Question

An illustration of a coordinate plane with four quadrants shows the x-axis extending from negative 9 to 9 in increments of 1 and the y-axis extending from negative 11 to 11 in increments of 1. A triangle A B C is plotted on the graph. The triangle A B C has its vertices marked with closed points at A left parenthesis 2 comma negative 1 right parenthesis, B left parenthesis 6 comma negative 3 right parenthesis, and C left parenthesis 1 comma negative 3 right parenthesis.

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

Answer 1

To reflect the triangle ABC across the x-axis and then the y-axis, we can follow these steps for each vertex:

1. **Reflecting across the x-axis**: When a point \((x, y)\) is reflected across the x-axis, its coordinates change to \((x, -y)\).

2. **Reflecting across the y-axis**: When a point \((x, y)\) is reflected across the y-axis, its coordinates change to \((-x, y)\).

Now let’s find the reflections for each vertex of triangle ABC.

### Vertices of Triangle ABC
- A \((2, -1)\)
- B \((6, -3)\)
- C \((1, -3)\)

### Step-by-step Reflection

1. **Vertex A(2, -1)**:
- Reflecting across the x-axis gives:
\[
A' = (2, -(-1)) = (2, 1)
\]
- Reflecting across the y-axis gives:
\[
A'' = (-2, 1)
\]

2. **Vertex B(6, -3)**:
- Reflecting across the x-axis gives:
\[
B' = (6, -(-3)) = (6, 3)
\]
- Reflecting across the y-axis gives:
\[
B'' = (-6, 3)
\]

3. **Vertex C(1, -3)**:
- Reflecting across the x-axis gives:
\[
C' = (1, -(-3)) = (1, 3)
\]
- Reflecting across the y-axis gives:
\[
C'' = (-1, 3)
\]

### Final Coordinates of the Reflected Triangle A′B′C′
Thus, the vertices of triangle A′B′C′ after being reflected across the x-axis and then the y-axis are:

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

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