Question

In a $△ABC,$ if $O,H$ and $G$ represents circum-centre, orthocentre and centroid respectively then $HG:GO=$

• A. $3:1$
• B. $2:1$
• C. $1:2$
• D. $1:3$

Answer 1

The correct option is B $2:1$

AD is altitude drawn from vertex A to BC.
H is orthocenter which lies on AD
AE is the median on side BC such that BE=EC.
OE is perpendicular on BC.

$\therefore AD\parallel OE$ and AE is a transversal.

In $\Delta AGH\text{and}\Delta EGO$ we get;

$\angle GAH=\angle GEO$ [alternate interior angles]
$\angle AGH=\angle EGO$[vertically opposite angle]
$\therefore \Delta AGH\sim \Delta EGO$ by AA

So the sides will be in equal proportion.

Also $AG=2GE$[since G is the centroid which divides the median in 2:1 ratio]

Now;

$\frac{AG}{GH}=\frac{EG}{OG}⟹\frac{AG}{EG}=\frac{GH}{GO}=\frac{2}{1}$
Hence $HG:HO=2:1$