To determine the correct sequence of transformations that results in the transformation from triangle ABC to triangle DEF, we will analyze the given options step by step.
-
Reflecting across the y-axis: This transformation changes the coordinates \((x, y)\) of each point to \((-x, y)\).
-
Shifting 2 units to the right and 2 units up: The coordinates \((x, y)\) transform to \((x + 2, y + 2)\).
-
Rotating 180° (clockwise or counterclockwise): This transformation changes the coordinates \((x, y)\) to \((-x, -y)\).
-
Shifting 2 units to the left and 2 units up: This transformation changes the coordinates \((x, y)\) to \((x - 2, y + 2)\).
Now let's analyze the provided options:
- Option A: Reflecting ABC across the y-axis, then shifting 2 units to the right and 2 units up. The transformation can be written as:
- Reflect: \((x, y) \rightarrow (-x, y)\)
- Shift: \((-x, y) \rightarrow (-x + 2, y + 2)\)
- Option B: Shifting 2 units to the right and 2 units up, then rotating 180°. The transformation can be written as:
- Shift: \((x, y) \rightarrow (x + 2, y + 2)\)
- Rotate: \((x + 2, y + 2) \rightarrow (-(x + 2), -(y + 2)) = (-x - 2, -y - 2)\)
- Option C: Rotating -180° (or 180° counterclockwise) then shifting 2 units to the right and 2 units up. The transformation can be written as:
- Rotate: \((x, y) \rightarrow (-x, -y)\)
- Shift: \((-x, -y) \rightarrow (-x + 2, -y + 2)\)
- Option D: Shifting 2 units to the left and 2 units up, then reflecting across the y-axis. The transformation can be written as:
- Shift: \((x, y) \rightarrow (x - 2, y + 2)\)
- Reflect: \((x - 2, y + 2) \rightarrow (-(x - 2), y + 2) = (-x + 2, y + 2)\)
To find which of these transformations is equivalent to the transformation from ABC to DEF, we would need to see the specific coordinates of triangles ABC and DEF.
If you can provide the coordinates of triangles ABC and DEF, we can precisely calculate the transformations. If the final transformation matches exactly one of these calculated transformations, that would be the correct option.