A triangle is rotated $$ 180^{\circ} $$ about the origin. Its image is reflected in the $$ x $$-axis. The vertices of the final triangle are $$ (-4,-4),(-2,-4) $$, and $$ (-3,-1) $$. What

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It appears that the problem statement is incomplete, ending with "What." However, based on the given information, we can infer that the likely question is to determine the original coordinates of the triangle before the transformations (rotation and reflection) were applied.

Let's break down the problem step by step:

1. **Given Transformations:**
   - The original triangle is rotated $$180^{\circ}$$ about the origin.
   - The resulting image is then reflected in the $$x$$-axis.
   - The final coordinates after these transformations are $$ (-4, -4) $$, $$ (-2, -4) $$, and $$ (-3, -1) $$.

2. **Objective:**
   - Find the original coordinates of the triangle before any transformations were applied.

3. **Solution Approach:**
   - **Reverse the Transformations:** To find the original triangle, we need to reverse the order of transformations applied.
     - First, reverse the reflection in the $$x$$-axis.
     - Then, reverse the rotation of $$180^{\circ}$$ about the origin.

4. **Step-by-Step Calculation:**
   
   **a. Reverse the Reflection in the $$x$$-axis:**
   - Reflection over the $$x$$-axis changes the sign of the $$y$$-coordinates.
   - Apply this to each vertex of the final triangle:
     - $$ (-4, -4) $$ becomes $$ (-4, 4) $$
     - $$ (-2, -4) $$ becomes $$ (-2, 4) $$
     - $$ (-3, -1) $$ becomes $$ (-3, 1) $$

**b. Reverse the $$180^{\circ}$$ Rotation:**
   - A rotation of $$180^{\circ}$$ about the origin transforms a point $$(x, y)$$ to $$(-x, -y)$$.
   - To reverse this, we apply the same rotation again (since rotating twice by $$180^{\circ}$$ brings the point back to its original position):
     - $$ (-4, 4) $$ becomes $$ (4, -4) $$
     - $$ (-2, 4) $$ becomes $$ (2, -4) $$
     - $$ (-3, 1) $$ becomes $$ (3, -1) $$

5. **Conclusion:**
   - The original coordinates of the triangle before any transformations were applied are:
     - $$ (4, -4) $$
     - $$ (2, -4) $$
     - $$ (3, -1) $$

**Final Answer:**
The original triangle had vertices at (4, –4), (2, –4), and (3, –1).
The original triangle had vertices at (4, –4), (2, –4), and (3, –1).

A triangle is rotated \( 180^{\circ} \) about the origin. Its image is reflected in the \( x \)-axis. The vertices of the final triangle are \( (-4,-4),(-2,-4) \), and \( (-3,-1) \). What

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