(3,-1)$
• D. $(9,5)$

Answer 1

The correct option is A $(-3,5)$

In any triangle, orthocentre, centroid and circumcentre are collinear and centroid divides the line joining orthocentre and circumcenter in the ratio $2:1.$

Let the orthocentre be $(x,y)$.

Using the section formula, $(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Substituting $(x_1,y_1)=(x,y)$ and $(x_2,y_2)=(6,2)$ and $m=2,n=1$ in the section formula, we get the centroid $=(\frac{2\left(6\right)+1\left(x\right)}{2+1},\frac{2\left(2\right)+1\left(y\right)}{2+1})=(\frac{x+12}{3},\frac{y+4}{3})$

Given centroid $=(3,3)$

$=>(\frac{x+12}{3},\frac{y+4}{3})=(3,3)$

$=>\frac{x+12}{3}=3;\frac{y+4}{3}=3$

$=>x+12=9;y+4=9$

$=>x=-3;y=5$

Hence, orthocentre $=(-3,5)$