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This problem involves using the **geometric mean theorem** for a right triangle with an altitude drawn to the hypotenuse. Here’s how to solve it step by step:

### Key Information:
- \( AD = 12 \)
- \( DC = 16 \)
- \( AC = AD + DC = 12 + 16 = 28 \)
- Altitude \( BD \) forms two similar right triangles.

According to the **geometric mean theorem**:

\[
BD = \sqrt{AD \cdot DC}
\]

### Step-by-Step Solution:
1. Substitute the values of \( AD \) and \( DC \) into the formula:
   \[
   BD = \sqrt{12 \cdot 16}
   \]

2. Simplify the product:
   \[
   BD = \sqrt{192}
   \]

3. Simplify \( \sqrt{192} \) to its simplest radical form:
   \[
   \sqrt{192} = \sqrt{64 \cdot 3} = 8\sqrt{3}
   \]

### Final Answer:
\[
BD = 8\sqrt{3}
\]

Would you like further clarification or details about the steps?  

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### Relative Questions:
1. How does the geometric mean theorem apply to right triangles?
2. What are other properties of similar triangles in this setup?
3. How can you simplify square roots systematically?
4. What are the conditions for triangles to be similar?
5. How is the hypotenuse related to the legs and altitude in a right triangle?

### Tip:
Always verify whether the given lengths satisfy the properties of right triangles (e.g., \( AC \) as the hypotenuse equals \( AD + DC \)).

Given right triangle ABC with altitude BD drawn to hypotenuse AC: If AD = 12 and DC = 16, what is the length of BD in simplest radical form?

Learn how to solve a right triangle problem by applying the geometric mean theorem. Find the length of altitude BD when AD = 12 and DC = 16. Perfect for Grades 8-10, this step-by-step guide simplifies the square root to provide the answer in simplest radical form.

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