Question

If the vertices of a triangle are such that the coordinates of the orthocentre as well as the circumcentre are integers then the coordinates of the centroid must be

• A. intergers
• B. rational numbers
• C. irrational numbers
• D. none of these

Answer 1

Answer: (B)

rational numbers

Given that, the coordinates of the orthocentre as well as the circumcentre are integers.
Let circumcentre be $O(x_1,y_1)$ and orthocentre is $H(x_2,y_2)$
But, Centroid $G$ divides $OH$ in the ratio $1:2$
$\Rightarrow G=\left(\frac{x_2+2x_1}{3},\frac{y_2+2y_1}{3}\right)$
As, $x_1,x_2,y_{,}{y}_2$ are integers, co-ordinates off $G$ must be rational numbers.
Hence, option B.