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To solve for \( x \) and side \( BC \) given that Triangle \( ABC \) is similar to Triangle \( FED \), we can set up a proportion between corresponding sides due to the similarity of the triangles.

### Step 1: Set up the proportion
Since triangles \( ABC \) and \( FED \) are similar, their corresponding sides are proportional:
\[
\frac{AB}{FD} = \frac{BC}{FE} = \frac{AC}{DE}
\]

Using the given measurements:
- \( AB = 77 \)
- \( FD = 30 \)
- \( BC = 11x + 11 \)
- \( FE = 21 \)

### Step 2: Form the equation
From the proportions, we can use:
\[
\frac{AB}{FD} = \frac{BC}{FE}
\]
Substitute the values:
\[
\frac{77}{30} = \frac{11x + 11}{21}
\]

### Step 3: Solve for \( x \)
Cross-multiply to solve for \( x \):
\[
77 \cdot 21 = 30 \cdot (11x + 11)
\]
\[
1617 = 330x + 330
\]
Subtract 330 from both sides:
\[
1287 = 330x
\]
Now, divide by 330:
\[
x = \frac{1287}{330} = 3.9
\]

### Step 4: Substitute \( x \) to find \( BC \)
Now, substitute \( x = 3.9 \) into the expression for \( BC \):
\[
BC = 11(3.9) + 11
\]
\[
BC = 42.9 + 11
\]
\[
BC = 53.9
\]

### Solution Summary
- \( x = 3.9 \)
- \( BC = 53.9 \)

Would you like further details on any of the steps? 

Here are some related questions to deepen understanding:
1. How do similar triangles help in solving for unknown sides?
2. What other methods could be used to verify the similarity of triangles?
3. How would the solution change if one of the side lengths were different?
4. How can we apply this concept to solve real-world problems?
5. What are the key properties of similar triangles?

**Tip:** Always check if triangles are similar by comparing angles or side ratios before setting up proportions.

Solve for x and side BC. Triangle ABC is similar to Triangle FED.

Learn how to solve for the unknown variable x and side length BC in a problem involving similar triangles ABC and FED. This step-by-step solution explains how to use triangle similarity and proportions to find x and side BC. Suitable for students in Grades 8-10.

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