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The problem describes a transformation of triangle \( ABC \) that involves a 180° rotation about the origin followed by a translation 1 unit to the right, forming triangle \( A'B'C' \).

Let's break down the transformation and solve for the true statements.

### 1. **180° Rotation about the Origin**:
- The rule for a 180° rotation is: \( (x, y) \rightarrow (-x, -y) \).
  
### 2. **Translation 1 Unit to the Right**:
- The translation rule: After rotation, the transformation adds 1 to the x-coordinate, meaning \( (x, y) \rightarrow (x + 1, y) \).

### Let's now verify each statement:

#### A) **The transformation is given by \( (x, y) \rightarrow (x - 1, -y) \)**:
- This is **incorrect** because the rule for 180° rotation is \( (x, y) \rightarrow (-x, -y) \), and the translation moves the point right, so the correct transformation includes an addition to the x-coordinate, not a subtraction.

#### B) **The transformation is given by \( (x, y) \rightarrow (-x + 1, -y) \)**:
- This is **correct**. After a 180° rotation, we apply \( (-x, -y) \) and then shift 1 unit right, giving \( (-x + 1, -y) \).

#### C) **The coordinates of \( C' \) are (2, 2)**:
- Let's verify by first applying the transformations to point \( C \).
  - \( C(-4, -3) \rightarrow (4, 3) \) after 180° rotation.
  - Then, translate 1 unit right: \( (4, 3) \rightarrow (5, 3) \).
  - So, the coordinates of \( C' \) are \( (5, 3) \), not \( (2, 2) \). This statement is **incorrect**.

#### D) **The coordinates of \( B' \) are (1, -2)**:
- Let's check point \( B \):
  - \( B(-1, 3) \rightarrow (1, -3) \) after 180° rotation.
  - Then, translate 1 unit right: \( (1, -3) \rightarrow (2, -3) \).
  - So, the coordinates of \( B' \) are \( (2, -3) \), not \( (1, -2) \). This statement is **incorrect**.

#### E) **The coordinates of \( A' \) are (3, -3)**:
- Let's check point \( A \):
  - \( A(-3, 4) \rightarrow (3, -4) \) after 180° rotation.
  - Then, translate 1 unit right: \( (3, -4) \rightarrow (4, -4) \).
  - So, the coordinates of \( A' \) are \( (4, -4) \), not \( (3, -3) \). This statement is **incorrect**.

### Conclusion:
The only correct statement is **B)**. The transformation is given by \( (x, y) \rightarrow (-x + 1, -y) \).

Would you like more details or clarification on any part? Here are some follow-up questions to explore further:

1. How do different types of transformations (rotations, reflections) change coordinates?
2. What is the general rule for translating points in the coordinate plane?
3. How does a 180° rotation affect points symmetrically?
4. Can you apply the transformations to other figures, such as quadrilaterals?
5. How would the transformation change if it involved reflection instead of rotation?

**Tip:** Always remember to apply transformations step by step, especially when dealing with multiple operations like rotations and translations!

Triangle ABC, shown on the coordinate grid, is rotated 180° about the origin and then translated 1 unit to the right to form Triangle A'B'C'. Which statements are true?

Explore how triangle ABC undergoes a 180° rotation around the origin and is then translated 1 unit to the right. Learn about transformations and verify the coordinates of the new triangle A'B'C'. Suitable for Grades 8-10.

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