Let's analyze the rotation transformation of the triangle with vertices [tex]\(L(2, 2)\)[/tex], [tex]\(M(4, 4)\)[/tex], and [tex]\(N(1, 6)\)[/tex] according to the given rule: a 180-degree rotation around the origin. This transformation can be described by the rule [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
### Steps:
1. Transforming Vertex [tex]\(L(2, 2)\)[/tex]:
- Apply the transformation rule:
[tex]\[
L' = (-x, -y) = (-2, -2)
\][/tex]
- Thus, the coordinates of [tex]\(L'\)[/tex] are [tex]\((-2, -2)\)[/tex].
2. Transforming Vertex [tex]\(M(4, 4)\)[/tex]:
- Apply the transformation rule:
[tex]\[
M' = (-x, -y) = (-4, -4)
\][/tex]
- Thus, the coordinates of [tex]\(M'\)[/tex] are [tex]\((-4, -4)\)[/tex].
3. Transforming Vertex [tex]\(N(1, 6)\)[/tex]:
- Apply the transformation rule:
[tex]\[
N' = (-x, -y) = (-1, -6)
\][/tex]
- Thus, the coordinates of [tex]\(N'\)[/tex] are [tex]\((-1, -6)\)[/tex].
### Evaluation of Given Statements:
- Statement 1: The coordinates of [tex]\(L'\)[/tex] are [tex]\((-2, -2)\)[/tex].
- This statement is true as we calculated.
- Statement 2: The coordinates of [tex]\(M'\)[/tex] are [tex]\((-4, 4)\)[/tex].
- This statement is false. The correct coordinates of [tex]\(M'\)[/tex] are [tex]\((-4, -4)\)[/tex].
- Statement 3: The coordinates of [tex]\(N'\)[/tex] are [tex]\((6, -1)\)[/tex].
- This statement is false. The correct coordinates of [tex]\(N'\)[/tex] are [tex]\((-1, -6)\)[/tex].
- Statement 4: The coordinates of [tex]\(N'\)[/tex] are [tex]\((-1, -6)\)[/tex].
- This statement is true as we calculated.
### Conclusion:
The correct true statements based on our calculations are:
1. The coordinates of [tex]\(L'\)[/tex] are [tex]\((-2, -2)\)[/tex].
4. The coordinates of [tex]\(N'\)[/tex] are [tex]\((-1, -6)\)[/tex].
These are the valid true statements regarding the 180-degree rotation transformation of the given triangle.
