let ABC be a triangle right angled at B.Let AD be the angle bisector of A.suppose AC=6cm and the area of the triangle ADC is 10cm^2, wat is length of BD?

let AB = a

BC = b

pythagoras thm

a^2 + b^2 = 36

also area 1/2ab = 10    => b=20/a  put in above eqn

a^2 + 400/a^2 = 36

a^4 - 36a^2 +400 = 0

discriminant 36*36 - 4*400 = -304 negative which give imaginary values

ans:3.33cm

let AB=h     BD=l   DC=x AC=6cm

in Triangle ADC

DC is base BUt Height Would be AB

Hence area would be 1/2hx=10

hx=20

but as AD is internal angle bisector   then using internal angle bisector theorem

AB/AC=BD/DC

AB.dC=BD.AC

hx=6l

but hx=20

hence l=10/3=3.33cm