[tex]$\triangle CAR$[/tex] has coordinates [tex]$C(2,4)$[/tex], [tex]$A(1,1)$[/tex], and [tex]$R(3,0)$[/tex]. A translation maps point [tex]$C$[/tex] to [tex]$C'(3,2)$[/tex]. Find the coordinates of [tex]$A'$[/tex] and [tex]$R'$[/tex] under this translation.

A. [tex]$A'(4,-2); R'(2,-1)$[/tex]
B. [tex]$A'(-1,0); R'(-2,2)$[/tex]
C. [tex]$A'(2,-1); R'(4,-2)$[/tex]
D. [tex]$A'(-2,2); R'(2,-2)$[/tex]



Answer :

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]