The problem involves two geometric transformations — a 270° counterclockwise rotation and a 180° rotation about the origin. Let's solve the transformations for both shapes.

Problem 1: Rectangle WXYZ

• Vertices: W(-8, -4), X(-2, -1), Y(0, -5), Z(-6, -8)
• Transformation: 270° counterclockwise rotation about the origin

To perform a 270° counterclockwise rotation on any point (x, y), the transformation formula is: $(x, y) \to (y, -x)$ Using this formula for each point:

1. W(-8, -4): $W' = (-4, 8)$

2. X(-2, -1): $X' = (-1, 2)$

3. Y(0, -5): $Y' = (-5, 0)$

4. Z(-6, -8): $Z' = (-8, 6)$

Problem 2: Triangle RST

• Vertices: R(2, 6), S(6, 7), T(8, 1)
• Transformation: 180° rotation about the origin

To perform a 180° rotation on any point (x, y), the transformation formula is: $(x, y) \to (-x, -y)$ Using this formula for each point:

1. R(2, 6): $R' = (-2, -6)$

2. S(6, 7): $S' = (-6, -7)$

3. T(8, 1): $T' = (-8, -1)$

Summary of Coordinates:

• For the rectangle WXYZ after the 270° rotation:

  • W'(-4, 8)
  • X'(-1, 2)
  • Y'(-5, 0)
  • Z'(-8, 6)

• For the triangle RST after the 180° rotation:

  • R'(-2, -6)
  • S'(-6, -7)
  • T'(-8, -1)

Would you like further explanations or clarifications on these transformations? Feel free to ask any questions!

Related Questions:

1. What are other rotation formulas for different angles (90°, 180°, 360°)?
2. How would the coordinates change for a reflection instead of a rotation?
3. What is the effect of rotating a shape by 270° clockwise instead of counterclockwise?
4. Can the transformations be visualized using graph paper or coordinate geometry tools?
5. How do these transformations affect the area or shape of a polygon?

Tip: Always remember the specific transformation rules for rotations around the origin to quickly solve problems involving geometric transformations.