Question

Reflect triangle ABC
𝐴
𝐵
𝐶
with respect to the x
𝑥
-axis and the y
𝑦
-axis. What are the vertices of triangle A′B′C′
𝐴
′
𝐵
′
𝐶
′
?

(1 point)
Responses

The vertices of triangle A′B′C′
𝐴
′
𝐵
′
𝐶
′
are A′(−2,−1)
𝐴
′
(
−
2
,
−
1
)
, B′(−6,−3)
𝐵
′
(
−
6
,
−
3
)
, and C′(−1,−3)
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′
𝐴
′
𝐵
′
𝐶
′
are A′(−2,1)
𝐴
′
(
−
2
,
1
)
, B′(−6,3)
𝐵
′
(
−
6
,
3
)
, and C′(−1,3)
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 1 right parenthesis ,

Answer 1

To reflect triangle ABC with respect to the x-axis and the y-axis, we will follow these steps for each vertex (A, B, C) of the triangle.

1. **Reflection with respect to the x-axis:** This will change the sign of the y-coordinate.
- If the original coordinates of any point are \((x, y)\), after reflection over the x-axis, the new coordinates will be \((x, -y)\).

2. **Reflection with respect to the y-axis:** This will change the sign of the x-coordinate.
- After reflecting over the y-axis, the coordinates become \((-x, y)\).

Combining these two reflections, the coordinates of a point after both reflections will be:
\[
(x, y) \to (x, -y) \to (-x, -y).
\]

Let's denote the original vertices of triangle ABC as:
- A(x₁, y₁)
- B(x₂, y₂)
- C(x₃, y₃)

After reflecting both over the x-axis and the y-axis, their new coordinates (A', B', C') will be:
- A'(-x₁, -y₁)
- B'(-x₂, -y₂)
- C'(-x₃, -y₃)

Now, if we have specific points for triangle ABC, we can apply this rule to find A', B', and C'.

If we take your example where triangle ABC has these vertices:
- A(-2, 1)
- B(-6, 3)
- C(-1, 3)

Performing the reflections:
- For A(-2, 1):
- Reflect over the x-axis: A''(-2, -1)
- Reflect over the y-axis: A' (2, -1)

- For B(-6, 3):
- Reflect over the x-axis: B''(-6, -3)
- Reflect over the y-axis: B' (6, -3)

- For C(-1, 3):
- Reflect over the x-axis: C''(-1, -3)
- Reflect over the y-axis: C' (1, -3)

So the final vertices of triangle A′B′C′ are:
- A′(2, -1)
- B′(6, -3)
- C′(1, -3)

Please check the coordinates of the original triangle ABC to ensure that we calculate the transformations correctly.