To reflect a triangle with vertices A, B, and C with respect to the x-axis and y-axis, we need the coordinates of the vertices of triangle ABC. Let's say the vertices are given as follows:
- Vertex A = (x₁, y₁)
- Vertex B = (x₂, y₂)
- Vertex C = (x₃, y₃)
Reflection with respect to the x-axis:
When reflecting a point across the x-axis, the x-coordinate remains the same while the y-coordinate changes sign. The new coordinates will be:
- A' = (x₁, -y₁)
- B' = (x₂, -y₂)
- C' = (x₃, -y₃)
Reflection with respect to the y-axis:
When reflecting a point across the y-axis, the y-coordinate remains the same while the x-coordinate changes sign. The new coordinates will be:
- A'' = (-x₁, y₁)
- B'' = (-x₂, y₂)
- C'' = (-x₃, y₃)
Combined Reflection:
If we reflect the triangle ABC first across the x-axis and then across the y-axis (or vice versa), the coordinates of the vertices transform as follows:
-
Reflect across the x-axis:
- A' = (x₁, -y₁)
- B' = (x₂, -y₂)
- C' = (x₃, -y₃)
-
Then reflect across the y-axis:
- A'' = (-x₁, -y₁)
- B'' = (-x₂, -y₂)
- C'' = (-x₃, -y₃)
So, if you wanted to get the vertices of triangle A' B' C' after both reflections, the coordinates of vertices A', B', and C' will be:
- A' = (-x₁, -y₁)
- B' = (-x₂, -y₂)
- C' = (-x₃, -y₃)
Therefore, the vertices of triangle A'B'C' after reflecting triangle ABC with respect to both the x-axis and y-axis will be A' = (-x₁, -y₁), B' = (-x₂, -y₂), C' = (-x₃, -y₃).