In trapezoid ABCD with bases AB and DC, diagonals intersect at point O. Find the length of diagonal BD, if BO=6 cm andAO/AC=3/4

Angle  AOB   =  Angle COD

Angle ABO = Angle CDO

So...by AA congruency....triangle AOB  is similar to triangle COD

Since  AO / AC = 3/4 ......then CO must be (1/3)AO  ⇒  3CO = AO

So

OD/ CO = OB / AO

OD / CO = 6/ (3CO)

OD / 1  = 6/3

OD = 2

So  BD =  BO + OD   =  6 + 2   =   8