To find the coordinates of [tex]\(O'\)[/tex] and [tex]\(G'\)[/tex] under the given translation, follow these steps:
1. Identify the translation vector:
- We need to translate point [tex]\(D(3,2)\)[/tex] to [tex]\(D'(2,4)\)[/tex].
- Calculate the translation vector by subtracting the coordinates of [tex]\(D\)[/tex] from [tex]\(D'\)[/tex]:
[tex]\[
\text{Translation vector} = (2 - 3, 4 - 2) = (-1, 2)
\][/tex]
2. Apply the translation vector to point [tex]\(O\)[/tex]:
- The initial coordinates of [tex]\(O\)[/tex] are [tex]\(O(2, -4)\)[/tex].
- Add the translation vector [tex]\((-1, 2)\)[/tex] to the coordinates of [tex]\(O\)[/tex]:
[tex]\[
O' = (2 + (-1), -4 + 2) = (1, -2)
\][/tex]
3. Apply the translation vector to point [tex]\(G\)[/tex]:
- The initial coordinates of [tex]\(G\)[/tex] are [tex]\(G(-1, -1)\)[/tex].
- Add the translation vector [tex]\((-1, 2)\)[/tex] to the coordinates of [tex]\(G\)[/tex]:
[tex]\[
G' = (-1 + (-1), -1 + 2) = (-2, 1)
\][/tex]
Hence, the coordinates of [tex]\(O'\)[/tex] and [tex]\(G'\)[/tex] after the translation are [tex]\(O'(1, -2)\)[/tex] and [tex]\(G'(-2, 1)\)[/tex] respectively.
Therefore, the correct answer is:
[tex]\[
O'(1, -2); G'(-2, 1)
\][/tex]
