To determine the composition transformation that maps triangle LMN to triangle L" M" N", let's first analyze the coordinates of the original triangle and the target triangle.
The original triangle has vertices:
- L(1, 1)
- M(7, 2)
- N(5, 7)
The target triangle has vertices:
- L"(2, -1)
- M"(-4, 0)
- N"(-2, 5)
Step 1: Find the Transformation
- Find the difference between corresponding vertices:
-
L to L":
\(L(1,1) \rightarrow L"(2,-1)\)
Transformation:
x-coordinate: \(1 \to 2\) (move right by 1)
y-coordinate: \(1 \to -1\) (move down by 2) -
M to M":
\(M(7,2) \rightarrow M"(-4,0)\)
Transformation:
x-coordinate: \(7 \to -4\) (move left by 11)
y-coordinate: \(2 \to 0\) (move down by 2) -
N to N":
\(N(5,7) \rightarrow N"(-2,5)\)
Transformation:
x-coordinate: \(5 \to -2\) (move left by 7)
y-coordinate: \(7 \to 5\) (move down by 2)
-
Step 2: Analyzing the transformations:
From the transformations above, it looks like there is an x-reflection involved. This can be inferred from observing that the x-coordinates of L and L" have both an opposite sign.
Let's verify for some symmetry:
-
Taking L to L":
- Reflect across the x-axis: L(1,1) becomes L'(1,-1)
- Then moving to the right by 1 will lead from L'(1,-1) to (2,-1).
-
For M and N, the same reflection would result in:
- Reflecting M(7,2) gives M'(7,-2)
- Moving left by 11 will again place M' at (-4,0).
-
For N, reflecting gives N'(5,-7), and moving left skews the true negative.
Step 3: Final interpretation and selection of the Appropriate Option:
Overall, we essentially reflect across the x-axis, and explore sufficing movements.
However, none of the options present: facts of also moving left or right—notably due to occurred inconsistencies.
After the exercise and exploring compositions: We can derive a pattern resembling:
- Reflection across the y-axis and adjusting by a subsequent amount verifies enough best clears into differing spots.
Thus the correct mapping aligns primarily with Option B: "a reflection across the y-axis and a move to the right by 3 and down by 2", while discovered mismatches provided outside valid options through transformed inconsistencies.
With all said, I hope this breakdown clears the overall complexity of triangle transformations encountered amid reflections and movements.
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