solve this---**Question Number:** 19
**Chart/Diagram Description:**
* **Type:** Geometric figure (Triangle).
* **Main Elements:**
* Triangle ABC is shown.
* Point D lies on segment AC.
* Segment BD is drawn from vertex B to point D on AC.
* An angle symbol indicates that BD is perpendicular to AC at point D (angle BDA and angle BDC are 90 degrees).
* Angle ABD is labeled as 30 degrees.
* Angle C (angle BCD) is labeled as 60 degrees.
* Side length BC is labeled as 12.
* Vertices are labeled A, B, C, and the intersection point on AC is labeled D.
**Question Stem:**
In $\triangle ABC$ above, what is the length of $\overline{AD}$ ?
**Options:**
A) 4
B) 6
C) $6\sqrt{2}$
D) $6\sqrt{3}$
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In this problem, we have triangle ABC with point D on side AC. Line BD is perpendicular to AC at point D. We're given that angle ABD is 30 degrees, angle C is 60 degrees, and the length of BC is 12 units. Our task is to find the length of segment AD. Let's analyze the triangles formed by these points to solve the problem.
Let's solve this step by step. First, we analyze triangle BDC. We know angle BDC is 90 degrees since BD is perpendicular to AC. We're given that angle C is 60 degrees. In a triangle, angles sum to 180 degrees, so angle DBC must be 30 degrees. This means triangle BDC is a 30-60-90 triangle. In such a triangle, if the hypotenuse BC is 12, then the side opposite the 30-degree angle, which is DC, equals half the hypotenuse, so DC equals 6. Now, since D is on AC, we know that AD plus DC equals AC. Since DC is 6, and AC equals 12, we can determine that AD equals 6. Therefore, the length of AD is 6, which corresponds to answer choice B.
Let's analyze triangle BDC. We know that angle BDC is 90 degrees because BD is perpendicular to AC. We're given that angle C is 60 degrees. Since the angles in a triangle sum to 180 degrees, we can calculate that angle DBC equals 30 degrees. This means triangle BDC is a 30-60-90 triangle, which has special properties we can use to find the length of AD.
Now that we've identified triangle BDC as a 30-60-90 triangle, we can use its special properties. In such a triangle, the side opposite the 30-degree angle equals half the length of the hypotenuse. We know that BC is 12 units, and DC is opposite to the 30-degree angle at B. Therefore, DC equals half of BC, which is 6 units. This is a key step in finding the length of AD.
Now we can find the length of AD. Since point D lies on segment AC, we know that AC equals AD plus DC. We've already determined that DC equals 6 units. Looking at the entire triangle ABC, we can determine that AC equals 12 units. Therefore, AD equals AC minus DC, which is 12 minus 6, giving us 6 units. So the length of AD is 6, which corresponds to answer choice B.
Let's summarize our solution. We started by analyzing triangle BDC, which we identified as a 30-60-90 triangle. Using the special property of such triangles, where the side opposite to the 30-degree angle equals half the hypotenuse, we determined that DC equals 6 units, which is half of BC's length of 12 units. Since point D lies on segment AC, we know that AC equals AD plus DC. We found that AC equals 12 units, and since DC equals 6 units, AD must equal 6 units. Therefore, the answer is B, 6.