triangle geometry

Triangle ACB  is similar to triangle ADE

Let  AE  = 6 

AE / DE = 3

Then DE =  2

Then  AD  =  sqrt  (6^2 + 2^2)  = sqrt 40 =  2sqrt 10

So AC = 1/2 AD =     sqrt 10

And AB =  (1/2) AE =    3

So

BC =  sqrt [  (sqrt (10)^2  - 3^2  ]   =  sqrt 1  =  1

So

AD / BC =   2 sqrt ( 10  )  /  1 =     2 sqrt 10

EDIT to correct a previous error   !!!

Let \(DE = x\)

We know that \(AE = 3x\) and \(AD = \sqrt{10}x\)

We also know that \(\triangle ABC\) is similar to \(\triangle ADE\)

Because the scale factor is 2, we know that \(CB = 0.5x\)

Thus, the ratio is \({\sqrt{10} \over 0.5 } = \color{brown}\boxed{2\sqrt{10}}\)