To solve this problem, we need to determine the translation vector that maps point [tex]\(T(-3,4)\)[/tex] to [tex]\(T'(-1,1)\)[/tex]. Then, we apply this translation vector to the other points [tex]\(O(-4,1)\)[/tex] and [tex]\(Y(-2,3)\)[/tex] to find their new coordinates [tex]\(O'\)[/tex] and [tex]\(Y'\)[/tex].
Step 1: Determine the translation vector
The translation vector can be found by calculating the difference in the coordinates of [tex]\(T'(-1,1)\)[/tex] and [tex]\(T(-3,4)\)[/tex].
Translation vector [tex]\( \vec{v} = \left( \Delta x, \Delta y \right) \)[/tex]
Where,
[tex]\[ \Delta x = -1 - (-3) = -1 + 3 = 2 \][/tex]
[tex]\[ \Delta y = 1 - 4 = -3 \][/tex]
So, the translation vector is [tex]\(\vec{v} = (2, -3)\)[/tex].
Step 2: Apply the translation vector to point [tex]\(O(-4,1)\)[/tex]
To translate [tex]\(O(-4,1)\)[/tex] using the vector [tex]\(\vec{v} = (2, -3)\)[/tex], we add the components of the vector to the coordinates of [tex]\(O\)[/tex]:
[tex]\[ O' = (O_x + \Delta x, O_y + \Delta y) \][/tex]
[tex]\[ O' = (-4 + 2, 1 - 3) \][/tex]
[tex]\[ O' = (-2, -2) \][/tex]
Step 3: Apply the translation vector to point [tex]\(Y(-2,3)\)[/tex]
Similarly, to translate [tex]\(Y(-2,3)\)[/tex] using the vector [tex]\(\vec{v} = (2, -3)\)[/tex], we add the components of the vector to the coordinates of [tex]\(Y\)[/tex]:
[tex]\[ Y' = (Y_x + \Delta x, Y_y + \Delta y) \][/tex]
[tex]\[ Y' = (-2 + 2, 3 - 3) \][/tex]
[tex]\[ Y' = (0, 0) \][/tex]
Conclusion
Thus, under the given translation, the new coordinates of [tex]\(O'\)[/tex] and [tex]\(Y'\)[/tex] are:
[tex]\[ O'(-2, -2) \][/tex]
[tex]\[ Y'(0, 0) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{O'(-2, -2); Y'(0, 0)} \][/tex]
This corresponds to the second choice given in the problem:
[tex]\[ O'(-2,-2) ; Y'(0,0) \][/tex]
