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To solve this problem, we need to map \(\triangle ABC\) to \(\triangle DEF\) using rigid motions such as rotations and translations. We are given that the figures are congruent, meaning that the size and shape are identical, but their orientation and position may differ.

### Step 1: Analyze the coordinates of the triangles
From the graph:
- Coordinates of \(\triangle ABC\):
  - \( A = (6, -1) \)
  - \( B = (1, 3) \)
  - \( C = (2, -4) \)
  
- Coordinates of \(\triangle DEF\):
  - \( D = (-5, 4) \)
  - \( E = (0, 8) \)
  - \( F = (-1, 1) \)

### Step 2: Determine the rigid transformations
1. **Rotation:**
   By observing the shapes and their positioning, it looks like \(\triangle ABC\) needs a **90-degree counterclockwise rotation** around the origin to match the orientation of \(\triangle DEF\).
   - Coordinate rule for a 90-degree counterclockwise rotation:
     \[
     (x, y) \to (-y, x)
     \]
   Applying this rule:
   - \( A(6, -1) \to A' = (1, 6) \)
   - \( B(1, 3) \to B' = (-3, 1) \)
   - \( C(2, -4) \to C' = (4, 2) \)

   After this rotation, the triangle is still not in the same position as \(\triangle DEF\), so we need to follow up with a translation.

2. **Translation:**
   We now translate the rotated triangle so that its vertices match \(\triangle DEF\). To do this, we can calculate the vector from a corresponding pair of points, say \( B'(-3, 1) \) to \( D(-5, 4) \).
   - Translation vector:
     \[
     \text{Translation vector} = D - B' = (-5 - (-3), 4 - 1) = (-2, 3)
     \]

   Therefore, the coordinate notation for the translation is:
   \[
   (x, y) \to (x - 2, y + 3)
   \]

### Final answer:
- **Rotation:** 90-degree counterclockwise rotation around the origin.
  - Coordinate rule: \((x, y) \to (-y, x)\)
- **Translation:** By \((-2, 3)\).
  - Coordinate rule: \((x, y) \to (x - 2, y + 3)\)

Let me know if you'd like further clarifications!

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Here are 5 related questions that expand on this concept:
1. How would the transformation rules change if the rotation was 90 degrees clockwise?
2. Can you map \(\triangle ABC\) to \(\triangle DEF\) using reflections instead of rotations?
3. What is the difference between a rigid transformation and a non-rigid transformation?
4. How can you determine if two figures are congruent without performing the transformation?
5. How would the transformation rules change if the rotation was around a point other than the origin?

**Tip:** When working with rotations, always pay attention to the direction (clockwise or counterclockwise) and the center of rotation.

The figures shown are congruent. Find a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations you use.

This problem involves mapping congruent figures using rigid transformations such as 90-degree counterclockwise rotation and translation. Learn how to apply coordinate rules for these transformations step by step, suitable for students in Grades 8-10.

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