https://file.aiquickdraw.com/mathbot/file/1736956234480t83d0l1y.jpg

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!

Given right triangle ABC with altitude BD drawn to hypotenuse AC. If AD = 11 and DC = 7, what is the length of BD in simplest radical form?

Learn how to find the altitude (BD) of a right triangle using the geometric mean theorem. This problem involves the segments of the hypotenuse (AD = 11, DC = 7) and step-by-step calculations to simplify BD into radical form. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation