To solve this problem, we use the geometric mean theorem (also known as the altitude-on-hypotenuse theorem), which states:

In a right triangle, the altitude drawn to the hypotenuse creates two smaller right triangles that are similar to each other and to the original triangle. The length of the altitude (BD in this case) is the geometric mean of the lengths of the two segments of the hypotenuse.

Given:

• $AD = 11$
• $DC = 7$
• Hypotenuse $AC = AD + DC = 18$

Step-by-Step Solution:

1. Apply the geometric mean formula for the altitude: $BD = \sqrt{AD \cdot DC}$ Substituting the given values: $BD = \sqrt{11 \cdot 7}$

2. Simplify the expression: $BD = \sqrt{77}$

3. Final Answer: The length of $BD$ is $\sqrt{77}$ in simplest radical form.

Would you like a detailed explanation of the geometric mean theorem or a verification using similarity properties? Here are 5 questions to explore further:

1. How does the geometric mean theorem relate to triangle similarity?
2. Can we calculate the area of the larger triangle using the given information?
3. What would the value of $BD$ be in decimal form?
4. How would the solution change if $AD$ and $DC$ were given in different units?
5. What are other applications of the geometric mean in right triangles?

Tip: Always verify the sum of the two hypotenuse segments equals the total hypotenuse before using the geometric mean theorem!