19 B 30° 12 60° A C D In ∆ABC above, what is the length of AD ? A) 4 B) 6 C) 6√2 D) 6√3 Diagram description: The image shows a triangle ABC, with a line segment BD drawn from vertex B to side AC. The line segment BD is perpendicular to AC, forming a right angle at point D. Angle ABC is split by the line BD with ABD being 30 degree. Angle BCA is 60 degrees and the length of BC is 12. The question asks for the length of AD.

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In this problem, we have triangle ABC with a perpendicular line BD drawn from vertex B to side AC. We're given that angle ABD is 30 degrees, angle BCA is 60 degrees, and the length of BC is 12 units. We need to find the length of AD. Let's start by finding the length of BD. We can use the right triangle BDC. In this triangle, we know that angle C is 60 degrees and the hypotenuse BC is 12 units. Using the sine function, we can write: sine of 60 degrees equals BD divided by BC. Solving for BD, we get BD equals BC times sine of 60 degrees, which is 12 times square root of 3 over 2, giving us BD equals 6 times square root of 3. Now, let's find the length of AD using the right triangle ABD. We know that angle ABD is 30 degrees and we just found that BD equals 6 times square root of 3. Using the tangent function, we can write: tangent of 30 degrees equals AD divided by BD. Solving for AD, we get AD equals BD times tangent of 30 degrees, which is 6 times square root of 3 times 1 over square root of 3. This simplifies to AD equals 6 units. There's an alternative way to solve this problem. We can recognize that triangle ABC is actually an equilateral triangle. Let's see why. In triangle ABD, angle BAD equals 180 minus 90 minus 30, which is 60 degrees. In triangle BDC, angle DBC equals 180 minus 90 minus 60, which is 30 degrees. Therefore, angle ABC equals angle ABD plus angle DBC, which is 30 plus 30, giving us 60 degrees. Since all angles in triangle ABC are 60 degrees, it is an equilateral triangle with all sides equal to 12 units. In an equilateral triangle, the altitude from a vertex to the opposite side bisects that side. Since BD is the altitude from B to AC, D is the midpoint of AC. Therefore, AD equals half of AC, which is half of 12, giving us 6 units. Let's summarize what we've learned. In triangle ABC, we found that the length of AD is 6 units. We used two different methods to solve this problem. In the first method, we used right triangles BDC and ABD along with trigonometric ratios to calculate BD and then AD. In the second method, we recognized that triangle ABC is an equilateral triangle where D is the midpoint of side AC. Both methods yielded the same result: AD equals 6 units. Therefore, the answer to the original question is option B: 6.