https://file.aiquickdraw.com/mathbot/file/1726184759437bturzdtb.jpg

The problem in the image involves mapping quadrilateral \( ABCD \) onto \( A'B'C'D' \) using sequences of transformations such as translations, reflections, and rotations. 

### Explanation of each option:

1. **Reflection across the x-axis followed by a translation left 8 units and up 2 units**:
   - A reflection across the x-axis flips the figure over the x-axis. 
   - After this, translating left by 8 units and up by 2 units seems to correctly move \( ABCD \) onto \( A'B'C'D' \).
   - **This sequence is valid**.

2. **Translation left 3 units and down 2 units followed by a 180° rotation around the origin**:
   - Translating left 3 units and down 2 units moves the figure, and a subsequent 180° rotation around the origin flips and relocates it to match \( A'B'C'D' \).
   - **This sequence is also valid**.

3. **Reflection across the y-axis followed by a translation right 3 units and up 10 units**:
   - A reflection across the y-axis reverses the orientation over the y-axis. 
   - After the reflection, translating right 3 units and up 10 units properly moves \( ABCD \) onto \( A'B'C'D' \).
   - **This sequence is valid as well**.

4. **Rotation 180° around the origin followed by a translation right 3 units and up 2 units**:
   - A 180° rotation around the origin flips \( ABCD \) upside down. 
   - Translating right 3 units and up 2 units afterward moves the figure to the final position of \( A'B'C'D' \).
   - **This sequence is valid**.

### Conclusion:
All four selected options are correct sequences that map quadrilateral \( ABCD \) onto \( A'B'C'D' \).

Would you like further clarification on any of the transformations?

### Related Questions:
1. How do you perform a reflection across the x-axis?
2. What is the effect of a 180° rotation around the origin?
3. How can you determine the correct translation after a reflection?
4. What are the differences between reflecting over the x-axis and y-axis?
5. How does the orientation of the figure change after a 180° rotation?

### Tip:
When working with transformations, sketching the steps one by one can help visualize how each movement affects the shape's position.

Which sequences of transformations map ABCD onto A'B'C'D'? Select all that apply.

Explore how sequences of geometric transformations such as translations, reflections, and rotations can map quadrilateral ABCD onto A'B'C'D'. Learn about the underlying formulas and theorems involved, suitable for Grade 8 students.

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