Question

If the orthocentre and circumcentre of a triangle are respectively $\left(1,1\right)$ and $\left(3,2\right)$, then the coordinates of its centroid are

• A. $\left(\frac{7}{3},\frac{5}{3}\right)$
• B. $\left(\frac{5}{3},\frac{7}{3}\right)$
• C. $\left(7,5\right)$
• D. None of these

Answer 1

The correct option is A

$\left(\frac{7}{3},\frac{5}{3}\right)$

Explanation for the correct option.

The line created by joining the circumcentre and orthocentre of a triangle is divided in the ratio $2:1$ by the centroid of a triangle.

The coordinates of the circumcentre and orthocentre are given, then to find the coordinates of the centroid we need to use the formula,

$\begin{array}{l}(\frac{{\mathrm{mx}}_2+{\mathrm{nx}}_1}{m+n},\frac{{\mathrm{my}}_2+{\mathrm{ny}}_1}{m+n})\end{array}$, where $m:n$ is the ratio in which the line is divided that is $2:1$, $\left(x_1,y_1\right)\text{and}\left(x_2,y_2\right)$ are the coordinates of orthocentre and circumcentre respectively.

The coordinates of the centroid are,

$\begin{array}{rcl}\therefore (\frac{{\mathrm{mx}}_2+{\mathrm{nx}}_1}{\mathrm{m}+\mathrm{n}},\frac{{\mathrm{my}}_2+{\mathrm{ny}}_1}{\mathrm{m}+\mathrm{n}})& =& (\frac{2\left(3\right)+1\left(1\right)}{2+1},\frac{2\left(2\right)+1\left(1\right)}{2+1})\\ & =& \left(\frac{7}{3},\frac{5}{3}\right)\end{array}$

Hence, the correct option is (A).