https://file.aiquickdraw.com/mathbot/file/1727463625273c08cjb9e.jpg

From the image, it appears that the question involves a geometric transformation applied to the points \( Q \), \( R \), and \( S \) shown on the graph.

The selected answer is:
- **"A 180° rotation about the origin, then a translation along the vector \(\langle 2, 2 \rangle\)"**

Let's break down this transformation:

1. **180° Rotation about the Origin**:
   - A 180° rotation about the origin will change each point \((x, y)\) to \((-x, -y)\).
   - Applying this to the points:
     - For point \( Q \), if its original coordinates are \( Q(x_1, y_1) \), after rotation, it will become \( Q'(-x_1, -y_1) \).
     - Similarly, the points \( R \) and \( S \) will also be reflected over the origin to their respective new positions.

2. **Translation along the vector \(\langle 2, 2 \rangle\)**:
   - After rotating the points, this translation moves every point 2 units right and 2 units up.
   - This means the new coordinates after rotation will be shifted accordingly.

The selected answer suggests this sequence of transformations results in the positions of \( Q \), \( R \), and \( S \) as shown.

Let me know if you'd like a more detailed explanation or if you have further questions!

### Here are 5 related questions you might explore:
1. How does a 90° clockwise rotation differ from a 180° rotation in terms of coordinate changes?
2. What are the general rules for rotating points by 90°, 180°, and 270° about the origin?
3. How do you determine the direction and magnitude of a vector used in translation?
4. What is the effect of reflecting a point across the x-axis and y-axis on its coordinates?
5. Can multiple transformations (rotation, translation, reflection) be combined into a single matrix?

### Tip:
When performing a sequence of transformations, it's important to apply them in the specified order, as changing the order can lead to different results.

Given the points Q, R, and S, identify the sequence of transformations that maps Q to R and S, from the following options: a translation along the vector 〈5, −2〉 and reflection in the x-axis, 90° clockwise rotation and reflection in the x-axis, 180° rotation and translation along 〈2, 2〉, or reflection in the x-axis and translation along 〈2, 6〉.

Understand how to apply a 180° rotation followed by a translation on the coordinate plane to points Q, R, and S. This problem involves rotation, translation, and reflection, and is designed for students in Grades 8-10.

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