Let's determine the translation rule used to move triangle ABC to triangle [tex]\( A'B'C' \)[/tex]. In a translation, each point [tex]\( (x, y) \)[/tex] of the original figure moves to a new position [tex]\( (x', y') \)[/tex] defined by the translation rule [tex]\( T_{(dx, dy)}(x, y) = (x + dx, y + dy) \)[/tex].
### Step-by-Step Process:
1. Identify the Original and Translated Coordinates:
- The original coordinates of the vertices of triangle ABC are:
- [tex]\( A(7, -4) \)[/tex]
- [tex]\( B(10, 3) \)[/tex]
- [tex]\( C(6, 1) \)[/tex]
- The coordinates of the translated vertices are:
- [tex]\( A'(5, 1) \)[/tex]
- [tex]\( B'(8, 8) \)[/tex]
- [tex]\( C'(4, 6) \)[/tex]
2. Calculate the Translation Vector:
- To find the translation vector [tex]\( (dx, dy) \)[/tex], we compare the coordinates of one original point with its corresponding translated point. We will use point [tex]\( A \)[/tex] and [tex]\( A' \)[/tex].
- For point [tex]\( A \)[/tex] and [tex]\( A' \)[/tex]:
[tex]\[
dx = A'_x - A_x = 5 - 7 = -2
\][/tex]
[tex]\[
dy = A'_y - A_y = 1 - (-4) = 1 + 4 = 5
\][/tex]
- Therefore, the translation vector is [tex]\( (dx, dy) = (-2, 5) \)[/tex].
3. Verify the Translation Rule:
- To ensure our calculation is correct, we can verify it using other vertices:
- For point [tex]\( B \)[/tex] and [tex]\( B' \)[/tex]:
[tex]\[
dx = B'_x - B_x = 8 - 10 = -2
\][/tex]
[tex]\[
dy = B'_y - B_y = 8 - 3 = 5
\][/tex]
- For point [tex]\( C \)[/tex] and [tex]\( C' \)[/tex]:
[tex]\[
dx = C'_x - C_x = 4 - 6 = -2
\][/tex]
[tex]\[
dy = C'_y - C_y = 6 - 1 = 5
\][/tex]
- Both checks match [tex]\( dx = -2 \)[/tex] and [tex]\( dy = 5 \)[/tex].
Therefore, the rule Randy used for the translation is [tex]\( T_{-2, 5}(x, y) \)[/tex]. Thus, the correct answer is:
[tex]\[ T_{-2, 5}(x, y) \][/tex]
