To determine the translation vector [tex]\(T\)[/tex] that maps point [tex]\(A(7,2)\)[/tex] onto point [tex]\(A^{\prime}(5,5)\)[/tex], let's follow a step-by-step approach:
1. Identify the initial coordinates of point [tex]\(A\)[/tex]:
[tex]\[
A_x = 7, \quad A_y = 2
\][/tex]
2. Identify the coordinates of the translated point [tex]\(A'\)[/tex]:
[tex]\[
A'_x = 5, \quad A'_y = 5
\][/tex]
3. Calculate the translation vector components [tex]\(T_x\)[/tex] and [tex]\(T_y\)[/tex]:
- The x-component of the translation vector is the difference between the x-coordinates of [tex]\(A'\)[/tex] and [tex]\(A\)[/tex]:
[tex]\[
T_x = A'_x - A_x = 5 - 7 = -2
\][/tex]
- The y-component of the translation vector is the difference between the y-coordinates of [tex]\(A'\)[/tex] and [tex]\(A\)[/tex]:
[tex]\[
T_y = A'_y - A_y = 5 - 2 = 3
\][/tex]
4. Combine these components to write the translation vector [tex]\(T\)[/tex]:
[tex]\[
T = \langle -2, 3 \rangle
\][/tex]
Thus, the correct translation vector that maps point [tex]\(A(7,2)\)[/tex] to point [tex]\(A^{\prime}(5,5)\)[/tex] is:
[tex]\[
\boxed{T_{\langle -2, 3 \rangle}}
\][/tex]
Therefore, the correct answer is:
D. [tex]\(T_{<-2, 3>}\)[/tex]
