To determine which translation rule Randy used to draw the image of triangle [tex]\( ABC \)[/tex], follow these steps:
1. Identify the coordinates before and after translation:
- Before translation: [tex]\( A(7, -4) \)[/tex], [tex]\( B(10, 3) \)[/tex], and [tex]\( C(6, 1) \)[/tex].
- After translation: [tex]\( A'(5, 1) \)[/tex], [tex]\( B'(8, 8) \)[/tex], and [tex]\( C'(4, 6) \)[/tex].
2. Calculate the translation vector using point [tex]\( A \)[/tex] and [tex]\( A' \)[/tex]:
[tex]\[
\text{Translation vector} = (x' - x, y' - y)
\][/tex]
For point [tex]\( A \)[/tex] to [tex]\( A' \)[/tex]:
[tex]\[
\text{Translation vector} = (5 - 7, 1 - (-4)) = (-2, 5)
\][/tex]
3. Verify the translation vector with the other points [tex]\( B \)[/tex] and [tex]\( B' \)[/tex], and [tex]\( C \)[/tex] and [tex]\( C' \)[/tex]:
For point [tex]\( B \)[/tex] to [tex]\( B' \)[/tex]:
[tex]\[
\text{Translation vector} = (8 - 10, 8 - 3) = (-2, 5)
\][/tex]
For point [tex]\( C \)[/tex] to [tex]\( C' \)[/tex]:
[tex]\[
\text{Translation vector} = (4 - 6, 6 - 1) = (-2, 5)
\][/tex]
Since the translation vector [tex]\((-2, 5)\)[/tex] is the same for all three points, the translation is consistent.
4. Determine the translation rule:
The translation vector [tex]\((-2, 5)\)[/tex] tells us that each point on the figure is translated 2 units to the left and 5 units up. Therefore, the rule used by Randy is:
[tex]\[
T_{-2,5}(x, y)
\][/tex]
So, the correct translation rule Randy used is:
[tex]\[
\boxed{T_{-2,5}(x, y)}
\][/tex]
