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.

Question

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:

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