Question

Reflect triangle ABC with respect to the x -axis and the y -axis. What are the vertices of triangle A′B′C′ ? (1 point) Responses The vertices of triangle A′B′C′ are A′(−2,−1) , B′(−6,−3) , and C′(−1,−3) . The vertices of triangle upper A prime upper B prime upper C prime are upper A prime left parenthesis negative 2 comma negative 1 right parenthesis , upper B prime left parenthesis negative 6 comma negative 3 right parenthesis , and upper C prime left parenthesis negative 1 comma negative 3 right parenthesis . The vertices of triangle A′B′C′ are A′(−2,1) , B′(−1,3) , and C′(−6,3) . The vertices of triangle upper A prime upper B prime upper C prime are upper A prime left parenthesis negative 2 comma 1 right parenthesis , upper B prime left parenthesis negative 1 comma 3 right parenthesis , and upper C prime left parenthesis negative 6 comma 3 right parenthesis . The vertices of triangle A′B′C′ are A′(2,1) , B′(6,3) , and C′(1,3) . The vertices of triangle upper A prime upper B prime upper C prime are upper A prime left parenthesis 2 comma 1 right parenthesis , upper B prime left parenthesis 6 comma 3 right parenthesis , and upper C prime left parenthesis 1 comma 3 right parenthesis . The vertices of triangle A′B′C′ are A′(−2,1) , B′(−6,3) , and C′(−1,3) .

Answer 1

To reflect triangle ABC with respect to the x-axis and the y-axis, we need to apply the following transformations to each vertex of the triangle:

1. **Reflection for the x-axis**: The y-coordinate of each vertex changes sign.
2. **Reflection for the y-axis**: The x-coordinate of each vertex changes sign.

Let's denote the original vertices of triangle ABC as \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \). The transformations can be computed as follows:

- Reflecting point A:
- First reflect over the x-axis: A becomes \( A'(x_1, -y_1) \)
- Then reflect over the y-axis: A becomes \( A''(-x_1, -y_1) \)

- Following the same process for points B and C:
- B becomes \( B'(-x_2, -y_2) \)
- C becomes \( C'(-x_3, -y_3) \)

From your question, it seems the coordinates of triangle ABC are not provided in your original context. However, it seems there are several choices for the vertices after transformation. We can analyze the provided answers to deduce the correct transformation.

If we consider one of the responses, let's assume:

- If the vertices obtained after reflection are said to be \( A'(-2,1) \), \( B'(-1,3) \), and \( C'(-6,3) \):
- Then we'd reflect the original coordinates preceding these transformations to confirm if they fulfill the requirements.

Let's check each option:

1. **Coordinates A′(−2,−1), B′(−6,−3), and C′(−1,−3)** - reflects neither below nor to the opposite quadrants correctly according to the transformation rules.

2. **Coordinates A′(−2,1), B′(−1,3), and C′(−6,3)** - If this reflects the original coordinates accurately based on x, then they would seem plausible.

3. **Coordinates A′(2,1), B′(6,3), and C′(1,3)** - These denote a reflection that doesn't abide by the rules since both orders remain in the positive quadrant.

4. **Coordinates A′(−2,1), B′(−6,3), and C′(−1,3)** - Again, aligns with the transformation pattern we expect following reflection over both axes.

Given the equations and transformations, you would need the original vertices to confirm which option reflects correctly. However, from the transformations outlined, it would appear that the set of vertices closest representing a reflection of ABC are logically reflected vertices but without the original vertices specified, identifying exact coordinates would require more information.

In sum, confirm based on original coordinates; if reflection yields an output corresponding more closely to either option three or four.