Reflect triangle ABC

Question

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 ,
A-2,-1 B-6,-3 C-1,-3

Answer 1

To reflect triangle ABC across both the x-axis and the y-axis, we follow these rules for the reflection of points:

1. Reflection across the x-axis: The y-coordinate of each vertex changes sign, while the x-coordinate remains the same. If a point is (x, y), after reflection, it will become (x, -y).

2. Reflection across the y-axis: The x-coordinate of each vertex changes sign, while the y-coordinate remains the same. If a point is (x, y), after reflection, it will become (-x, y).

Assuming we start with the points \( A (x_A, y_A) \), \( B (x_B, y_B) \), and \( C (x_C, y_C) \):

Step 1: Reflect Across the x-axis

• Vertex A becomes \( A'(x_A, -y_A) \)
• Vertex B becomes \( B'(x_B, -y_B) \)
• Vertex C becomes \( C'(x_C, -y_C) \)

Step 2: Reflect Across the y-axis

• Vertex \( A' \) becomes \( A''(-x_A, -y_A) \)
• Vertex \( B' \) becomes \( B''(-x_B, -y_B) \)
• Vertex \( C' \) becomes \( C''(-x_C, -y_C) \)

Example Calculation

Let’s say:

• A (2, 1)
• B (6, 3)
• C (1, 3)

Step 1: Reflection across x-axis

• \( A'(2, -1) \)
• \( B'(6, -3) \)
• \( C'(1, -3) \)

Step 2: Reflection across y-axis

• \( A''(-2, -1) \)
• \( B''(-6, -3) \)
• \( C''(-1, -3) \)

Result

So the vertices of triangle A'B'C' after reflections are:

• A' (-2, -1)
• B' (-6, -3)
• C' (-1, -3)

Thus, the correct response matches with:

The vertices of triangle A'B'C' are A'(-2, -1), B'(-6, -3), and C'(-1, -3).