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The image shows two triangles, \( \triangle ABC \) and \( \triangle DEF \), where triangle ABC is rotated and translated to form triangle DEF. We are tasked with finding the length of side \( EF \).

Since the transformation involved is a rotation and translation, \( \triangle ABC \) and \( \triangle DEF \) are congruent triangles. This means that corresponding sides of the two triangles are equal in length.

In \( \triangle ABC \):
- \( AB = 4 \, \text{in.} \)
- \( BC = 6 \, \text{in.} \)
- \( AC = 7 \, \text{in.} \)

By congruence:
- \( EF \) corresponds to side \( AB \).

Thus, the length of \( EF \) is the same as the length of \( AB \), which is \( 4 \, \text{in.} \).

**Answer:**
The length of side \( EF \) is \( 4 \, \text{inches} \).

Would you like further clarification or details? Here are some related questions to consider:

1. What is the relationship between corresponding sides of congruent triangles?
2. How do we determine congruence between triangles after a rotation and translation?
3. Can a rotation and translation ever change the size of a triangle?
4. What other properties remain unchanged in congruent triangles?
5. How can we prove two triangles are congruent using geometric transformations?

**Tip:** When triangles are congruent, not only are their corresponding sides equal, but their corresponding angles are also identical.

What is the length, in inches (in.), of side EF in triangle DEF, where triangle ABC has been rotated and translated to form triangle DEF?

This problem involves identifying the length of side EF in triangle DEF after triangle ABC has been rotated and translated. Using the properties of congruent triangles, we learn that corresponding sides are equal. Ideal for students in Grades 6-8, the solution includes recognizing geometric transformations and applying congruence theorems.

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