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?

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!