here is my two cents

Hi Bunuel,

Small question here

Since the AC divides the triangle into three similar triangles. and we have side AD as 5. Cant we calculate ACD as a pyth triplet, hence 3-4-5 and then
AC/BC=CD/AC. This would give us BC??

This is why statement 2 is sufficient:-

Let us take alone the statement 2 - which says BC = CD.
Now,
If BC=CD , Acc. to the property "Median drawn to the hypotenuse of right triangle is half length of hypotenuse length" we know ,
AC=CD and AC = BC. - Because AC is half of the hypotenuse BD and C is the mid point of BD.

Hence we will now have two isosceles triangles - Triangle ABC and Triangle ADC.
Therefore, Angle BAC = Angle CBA and Angle CAD= Angle ADC.

Or we can say Angle ABC = Angle ADC

Hence we have an isosceles triangle ABD . and therefore side AD = side AB.
Now we can find the side BD.

We need to know the following properties for any question with a right triangle and a median from the right triangle:
1) "Median drawn to the hypotenuse of right triangle is half length of hypotenuse length"
2) "In the right triangle the median drawn to the hypotenuse divides the triangle in two isosceles triangles."

I hope this is correct . Bunuel , VeritasPrepKarishma , chetan2u It would be of great help if you can verify the above procedure is correct.