To determine the translation rule Randy used to draw the image of triangle [tex]\(ABC\)[/tex], we begin by examining the translation of each vertex.
### Translation of Point [tex]\(A\)[/tex]
Original coordinates of [tex]\(A\)[/tex]: [tex]\((7, -4)\)[/tex]
Translated coordinates of [tex]\(A'\)[/tex]: [tex]\((5, 1)\)[/tex]
To determine the translation vector for point [tex]\(A\)[/tex], we calculate the change in the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates:
- Change in [tex]\(x\)[/tex]: [tex]\(5 - 7 = -2\)[/tex]
- Change in [tex]\(y\)[/tex]: [tex]\(1 - (-4) = 1 + 4 = 5\)[/tex]
So, the translation for point [tex]\(A\)[/tex] is by [tex]\((-2, 5)\)[/tex].
### Translation of Point [tex]\(B\)[/tex]
Original coordinates of [tex]\(B\)[/tex]: [tex]\((10, 3)\)[/tex]
Translated coordinates of [tex]\(B'\)[/tex]: [tex]\((8, 8)\)[/tex]
To check if the same translation vector applies to point [tex]\(B\)[/tex], we calculate the change in the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates:
- Change in [tex]\(x\)[/tex]: [tex]\(8 - 10 = -2\)[/tex]
- Change in [tex]\(y\)[/tex]: [tex]\(8 - 3 = 5\)[/tex]
So, the translation for point [tex]\(B\)[/tex] is also by [tex]\((-2, 5)\)[/tex].
### Translation of Point [tex]\(C\)[/tex]
Original coordinates of [tex]\(C\)[/tex]: [tex]\((6, 1)\)[/tex]
Translated coordinates of [tex]\(C'\)[/tex]: [tex]\((4, 6)\)[/tex]
Finally, to check if the same translation vector applies to point [tex]\(C\)[/tex], we calculate the change in the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates:
- Change in [tex]\(x\)[/tex]: [tex]\(4 - 6 = -2\)[/tex]
- Change in [tex]\(y\)[/tex]: [tex]\(6 - 1 = 5\)[/tex]
So, the translation for point [tex]\(C\)[/tex] is by [tex]\((-2, 5)\)[/tex].
Since all three points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] translate in the same manner, the translation rule that Randy used is [tex]\( T_{-2, 5}(x, y) \)[/tex].
Answer:
[tex]\[ T_{-2, 5}(x, y) \][/tex]
