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The problem asks to find the perimeter of \( \triangle DEF \) in the given figure.

### Steps to solve the problem:

1. **Identify key segments**:
   - \( AB = AC = 7 \) (since \( AB = 3 + 5 \) and \( AC = 4 + 5 \)).
   - \( DE \) is parallel to \( BC \) and \( D \) and \( E \) are midpoints of \( AB \) and \( AC \), respectively, because the segments are symmetric, as seen in the triangle.
   - Therefore, \( DE = \frac{BC}{2} = \frac{10}{2} = 5 \).

2. **Lengths of the sides of \( \triangle DEF \)**:
   - \( DE = 5 \) (from the reasoning above).
   - \( DF = 3 \) (since \( D \) is the midpoint of \( AB \), the segment \( DF \) is half of \( BD \), which is given as 3).
   - \( EF = 4 \) (similar to \( DF \), \( EF \) is half of \( CE \), and \( CE = 4 \)).

3. **Perimeter calculation**:
   The perimeter of \( \triangle DEF \) is:
   \[
   \text{Perimeter} = DF + EF + DE = 3 + 4 + 5 = 12.
   \]

Thus, the perimeter of \( \triangle DEF \) is **12 units**.

Would you like a detailed explanation or further questions?

Here are 5 related questions:
1. How can we verify if \( D \) and \( E \) are indeed midpoints of \( AB \) and \( AC \)?
2. How would the perimeter change if \( BC \) were a different length?
3. What properties of triangles are useful when dealing with midpoints and parallel lines?
4. Can you derive the relationship between \( DE \) and \( BC \) using the midpoint theorem?
5. What role does symmetry play in determining the sides of \( \triangle DEF \)?

**Tip**: When dealing with midpoints and parallel lines in triangles, the midpoint theorem is a powerful tool, as it relates the segment joining midpoints to the third side.

What is the perimeter of triangle DEF in the figure?

Learn how to calculate the perimeter of triangle DEF in a geometric figure using the midpoint theorem and symmetry. This problem covers triangle geometry, parallel lines, and properties of midpoints, suitable for students in Grades 8-10.

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