https://file.aiquickdraw.com/mathbot/file/17271113599066fbtnfyk.jpg

The question asks to determine the coordinates of the preimage after reflecting the triangle both over the x-axis and the y-axis. Here's how we approach this:

### Step-by-Step Solution:

1. **Identify the vertices of the triangle from the image:**
   - Let's label the vertices of the triangle. Based on the grid:
     - Vertex A: \((3, 4)\)
     - Vertex B: \((5, 2)\)
     - Vertex C: \((1, 2)\)

2. **Reflection across the x-axis:**
   - When reflecting over the x-axis, the y-coordinates change sign, while the x-coordinates remain the same.
     - A': \((3, -4)\)
     - B': \((5, -2)\)
     - C': \((1, -2)\)

3. **Reflection across the y-axis:**
   - When reflecting over the y-axis, the x-coordinates change sign, while the y-coordinates remain the same.
     - A'': \((-3, -4)\)
     - B'': \((-5, -2)\)
     - C'': \((-1, -2)\)

### Conclusion:
The preimage coordinates after reflection across both the x-axis and y-axis are:
- A'' \((-3, -4)\)
- B'' \((-5, -2)\)
- C'' \((-1, -2)\)

Would you like further details or have any questions on this process?

Here are five related questions to explore further:
1. What happens when a shape is reflected only over the x-axis?
2. How do reflections in both axes impact the orientation of the shape?
3. Can you predict the preimage coordinates if only the y-axis reflection occurs?
4. How can translations and reflections be combined in geometric transformations?
5. What is the algebraic rule for reflecting a point over the origin?

**Tip:** For any geometric reflection, always remember which axis the reflection is happening over and which coordinate changes accordingly!

Determine the coordinates of the preimage given the image below and the composition of a reflection in the x-axis and a reflection in the y-axis.

This problem involves reflecting a triangle in the coordinate plane across both the x-axis and y-axis. Learn how to find the preimage coordinates after reflections with a detailed, step-by-step solution.

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