Final answer:
The task involved performing reflections over the lines X = -3 and Y = 5 for the vertices of a triangle with original coordinates A(-4, 4), B(4, 7), and C(-1, 3). After calculating, it was clear that none of the provided answer choices were correct, as each reflection needs to maintain the distance between the original points and the line of reflection.
Step-by-step explanation:
To find the coordinates of ΔA'B'C' after reflecting over the line X = -3, and then the coordinates of ΔA"B"C" after reflecting over the line Y = 5, we must apply the rules for reflections over a vertical and horizontal line, respectively.
For the reflection over X = -3, the X-coordinate of each point changes based on its distance from X = -3, while the Y-coordinate remains the same:
- A'(-4, 4) becomes A'(-2(3) + 4, 4) = A'(-6, 4)
- B(4, 7) becomes B'(2(3) + 4, 7) = B'(10, 7) - This is incorrect as reflected point should remain equidistant from the line of reflection
- C(-1, 3) becomes C'(-2(3) + 1, 3) = C'(-5, 3)
Next, we reflect these new points over Y = 5. In this case, the Y-coordinate of each point changes based on its distance from Y = 5, while the X-coordinate remains the same:
- A'(-6, 4) becomes A"(-6, 2(5) - 4) = A"(-6, 6) - Again this is incorrect as reflected point should remain equidistant from the line of reflection
The correct reflections over X = -3 and Y = 5 should result in coordinates that are equidistant from the line of reflection as the original points.
Reviewing the given choices, the correct coordinates after both reflections can only be:
- ΔA'B'C' = (-2, 4), (4, 7), (-5, 3)
- ΔA"B"C" = (-2, 6), (4, 3), (-5, 7)
So the correct answer from the provided options, after accounting for the proper reflections and distances from the lines of reflections, is none of the given options since none match the correct method for reflections over both lines.