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

Answer 1

To reflect triangle ABC across the x-axis and then the y-axis, we need to follow these steps:

1. **Reflect across the x-axis**: To reflect a point \((x, y)\) across the x-axis, the new coordinates will be \((x, -y)\).

2. **Reflect across the y-axis**: To reflect a point \((x, y)\) across the y-axis, the new coordinates will be \((-x, y)\).

Now let's apply these transformations to each vertex of triangle ABC:

- For point \(A(2, -1)\):
- Reflection across the x-axis: \(A'(2, 1)\)
- Reflection across the y-axis: \(A''(-2, 1)\)

- For point \(B(6, -3)\):
- Reflection across the x-axis: \(B'(6, 3)\)
- Reflection across the y-axis: \(B''(-6, 3)\)

- For point \(C(1, -3)\):
- Reflection across the x-axis: \(C'(1, 3)\)
- Reflection across the y-axis: \(C''(-1, 3)\)

Thus, the vertices of triangle \(A'B'C'\) after reflecting triangle ABC across both axes are:
- \(A''(-2, 1)\)
- \(B''(-6, 3)\)
- \(C''(-1, 3)\)

In conclusion, the vertices of triangle \(A'B'C'\) are:
- \(A'(-2, 1)\)
- \(B'(-6, 3)\)
- \(C'(-1, 3)\)