The figure below shows a quadrilateral ABCD in which AB =8cm,DC=12cm <BAD=45° <CBD=90° and <BCD=30° a) The length of BD b) The size of angle ADB

We are given $\angle CBD = 90^\circ$, $\angle BCD = 30^\circ$, and $DC = 12$ cm.
We need to find the length of $BD$. In $\triangle BCD$, $BD$ is opposite to $\angle BCD$, and $DC$ is the hypotenuse.

Since $\sin(30^\circ) = \frac{1}{2}$:

We know $AB = 8$ cm, $BD = 6$ cm (from part a), and $\angle BAD = 45^\circ$.
We need to find $\angle ADB$.

The Sine Rule states that for a triangle with sides $a, b, c$ and opposite angles $A, B, C$:

In $\triangle ABD$:

Since $\sin(45^\circ) = \frac{\sqrt{2}}{2}$:

The length of BD is $\boxed{6 \text{ cm}}$.
The size of angle ADB is $\boxed{70.53^\circ}$.