Question

The orthocentre O of triangle ABC is interior of triangle ABC.If OB=AC and AD=10cm where ADis an altitude then find AB.​

Answer 1

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.

∴AD∥OE and AE is a transversal.

∴∠GAH=∠GEO [alternate interior angles]

∠AGH=∠EGO[vertically opposite angle]

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

Now,

In ΔAGH and ΔEGO: we have,

∠GAH=∠GEO

∠AGH=∠EGO

∴ΔAGH∼ΔEGO by AA

∴ Sides will be in equal proportion.

GH

AG

=

OG

EG

⟹

EG

AG

=

GO

GH

=

1

2

⟹GH=2GO

Hence GO:HG=1:2