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?

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.