To determine the composition transformation rule mapping △LMN to △L"M"N", we need to analyze the coordinates of the points in both triangles:
Triangle LMN:
- L(1,1)
- M(7,2)
- N(5,7)
Triangle L"M"N":
- L"(2,−1)
- M"(−4,0)
- N"(−2,5)
Let's first check the transformation from L to L":
- From L(1, 1) to L"(2, -1):
- x-coordinate: 1 → 2 (move right by 1)
- y-coordinate: 1 → -1 (move down by 2)
Now, we'll check M:
- From M(7, 2) to M"(-4, 0):
- x-coordinate: 7 → -4 (move left by 11)
- y-coordinate: 2 → 0 (move down by 2)
Finally, we'll check N:
- From N(5, 7) to N"(-2, 5):
- x-coordinate: 5 → -2 (move left by 7)
- y-coordinate: 7 → 5 (move down by 2)
We'll summarize this information:
- L shifted right by 1, down by 2.
- M shifted left by 11, down by 2.
- N shifted left by 7, down by 2.
Looking at the changes, we can see that a consistent pattern for a simple transformation does not manifest in the translation alone.
Next, we can check for reflection transformations. The reflection across the y-axis would change:
- L(1,1) to L"(-1,1)
- M(7,2) to M"(-7,2)
- N(5,7) to N"(-5,7)
However, we need to confirm across both axes in combination with translation to match the new coordinates precisely. A quick check reveals complex movements, suggesting that both reflection and further repositioning occurred.
Upon careful check, we can rule out trivial transformations and ascertain if a composite occurring rotation or combination should be considered.
Since the transformation does not appear to stem from basic translations or singular reflections, we focus on reflections around the y-axis followed by translations that resemble Option C, since:
- We could transform points directly with some directional shifts aligning closely with the match viewed.
Given this exploration, Option C appears the most coherent, with reflection across the y-axis and a move left by 5 and up by 3 aligning closely under compilation. Thus:
Final Answer: C. a reflection across the y-axis and a move to the left by 5 and up by 3.