To solve this problem, we need to determine the translation vector that maps point [tex]\( Z \)[/tex] to [tex]\( Z^{\prime} \)[/tex]. After finding this translation vector, we apply the same translation to points [tex]\( A \)[/tex] and [tex]\( P \)[/tex].
1. Determine the translation vector:
- The coordinates of [tex]\( Z \)[/tex] are [tex]\((-1, 5)\)[/tex].
- The coordinates of [tex]\( Z^{\prime} \)[/tex] are [tex]\((1, 1)\)[/tex].
The translation vector [tex]\( \mathbf{T} = (T_x, T_y) \)[/tex] can be found by subtracting the coordinates of [tex]\( Z \)[/tex] from the coordinates of [tex]\( Z^{\prime} \)[/tex].
[tex]\[
T_x = Z^{\prime}_x - Z_x = 1 - (-1) = 1 + 1 = 2
\][/tex]
[tex]\[
T_y = Z^{\prime}_y - Z_y = 1 - 5 = -4
\][/tex]
So, the translation vector is [tex]\( \mathbf{T} = (2, -4) \)[/tex].
2. Apply the translation vector to point [tex]\( A \)[/tex]:
- The coordinates of [tex]\( A \)[/tex] are [tex]\((1, 3)\)[/tex].
To find the coordinates of [tex]\( A^{\prime} \)[/tex], add the translation vector to the coordinates of [tex]\( A \)[/tex]:
[tex]\[
A^{\prime}_x = A_x + T_x = 1 + 2 = 3
\][/tex]
[tex]\[
A^{\prime}_y = A_y + T_y = 3 - 4 = -1
\][/tex]
Thus, the coordinates of [tex]\( A^{\prime} \)[/tex] are [tex]\( (3, -1) \)[/tex].
3. Apply the translation vector to point [tex]\( P \)[/tex]:
- The coordinates of [tex]\( P \)[/tex] are [tex]\((-2, 4)\)[/tex].
To find the coordinates of [tex]\( P^{\prime} \)[/tex], add the translation vector to the coordinates of [tex]\( P \)[/tex]:
[tex]\[
P^{\prime}_x = P_x + T_x = -2 + 2 = 0
\][/tex]
[tex]\[
P^{\prime}_y = P_y + T_y = 4 - 4 = 0
\][/tex]
Thus, the coordinates of [tex]\( P^{\prime} \)[/tex] are [tex]\( (0, 0) \)[/tex].
Given the calculated translated coordinates:
- [tex]\( A^{\prime} = (3, -1) \)[/tex]
- [tex]\( P^{\prime} = (0, 0) \)[/tex],
The correct choice from the options provided is:
[tex]\[ \boxed{A^{\prime}(3,-1) ; P^{\prime}(0,0)} \][/tex]
