https://file.aiquickdraw.com/mathbot/file/1725512149369p54u2yw8.jpg

To determine the series of transformations that will map **Figure U** onto **Figure V**, I will analyze the positions of both figures on the coordinate plane.

### Observations:

- **Figure U** is located in the third quadrant, with a slant and specific vertices.
- **Figure V** is in the first quadrant, similarly slanted but at a different location.
  
### Steps for Transformation:

1. **Rotation**: It appears that Figure U can be rotated 180 degrees about the origin (or a 180° rotation, which flips a figure across the origin), as Figure V is roughly an upside-down and repositioned version of Figure U.

2. **Translation**: After rotation, we would need to translate the rotated figure upward and rightward to match the exact position of Figure V.

### Verification of the Steps:

Let me first list down the steps visually and explain the coordinates after each transformation.

1. **Rotation of 180° about the origin**: 
   - A 180° rotation transforms any point \((x, y)\) to \((-x, -y)\).
   - After rotation, Figure U should be flipped and its orientation will match Figure V.

2. **Translation**: We would then apply a translation to move the rotated figure to the correct position. The exact translation vector will be determined by the relative positions of a corresponding point on Figure V.

Would you like more details on calculating the precise coordinates or how to perform these transformations on the figure?

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**Follow-up questions:**

1. What is the exact translation vector required after rotation?
2. How does a 180° rotation about the origin change the coordinates of points?
3. Can a reflection accomplish the same result as rotation?
4. What is the general form of a rotation matrix for rotating points?
5. How do translations affect the coordinates of points on the plane?

**Tip:** In transformation problems, always apply one transformation at a time and check intermediate steps for accuracy!

Determine a series of transformations that would map Figure U onto Figure V.

Learn how to map Figure U onto Figure V using a series of transformations, including a 180° rotation about the origin and a translation. This problem involves key geometric concepts suitable for students in Grades 8-10.

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