To solve for $BD$ in the given triangle problem, let’s break down the information provided and use trigonometry.

Step-by-Step Solution:

1. Given Information:

   • $\triangle ABC$ is a right triangle with $\angle ACB = 90^\circ$.
   • $AD = 6$ cm, and $CD$ is perpendicular to $AB$.
   • $\angle ACD = 30^\circ$.

2. Objective:

   • Find the length of $BD$.

3. Analysis Using Trigonometry: Since $\triangle ACD$ is a right triangle with $\angle ACD = 30^\circ$, we can apply trigonometric ratios.

   In a right triangle, if one angle is $30^\circ$, the side opposite this angle is half of the hypotenuse.

   Here:

   • $\angle ACD = 30^\circ$.
   • $AD$ is opposite $\angle ACD$, so $AD = 6$ cm.

   Since $AD$ is opposite the $30^\circ$ angle, $CD$ is the hypotenuse of $\triangle ACD$.

4. Finding $CD$: Using the property of a 30-60-90 triangle: $CD = 2 \times AD = 2 \times 6 = 12 \text{ cm}$

5. Using Pythagoras in $\triangle BCD$: In $\triangle BCD$, since $\angle ACB = 90^\circ$, $AB$ is the hypotenuse. We want to find $BD$, and we now know:

   • $CD = 12$ cm.

   Since $\triangle ABC$ and $\triangle ACD$ are right triangles sharing $CD$, we could apply trigonometric or geometric principles to solve for $BD$, but given this specific setup, let's use the calculated measurements and trigonometry as needed to find