To find the coordinates of the translated quadrilateral \( A'B'C'D' \) after translating quadrilateral \( ABCD \) by 2 units to the right and 4 units up, we will apply the following transformations to each vertex coordinate:
- Translate 2 units to the right: This means we add 2 to the x-coordinate of each vertex.
- Translate 4 units up: This means we add 4 to the y-coordinate of each vertex.
The coordinates of the original quadrilateral \( ABCD \) are:
- \( A(2, 2) \)
- \( B(6, 4) \)
- \( C(10, 4) \)
- \( D(6, 2) \)
Let's find the new coordinates for each vertex:
-
Vertex A (2, 2): \[ A' = (2 + 2, 2 + 4) = (4, 6) \]
-
Vertex B (6, 4): \[ B' = (6 + 2, 4 + 4) = (8, 8) \]
-
Vertex C (10, 4): \[ C' = (10 + 2, 4 + 4) = (12, 8) \]
-
Vertex D (6, 2): \[ D' = (6 + 2, 2 + 4) = (8, 6) \]
After the translation, the new coordinates for the translated quadrilateral \( A'B'C'D' \) are:
- \( A'(4, 6) \)
- \( B'(8, 8) \)
- \( C'(12, 8) \)
- \( D'(8, 6) \)
Now we can check the provided options against these calculated coordinates:
The options are:
- \( A'(7, -1), B'(10, 1), C'(12, -1), D'(10, -3) \)
- \( A'(3, 7), B'(6, 9), C'(8, 7), D'(6, 5) \)
- \( A'(7, 7), B'(10, 9), C'(12, 7), D'(10, 5) \)
- \( A'(3, -1), B'(6, 1), C'(8, -1), D'(6, -3) \)
None of the options match the calculated coordinates of \( A'(4, 6), B'(8, 8), C'(12, 8), D'(8, 6) \).
However, there seems to have been a misunderstanding in summing up the transformations and the options provided, as none appear to reflect the correct coordinates after translating quadrilateral ABCD.
In conclusion, based on the provided options, there are no correct responses for the new coordinates of quadrilateral \( A'B'C'D' \) after translation.