https://file.aiquickdraw.com/mathbot/file/1730927299149j4k09j6p.jpg

To solve this problem, let's analyze each transformation sequence given and how they might map two similar figures from one to another.

### Transformation Sequence Analysis

1. **First Sequence:**
   - **Dilation with scale factor \( \frac{1}{2} \)** centered at \( (0, 0) \):
     - This transformation reduces the size of a figure by half, scaling all points toward the origin.
   - **Reflection across the x-axis**:
     - This transformation flips the figure over the x-axis. If a point on the figure was at \( (x, y) \), after reflection, it would be at \( (x, -y) \).

   The combination of these transformations would map any point \( (x, y) \) to \( \left(\frac{x}{2}, -\frac{y}{2}\right) \).

2. **Second Sequence:**
   - **Dilation given by \( (x, y) \to (2x, 2y) \)** centered at \( (0, 0) \):
     - This transformation enlarges the figure by a factor of 2, moving all points away from the origin by twice their distance.
   - **Translation 12 units left and 2 units up**:
     - This transformation shifts all points 12 units in the negative x-direction and 2 units in the positive y-direction. So, a point \( (x, y) \) would be mapped to \( (x - 12, y + 2) \).

   The combined result of these transformations would take any point \( (x, y) \) to \( (2x - 12, 2y + 2) \).

### Conclusion:
These two transformation sequences are different types of transformations (one involving a reduction and reflection, the other involving an enlargement and translation), and they would map different initial figures into transformed figures that are similar but differ in orientation, position, and size.

To match two similar figures from each sequence of transformations, look for:
- A figure that could be shrunk and reflected for the first transformation.
- A figure that could be enlarged and translated for the second transformation.

Would you like additional details on specific transformations, or do you have questions about any part of this solution?

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1. What is a dilation in geometric transformations, and how does it affect the figure?
2. How does a reflection across the x-axis alter the coordinates of a figure?
3. What effect does translation have on a figure, and how do we calculate it?
4. Why would a scale factor of \( \frac{1}{2} \) shrink a figure instead of enlarging it?
5. How can combinations of transformations change the orientation and size of figures?

**Tip:** When analyzing transformations, start by applying each step individually to a sample point to see the cumulative effect clearly.

Which pair of similar figures can be mapped from one to the other by the given sequence of transformations?
1. A dilation with scale factor 1/2 and center of dilation (0, 0), followed by a reflection across the x-axis.
2. A dilation (x, y) → (2x, 2y) with a center of dilation (0, 0), followed by a translation 12 units left and 2 units up.

Explore how to map similar figures through transformations including dilation with different scale factors, reflection, and translation. Understand the effect of each transformation on size, orientation, and position, suitable for Grades 8-10.

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