Question

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

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

Answer 1

The correct option is A $1:2$

AD is altitude drawn from vertex A to side 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.
$\therefore \angle GAH=\angle GEO$ [alternate interior angles]
$\angle AGH=\angle EGO$[vertically opposite angle]
$AG=2GE$[since G is the centroid which divides the median in 2:1 ratio]
Now,
In $\Delta AGH\text{and}\Delta EGO$: we have,
$\angle GAH=\angle GEO$
$\angle AGH=\angle EGO$
$\therefore \Delta AGH\sim \Delta EGO$ by AA
$\therefore$ Sides will be in equal proportion.
$\frac{AG}{GH}=\frac{EG}{OG}⟹\frac{AG}{EG}=\frac{GH}{GO}=\frac{2}{1}$
$⟹GH=2GO$
Hence $GO:HG=1:2$