Math Problem Statement

Parallelogram W'X'Y'Z' is the image of parallelogram WXYZ under a rotation 90° counterclockwise about the origin followed by a reflection. Write the rules for the rotation and reflection. Rotation: (x,y)↦ , x Reflection: (x,y)↦ ,

Solution

To solve this, we need to determine the specific transformations involved in the problem:

1. Rotation (90° counterclockwise about the origin):

A 90° counterclockwise rotation around the origin transforms any point (x,y)(x, y) to the new coordinates: (x,y)(y,x)(x, y) \to (-y, x) This is the rule for the 90° counterclockwise rotation.

2. Reflection:

Since we don't know yet which axis the reflection occurs across (e.g., x-axis, y-axis, line y=xy = x, etc.), we need more details to define this. However, a common reflection is across the y-axis. The reflection of (x,y)(x, y) across the y-axis would be: (x,y)(x,y)(x, y) \to (-x, y) If the reflection is across a different axis, the rule would change accordingly. For example, a reflection across the x-axis would be: (x,y)(x,y)(x, y) \to (x, -y) And if it's across the line y=xy = x, the reflection rule would be: (x,y)(y,x)(x, y) \to (y, x)

To summarize:

Rotation Rule:

(x,y)(y,x)(x, y) \to (-y, x)

Reflection Rule:

  • If the reflection is across the y-axis: (x,y)(x,y)(x, y) \to (-x, y)
  • If the reflection is across another axis, the rule would change accordingly.

Would you like to provide more details about the reflection's axis, or would you like help analyzing the diagram for a more precise answer?

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Math Problem Analysis

Mathematical Concepts

Geometry
Transformations
Rotation
Reflection

Formulas

Rotation Rule: (x, y) → (-y, x)
Reflection across y-axis: (x, y) → (-x, y)
Reflection across x-axis: (x, y) → (x, -y)
Reflection across y = x: (x, y) → (y, x)

Theorems

Rotation Theorem
Reflection Theorem

Suitable Grade Level

Grades 9-11