If Figure $$ F $$ is rotated $$ 180^{\circ} $$ and dilated by a factor of $$ \frac{1}{2} $$, which new figure could be produced?

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When transforming a figure through a rotation and dilation, it's important to understand each step separately and then see how they combine.

1. **Rotation by $$ 180^{\circ} $$:**
   - **Effect:** Rotating a figure by $$ 180^{\circ} $$ around the origin (assuming the standard coordinate system) turns it upside down. Every point $$(x, y)$$ in the original figure will be mapped to $$(-x, -y)$$.
   - **Visual Outcome:** The figure appears flipped both horizontally and vertically.

2. **Dilation by a Factor of $$ \frac{1}{2} $$:**
   - **Effect:** Dilating a figure by a factor of $$ \frac{1}{2} $$ scales it down to half its original size. Every point $$(x, y)$$ in the rotated figure will be mapped to $$\left(\frac{x}{2}, \frac{y}{2}\right)$$.
   - **Visual Outcome:** The figure becomes smaller, maintaining its shape and proportions but with all dimensions reduced by half.

**Combining Both Transformations:**
- **Sequential Application:** First, the figure is rotated $$ 180^{\circ} $$, and then this rotated figure is scaled down by a factor of $$ \frac{1}{2} $$.
- **Resulting Transformation:** The new figure will be an upside-down version of the original, reduced to half its size.

**Example Illustration:**
- Suppose the original figure $$ F $$ has a point at $$(2, 3)$$.
  - After a $$ 180^{\circ} $$ rotation: $$(-2, -3)$$.
  - After dilation by $$ \frac{1}{2} $$: $$(-1, -1.5)$$.

**Conclusion:**
The new figure produced by rotating $$ F $$ $$ 180^{\circ} $$ and then dilating it by a factor of $$ \frac{1}{2} $$ will be a mirror image of $$ F $$ that is half the size of the original.

**Visual Representation:**

![Transformed Figure](https://i.imgur.com/yourimage.png) *(Replace with an appropriate image showing the rotated and scaled figure.)*

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*If you have specific options or figures to choose from, please provide them, and I can help identify which one matches the described transformation.*
After rotating Figure $$ F $$ $$ 180^{\circ} $$ and dilating it by a factor of $$ \frac{1}{2} $$, the new figure will be an upside-down, smaller version of the original.

If Figure is rotated and dilated by a factor of , which new figure could be
produced?

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If Figure is rotated and dilated by a factor of , which new figure could be produced?