https://r2.erweima.ai/mathbot/image/1726769120505z78dwfmr.jpg

The problem asks to find which figure shows the triangle reflected over the \( x \)-axis first and then reflected over the \( y \)-axis.

1. **Reflection over the \( x \)-axis**:
   - When a point is reflected over the \( x \)-axis, its \( y \)-coordinate changes sign, while the \( x \)-coordinate remains the same.
   - The original triangle is in the first quadrant, so after reflecting over the \( x \)-axis, the triangle will move to the fourth quadrant.

2. **Reflection over the \( y \)-axis**:
   - When a point is reflected over the \( y \)-axis, its \( x \)-coordinate changes sign, while the \( y \)-coordinate remains the same.
   - After reflecting the triangle in the fourth quadrant over the \( y \)-axis, it will move to the third quadrant.

Now, let's review the figures:
- **Option A**: The triangle is in the first quadrant, which doesn't match the expected result.
- **Option B**: The triangle is in the second quadrant, incorrect.
- **Option C**: The triangle is in the third quadrant, this is correct. It matches the result of reflecting first over the \( x \)-axis, then over the \( y \)-axis.
- **Option D**: The triangle is in the fourth quadrant, incorrect.
- **Option E**: The triangle is in the first quadrant again, incorrect.

Thus, the correct answer is **Option C**.

Would you like more details or have any questions?

Here are 5 relative questions to explore further:
1. What is the formula for reflecting a point over the \( x \)-axis and \( y \)-axis?
2. How do the coordinates of a triangle's vertices change under reflection?
3. What happens when you reflect a triangle over both axes at once?
4. Can reflection be considered a type of transformation? Why?
5. What other transformations, besides reflection, can be performed on geometric figures?

**Tip**: Reflections change the sign of coordinates but keep the shape and size of the object unchanged!

Which figure shows the triangle below reflected over the x-axis, then reflected over the y-axis?

Explore how reflecting a triangle first over the x-axis and then over the y-axis changes its position. This detailed analysis covers geometric transformations, specifically reflections, and includes formulas and properties used.

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