https://r2.erweima.ai/mathbot/image/172625695433023y3pe7q.jpg

To determine which step will cause the rectangle to map onto itself, let's break down the process:

1. **Identify the rectangle's position**: From the image, the rectangle is located in the first quadrant. Its corners appear to be roughly around points \((3, 4)\), \((5, 4)\), \((5, 2)\), and \((3, 2)\). It is centered closer to the y-axis than the x-axis.

2. **Reflection across the y-axis**: Reflecting across the y-axis will change the x-coordinates of each vertex. The new coordinates after reflection would be:
   - \((3, 4)\) becomes \((-3, 4)\)
   - \((5, 4)\) becomes \((-5, 4)\)
   - \((5, 2)\) becomes \((-5, 2)\)
   - \((3, 2)\) becomes \((-3, 2)\)

3. **Horizontal shifting**: After the reflection, the rectangle would need to be shifted to the right by a certain number of units to return to its original position.

Let's now evaluate each option:
- **Option A: Reflect across the y-axis, then shift 9 units right**. 
   - After reflecting, the rectangle's coordinates become \((-3, 4)\), \((-5, 4)\), \((-5, 2)\), and \((-3, 2)\).
   - Shifting these points 9 units right gives:
     - \((-3, 4)\) becomes \((6, 4)\)
     - \((-5, 4)\) becomes \((4, 4)\)
     - \((-5, 2)\) becomes \((4, 2)\)
     - \((-3, 2)\) becomes \((6, 2)\)

   The new position is not the same as the original, so **A is incorrect**.

- **Option B: Reflect across the y-axis, then shift 8 units right**.
   - After reflecting, the rectangle's coordinates become \((-3, 4)\), \((-5, 4)\), \((-5, 2)\), and \((-3, 2)\).
   - Shifting these points 8 units right gives:
     - \((-3, 4)\) becomes \((5, 4)\)
     - \((-5, 4)\) becomes \((3, 4)\)
     - \((-5, 2)\) becomes \((3, 2)\)
     - \((-3, 2)\) becomes \((5, 2)\)

   This results in the rectangle mapping back to its original position, so **B is correct**.

- **Option C**: Shifting by 7 units would not be enough to return the rectangle to its original position.

- **Option D**: Shifting by 6 units would also not return the rectangle to its original position.

Thus, the correct answer is **B: Reflect across the y-axis, then shift 8 units right**.

Let me know if you'd like further details or have any questions.

Here are some related questions for practice:
1. What happens to a figure's coordinates after a reflection across the x-axis?
2. How does translating a figure horizontally affect its coordinates?
3. What is the general rule for reflecting a point across the y-axis?
4. How do you determine the number of units to shift after a reflection to return a figure to its original position?
5. How would reflecting across both axes change the position of the rectangle?

**Tip**: When working with reflections, always carefully track how the coordinates change and use translations to realign figures to their original position.

Which of the steps will cause the rectangle to map onto itself?

This problem explores how reflecting a rectangle across the y-axis and translating it by specific units can map the rectangle onto itself. Learn step-by-step how to solve this geometric transformation problem, including reflections and horizontal shifts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation