To determine the rule used to translate triangle [tex]\(LMN\)[/tex], let's focus on the movement of point [tex]\(L\)[/tex], which initially has coordinates [tex]\(L(7, -3)\)[/tex] and is translated to [tex]\(L'(-1, 8)\)[/tex].
Step-by-step, we analyze the changes in the coordinates:
1. Calculate the change in the x-coordinate:
- The original x-coordinate of [tex]\(L\)[/tex] is [tex]\(7\)[/tex].
- The translated x-coordinate of [tex]\(L'\)[/tex] is [tex]\(-1\)[/tex].
- The difference (change in x) can be calculated as:
[tex]\[
\Delta x = L'_{x} - L_{x} = -1 - 7 = -8
\][/tex]
2. Calculate the change in the y-coordinate:
- The original y-coordinate of [tex]\(L\)[/tex] is [tex]\(-3\)[/tex].
- The translated y-coordinate of [tex]\(L'\)[/tex] is [tex]\(8\)[/tex].
- The difference (change in y) can be calculated as:
[tex]\[
\Delta y = L'_{y} - L_{y} = 8 - (-3) = 8 + 3 = 11
\][/tex]
3. Establish the translation rule:
Based on the changes calculated, the rule of translation for this transformation can be formulated. Each point [tex]\((x, y)\)[/tex] on the triangle [tex]\(LMN\)[/tex] is translated by [tex]\(\Delta x = -8\)[/tex] and [tex]\(\Delta y = 11\)[/tex]. Therefore, the translation rule is:
[tex]\[
(x, y) \rightarrow (x - 8, y + 11)
\][/tex]
Since we have computed the changes [tex]\(\Delta x = -8\)[/tex] and [tex]\(\Delta y = 11\)[/tex], the correct translation rule is:
[tex]\[
(x, y) \rightarrow (x-8, y+11)
\][/tex]
So, the answer is:
[tex]\[
\boxed{(x, y) \rightarrow (x-8, y+11)}
\][/tex]
