Given that [tex]$\triangle ZAP$[/tex] has coordinates [tex]$Z(-1,5)$[/tex], [tex]$A(1,3)$[/tex], and [tex]$P(-2,4)$[/tex], a translation maps point [tex]$Z$[/tex] to [tex]$Z^{\prime}(1,1)$[/tex]. Find the coordinates of [tex]$A^{\prime}$[/tex] and [tex]$P^{\prime}$[/tex] under this translation.

A. [tex]$A^{\prime}(-1,-1) ; P^{\prime}(-4,0)$[/tex]
B. [tex]$A^{\prime}(0,0) ; P^{\prime}(3,-1)$[/tex]
C. [tex]$A^{\prime}(-4,0) ; P^{\prime}(-1,-1)$[/tex]
D. [tex]$A^{\prime}(3,-1) ; P^{\prime}(0,0)$[/tex]



Answer :

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]