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.