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To solve the given problem, we need to calculate the perimeter of triangle \( \triangle DEF \), which is formed by connecting the midpoints of the sides of triangle \( \triangle ABC \). The midpoints divide each side into two equal parts.

### Step-by-step Solution:

1. **Understand the Triangle Midsegment Theorem**:
   The midsegment of a triangle is a line segment connecting the midpoints of two sides of a triangle. It is always:
   - Parallel to the third side.
   - Half the length of the third side.

2. **Lengths of Triangle \( \triangle ABC \)**:
   The sides of \( \triangle ABC \) are labeled as:
   - \( AB = 8 \) (sum of two \( 4+4 \)),
   - \( AC = 6 \),
   - \( BC = 4 \).

3. **Midsegments of \( \triangle ABC \)**:
   - \( DE \) (parallel to \( AC \)) = \( \frac{1}{2} AC = \frac{1}{2} \times 6 = 3 \),
   - \( EF \) (parallel to \( BC \)) = \( \frac{1}{2} BC = \frac{1}{2} \times 4 = 2 \),
   - \( FD \) (parallel to \( AB \)) = \( \frac{1}{2} AB = \frac{1}{2} \times 8 = 4 \).

4. **Perimeter of \( \triangle DEF \)**:
   Add the midsegment lengths:
   \[
   \text{Perimeter of } \triangle DEF = DE + EF + FD = 3 + 2 + 4 = 9
   \]

### Final Answer:
The perimeter of \( \triangle DEF \) is **9 units**.

Would you like a detailed breakdown of any step or further clarifications?  

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### 5 Relative Questions:
1. How does the Triangle Midsegment Theorem help simplify perimeter calculations?
2. What properties do midsegments of a triangle always satisfy?
3. Can you determine the area of \( \triangle DEF \) if the area of \( \triangle ABC \) is known?
4. How would the perimeter change if triangle \( \triangle ABC \) were scalene?
5. Can this method apply to quadrilaterals formed by midpoints of a quadrilateral's sides?

### Tip:
In any triangle, connecting midpoints creates a smaller triangle similar to the original one with sides exactly half as long as the corresponding sides of the larger triangle.

Triangle DEF is formed by connecting the midpoints of the sides of triangle ABC. The lengths of the sides of triangle ABC are shown. Find the perimeter of triangle DEF. Figures not necessarily drawn to scale.

Explore how to calculate the perimeter of triangle DEF, formed by midpoints of triangle ABC. This problem demonstrates the use of the Triangle Midsegment Theorem, with step-by-step geometry solutions and key mathematical concepts. Suitable for Grades 7-9 students.

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