To solve this problem, we need to apply the transformation rule [tex]\( (x, y) \rightarrow (-x, -y) \)[/tex] to each vertex of the triangle and verify the new coordinates.
### Step-by-step Transformation:
1. Transforming [tex]\( L(2, 2) \)[/tex]:
- Apply the rule:
[tex]\[ L' = (-2, -2) \][/tex]
2. Transforming [tex]\( M(4, 4) \)[/tex]:
- Apply the rule:
[tex]\[ M' = (-4, -4) \][/tex]
3. Transforming [tex]\( N(1, 6) \)[/tex]:
- Apply the rule:
[tex]\[ N' = (-1, -6) \][/tex]
Now, we check the provided options against these transformed coordinates:
1. The rule for the transformation is [tex]\( (x, y) \rightarrow (-x, -y) \)[/tex].
- True. This is the given rule for a 180-degree rotation.
2. The coordinates of [tex]\( L' \)[/tex] are [tex]\( (-2, -2) \)[/tex].
- True. [tex]\( L' \)[/tex] was transformed from [tex]\( L(2, 2) \)[/tex] to [tex]\( (-2, -2) \)[/tex].
3. The coordinates of [tex]\( M' \)[/tex] are [tex]\( (-4, 4) \)[/tex].
- False. The correct coordinates of [tex]\( M' \)[/tex] are [tex]\( (-4, -4) \)[/tex].
4. The coordinates of [tex]\( N' \)[/tex] are [tex]\( (6, -1) \)[/tex].
- False. The correct coordinates of [tex]\( N' \)[/tex] are [tex]\( (-1, -6) \)[/tex].
5. The coordinates of [tex]\( N' \)[/tex] are [tex]\( (-1, -6) \)[/tex].
- True. This matches the transformed coordinates of [tex]\( N \)[/tex].
Therefore, the three true statements regarding the transformation are:
1. The rule for the transformation is [tex]\( (x, y) \rightarrow (-x, -y) \)[/tex].
2. The coordinates of [tex]\( L' \)[/tex] are [tex]\( (-2, -2) \)[/tex].
5. The coordinates of [tex]\( N' \)[/tex] are [tex]\( (-1, -6) \)[/tex].
So, the correct answers are:
- The rule for the transformation is [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
- The coordinates of [tex]\(L '\)[/tex] are [tex]\((-2, -2)\)[/tex].
- The coordinates of [tex]\(N '\)[/tex] are [tex]\((-1, -6)\)[/tex].
