To find the vertices of triangle A′B′C′ after reflecting triangle ABC with respect to the x-axis and then the y-axis, we must follow these steps:
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Reflecting over the x-axis: For any point (x, y), the reflection over the x-axis will be (x, -y).
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Reflecting over the y-axis: For any point (x, y), the reflection over the y-axis will be (-x, y).
Combining these two transformations:
- You first reflect over the x-axis to get (x, -y).
- Then you reflect over the y-axis to get (-x, -y).
Thus, the transformation of a point (x, y) after these two reflections will be (-x, -y).
Now, let’s denote the original vertices of triangle ABC as:
- A(x₁, y₁)
- B(x₂, y₂)
- C(x₃, y₃)
After reflecting across both axes, the new vertices A′, B′, and C′ will be:
- A′(-x₁, -y₁)
- B′(-x₂, -y₂)
- C′(-x₃, -y₃)
To determine the specific vertices of triangle A′B′C′, we need the original coordinates of points A, B, and C. Since the original coordinates of triangle ABC have not been provided, we cannot calculate the transformed vertices without that information.
However, based on the response options provided, the vertices of triangle A′B′C′ are likely created by reflecting triangle ABC.
If the original coordinates of triangle ABC were:
- A(2, 1)
- B(6, 3)
- C(1, 3)
Then the reflections would yield:
- A′(-2, -1)
- B′(-6, -3)
- C′(-1, -3)
None of the answer choices matches this exact result. We would need the original triangle coordinates to definitively identify the new triangle’s vertices. If you have specific coordinates for triangle ABC, please provide them, and I can assist you in determining the correct vertices of triangle A′B′C′.