To solve this problem, we need to perform a translation of the points [tex]\(A\)[/tex] and [tex]\(R\)[/tex] based on the translation vector obtained from the given transformation of point [tex]\(C\)[/tex] to [tex]\(C'\)[/tex].
1. Determine the Translation Vector:
Given that point [tex]\(C\)[/tex] maps to [tex]\(C'\)[/tex], with coordinates [tex]\(C(2, 4)\)[/tex] to [tex]\(C'(3, 2)\)[/tex], we first find the translation vector [tex]\( \vec{T} \)[/tex] [tex]\( (tx, ty) \)[/tex].
The translation vector can be found by subtracting the coordinates of [tex]\(C\)[/tex] from those of [tex]\(C'\)[/tex]:
[tex]\[
tx = 3 - 2 = 1
\][/tex]
[tex]\[
ty = 2 - 4 = -2
\][/tex]
So, the translation vector [tex]\( \vec{T} = (1, -2) \)[/tex].
2. Translate Point [tex]\(A\)[/tex], Coordinates [tex]\( (1, 1) \)[/tex]:
Applying the translation vector [tex]\((1, -2)\)[/tex] involves adding these values to the coordinates of point [tex]\(A\)[/tex]:
[tex]\[
A^{\prime} = (A_x + tx, A_y + ty)
\][/tex]
[tex]\[
A^{\prime} = (1 + 1, 1 - 2)
\][/tex]
[tex]\[
A^{\prime} = (2, -1)
\][/tex]
3. Translate Point [tex]\(R\)[/tex], Coordinates [tex]\( (3, 0) \)[/tex]:
Applying the same translation vector [tex]\((1, -2)\)[/tex] to point [tex]\(R\)[/tex]:
[tex]\[
R^{\prime} = (R_x + tx, R_y + ty)
\][/tex]
[tex]\[
R^{\prime} = (3 + 1, 0 - 2)
\][/tex]
[tex]\[
R^{\prime} = (4, -2)
\][/tex]
Thus, the new coordinates after translation are:
- [tex]\(A^{\prime} = (2, -1)\)[/tex]
- [tex]\(R^{\prime} = (4, -2)\)[/tex]
From the given options:
1. [tex]\(A^{\prime}(4,-2); R^{\prime}(2,-1)\)[/tex]
2. [tex]\(A^{\prime}(-1,0); R^{\prime}(-2,2)\)[/tex]
3. [tex]\(A^{\prime}(2,-1); R(4,-2)\)[/tex]
4. [tex]\(A^{\prime}(-2,2); R(2,-2)\)[/tex]
The correct answer is:
[tex]\[
\boxed{A^{\prime}(2, -1) ; R(4, -2)}
\][/tex]
