https://file.aiquickdraw.com/mathbot/file/17274600710811ixp9rra.jpg

The problem involves transforming triangle \( \triangle ABC \) with given coordinates of \( A(3, 1) \), \( B(1, 8) \), and \( C(7, 2) \) to a new triangle \( \triangle A'B'C' \), as shown on the graph.

### Step-by-step analysis:
- **Original points:**
  - \( A(3, 1) \)
  - \( B(1, 8) \)
  - \( C(7, 2) \)

- **New points (after transformation):**
  - From the graph, we can estimate the transformed points as:
    - \( A'(-3, -1) \)
    - \( B'(-1, -8) \)
    - \( C'(-7, -2) \)

By comparing the coordinates of \( A \), \( B \), and \( C \) with their transformed counterparts \( A' \), \( B' \), and \( C' \), it seems that the transformation involves **reflection**.

### What type of reflection?

The transformation from \( (x, y) \) to \( (-x, -y) \) suggests a reflection across the origin, or equivalently, a **rotation of 180°** about the origin.

### Answer:
The transformation that occurred is **rotation by 180° about the origin**.

Would you like more details or further clarifications?

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Here are 5 related questions to deepen your understanding:

1. What is the general rule for rotating a point by 180° around the origin?
2. How would the transformation differ if it were a reflection over the x-axis instead?
3. Can you identify the coordinates of the new triangle after a reflection over the y-axis?
4. What are the effects of a 90° rotation about the origin on the triangle's coordinates?
5. How can you determine the center of rotation if it’s not the origin?

**Tip:** When analyzing transformations, always check for consistent changes in the coordinates to identify the transformation rule.

Which transformation happened to triangle ABC to form triangle A'B'C'? A is at (3, 1), B is at (1, 8), and C is at (7, 2).

Discover how triangle ABC with points A(3, 1), B(1, 8), and C(7, 2) undergoes a transformation to form triangle A'B'C' through a 180-degree rotation about the origin. Suitable for students in Grades 8-10.

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