Answer:
Reflection across the y-axis: A'(3, 1), B'(4, 5), C'(1, 3)
Translation of 3 units left and 5 units down: A''(0, -4), B''(1, 0), C''(-2, -2)
Step-by-step explanation:
To map triangle ABC to triangle A''B''C'', we can first reflect ΔABC in the y-axis to obtain ΔA'B'C', and then translate ΔA'B'C' 3 units left and 5 units down to get ΔA''B''C''.
Transformation 1: Reflection across the y-axis
When a figure is reflected across the y-axis, the x-coordinate of each point is negated and the y-coordinate remains unchanged:
[tex](x,y)\rightarrow (-x,y)[/tex]
Therefore, when triangle ABC is reflected across the y-axis, the coordinates of its vertices become:
Transformation 2: Translation of 3 units left and 5 units down
When a figure is translated 3 units left, we subtract 3 from its x-coordinate. When a figure is translated 5 units down, we subtract 5 from its y-coordinate:
[tex](x,y)\rightarrow (x-3,y-5)[/tex]
Therefore, when triangle A'B'C' is translated 3 units left and 5 units down, the coordinates of its vertices become:
- A'' = (3-3, 1-5) = (0, -4)
- B'' = (4-3, 5-5) = (1, 0)
- C'' = (1-3, 3-5) = (-2, -2)
