Let's analyze this triangle problem step by step. We have triangle ABC with a special point D on side AC. The segment BD is perpendicular to AC, forming a right angle. We're given that angle ABD measures 30 degrees, angle C measures 60 degrees, and the length of side BC is 12 units. Our goal is to find the length of segment AD.Now let's analyze triangle BDC. Since BD is perpendicular to AC, angle BDC is a right angle, measuring 90 degrees. We know angle C is 60 degrees and BC is 12. Using the sine ratio, sine of 60 degrees equals BD over BC, so BD equals 12 times sine 60 degrees, which is 12 times square root of 3 over 2, giving us 6 root 3. Using the cosine ratio, cosine of 60 degrees equals DC over BC, so DC equals 12 times cosine 60 degrees, which is 12 times one half, giving us 6.Next, let's examine triangle ABD. We know angle ABD is 30 degrees, and since BD is perpendicular to AC, angle ADB is 90 degrees. We already found that BD equals 6 root 3. Using the tangent ratio, tangent of 30 degrees equals AD over BD. Therefore, AD equals BD times tangent 30 degrees. Tangent of 30 degrees is 1 over root 3, which can be rationalized as root 3 over 3. So AD equals 6 root 3 times root 3 over 3, which simplifies to 6 times 3 over 3, giving us 6.Let's summarize our complete solution. From triangle BDC, we found that BD equals 6 root 3 and DC equals 6. From triangle ABD, using the tangent of 30 degrees, we calculated that AD equals 6. Therefore, the length of segment AD is 6 units. Looking at the multiple choice options, the correct answer is B, which is 6.