To solve the problem, let's first understand the transformation rule given. The rule states that a [tex]$180^{\circ}$[/tex] rotation around the origin will transform any point [tex]$(x, y)$[/tex] to [tex]$(-x, -y)$[/tex].
Now, let's apply this transformation rule to each vertex of the triangle:
1. For the vertex [tex]\( L(2, 2) \)[/tex]:
- Applying the transformation rule:
[tex]\[
L' = (-2, -2)
\][/tex]
- Therefore, the coordinates of [tex]\( L' \)[/tex] are [tex]\((-2, -2)\)[/tex].
2. For the vertex [tex]\( M(4, 4) \)[/tex]:
- Applying the transformation rule:
[tex]\[
M' = (-4, -4)
\][/tex]
- Therefore, the coordinates of [tex]\( M' \)[/tex] are [tex]\((-4, -4)\)[/tex].
3. For the vertex [tex]\( N(1, 6) \)[/tex]:
- Applying the transformation rule:
[tex]\[
N' = (-1, -6)
\][/tex]
- Therefore, the coordinates of [tex]\( N' \)[/tex] are [tex]\((-1, -6)\)[/tex].
Given these results, we can now identify the true statements from the options:
- 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].
Thus, 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].
3. The coordinates of [tex]\( N' \)[/tex] are [tex]\((-1, -6)\)[/tex].
