To solve this question, we start with the coordinates given for the points [tex]\( Z, A, \)[/tex] and [tex]\( P \)[/tex]. They are:
[tex]\[ Z(-1, 5), A(1, 3), P(-2, 4) \][/tex]
We know that the point [tex]\( Z \)[/tex] is translated to a new point [tex]\( Z(1,1) \)[/tex]. To find the correct translation, we need to determine the translation vector. The translation vector represents the change required in both coordinates (x and y) to move from the initial position of [tex]\( Z \)[/tex] to the new position of [tex]\( Z \)[/tex].
[tex]\[ Z(-1, 5) \rightarrow Z(1, 1) \][/tex]
To find the translation vector, we examine the change in both x and y coordinates:
[tex]\[ \Delta x = 1 - (-1) = 2 \][/tex]
[tex]\[ \Delta y = 1 - 5 = -4 \][/tex]
Thus, the translation vector is:
[tex]\[ (2, -4) \][/tex]
Next, we apply this translation vector to the coordinates of points [tex]\( A \)[/tex] and [tex]\( P \)[/tex].
For point [tex]\( A \)[/tex]:
[tex]\[ A(1, 3) \][/tex]
We add the translation vector to [tex]\( A \)[/tex] coordinates:
[tex]\[ A_x' = 1 + 2 = 3 \][/tex]
[tex]\[ A_y' = 3 + (-4) = -1 \][/tex]
So, the new coordinates of [tex]\( A' \)[/tex] are:
[tex]\[ A'(3, -1) \][/tex]
For point [tex]\( P \)[/tex]:
[tex]\[ P(-2, 4) \][/tex]
We add the translation vector to [tex]\( P \)[/tex] coordinates:
[tex]\[ P_x' = -2 + 2 = 0 \][/tex]
[tex]\[ P_y' = 4 + (-4) = 0 \][/tex]
So, the new coordinates of [tex]\( P' \)[/tex] are:
[tex]\[ P'(0, 0) \][/tex]
Therefore, the translated coordinates are:
[tex]\[ A'(3, -1) \text{ and } P'(0, 0) \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{A^{\prime}(3,-1) ; P(0,0)} \][/tex]
