Question

If ($\alpha ,\beta$), $(\overline{x},\overline{y})and(u,v)$are respectively coordinates of the circumcentre, centroid and orthocentre of a triangle.

• A. 3$\overline{x}=2\alpha +uand3\overline{y}=2\beta +v$
• B. 3$\overline{x}=2\alpha -uand3\overline{y}=2\beta -v$
• C. 3$\overline{x}=2\alpha -uand3\overline{y}=2\beta +v$
• D. 3$\overline{x}=2\alpha +uand3\overline{y}=2\beta -v$

Answer 1

The correct option is C 3$\overline{x}=2\alpha -uand3\overline{y}=2\beta +v$

We know that, the centroid of a triangle divides the segment joining the orthocentre and circumcentre internally in the ratio 2 : 1. Therefore,
$\overline{x}=\frac{2\alpha +u}{2+1}and\overline{y}=\frac{2\beta +v}{2+1}$
$\Rightarrow 3\overline{x}=2\alpha +uand\overline{y}=2\beta +v$