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We are given two right triangles, \( \triangle BCF \) and \( \triangle BDE \), sharing vertex \( F \). We need to solve for \( x \), the length of side \( DE \).

### Step-by-step solution:

1. **Using the Pythagorean Theorem in \( \triangle CBF \):**
   Since \( \triangle CBF \) is a right triangle, we can apply the Pythagorean theorem:
   \[
   BC^2 = CF^2 + BF^2
   \]
   Given that \( CF = 20 \) and \( BF = 24 \), we can solve for \( BC \):
   \[
   BC^2 = 20^2 + 24^2
   \]
   \[
   BC^2 = 400 + 576 = 976
   \]
   \[
   BC = \sqrt{976} = 31.24
   \]

2. **Similarity of Triangles \( \triangle BCF \) and \( \triangle BDE \):**
   Since \( \triangle BCF \) and \( \triangle BDE \) are similar (they share a common angle at \( B \), and both have right angles), we can set up the following proportion between corresponding sides:
   \[
   \frac{BD}{BC} = \frac{BE}{BF} = \frac{DE}{CF}
   \]

3. **Using the given lengths:**
   - \( BC = 31.24 \)
   - \( BD = 55 \)
   - \( BF = 24 \)
   - \( CF = 20 \)

   From the proportion \( \frac{BD}{BC} = \frac{DE}{CF} \), we get:
   \[
   \frac{55}{31.24} = \frac{x}{20}
   \]
   Solving for \( x \):
   \[
   x = \frac{55 \times 20}{31.24} = \frac{1100}{31.24} \approx 35.2
   \]

Thus, the value of \( x \) is approximately **35.2**.

Would you like further clarification, or do you have any other questions?

Here are 5 related questions for further exploration:
1. How can you prove that two triangles are similar based on their angles and sides?
2. What are the applications of the Pythagorean theorem in real-world problems?
3. How do the properties of right triangles help in solving geometry problems?
4. How does triangle similarity aid in solving for unknown side lengths?
5. Can you find another method to solve this problem, possibly using trigonometry?

**Tip:** When solving geometry problems, always check for triangle similarity or congruence, as they often provide shortcuts to finding unknown side lengths.

Solve for x in the given two right triangles, where triangle BCF has side lengths CF = 20 and BF = 24, and triangle BDE has BD = 55 and the triangles are similar. Solve for x, the length of DE.

Learn how to solve for x in a geometric problem involving two right triangles. Using the Pythagorean Theorem and Triangle Similarity, this problem breaks down solving for unknown side lengths. Suitable for high school students in Grades 9-12.

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