Question

If $(\alpha ,\beta ),(\overline{x},\overline{y})$ and (p , q) are the co - ordinates of the circumcentre, centroid and orthocentre of a triangle, then $3\overline{x}=2\alpha +pand3\overline{y}=2\beta +q$

• A. True
• B. False

Answer 1

The correct option is A True

Let,

$H,O\text{and}G$ be the orthocentre, circumcentre and centroid

of any triangle.

Then, these points are collinear.

Further, $G$ divides the line segment $HO$ from $H$ in the ratio $2:1$

Here,

$H(p,q),O(\alpha ,\beta )$ and $G(\overline{x},\overline{y})$

Then by the section formula:

$⟹\overline{y}=\frac{2\beta +q}{3}⟹3\overline{y}=2\beta +q$ and

$⟹\overline{x}=\frac{2\alpha +p}{3}⟹3\overline{x}=2\alpha +p$