Hi, I am kind of stuck at this question

I know how to solve the question in principle but somehow couldn’t quite wrap my head around why CB = x / (root 3). Shouldn’t CB = x because the opposite angle is 30 degree which is the smallest angle for this 30-60-90 triangle?

Thank you.

It depends on what you choose x as. In the reference solution, AB = x. You can choose CB = x, but that would result in the rest of the solution being a bit different (i.e, AB = \sqrt{3} x for instance).

Finally solve it although the pathway is quite convoluted

There are actually 2 ways to solve this problem but we have to be consistent with the approach. The question is tough because the answer dictates that it has to be expressed with AB in it.

For triangle ABD

AB = x

BD = (root 3)(x)

Since CD = BD - BC

BD = (root 3)(x)

BC has to be expressed in AB term given the options, therefore solve it using the ratio approach

For triangle ABC

BC/AB = x / (root 3)(x)

BC = AB / (root 3)

Therefore, DC = BD - BC

= (root 3)(x) - AB / (root 3)

earlier, we deduced that x = AB

= (root 3)(AB) - (AB)/(root 3)

= (3AB - AB) / (root 3)

= (2AB) / (root 3)

Damn, this question took me 30 min to figure out, surely fail the exam then if this level of difficulty is common