To solve this problem, we need to apply the transformation rule [tex]\( R_{0,180^\circ} \)[/tex] to each vertex of the triangle. The given transformation [tex]\( R_{0,180^\circ} \)[/tex] rotates each point by 180 degrees around the origin. Mathematically, this is represented by the transformation [tex]\( (x, y) \rightarrow (-x, -y) \)[/tex].
Let's apply this transformation to each vertex:
1. Vertex [tex]\( L(2,2) \)[/tex]:
[tex]\[
L' = (-2, -2)
\][/tex]
2. Vertex [tex]\( M(4,4) \)[/tex]:
[tex]\[
M' = (-4, -4)
\][/tex]
3. Vertex [tex]\( N(1,6) \)[/tex]:
[tex]\[
N' = (-1, -6)
\][/tex]
Now, let’s evaluate the provided statements:
1. The coordinates of [tex]\( N' \)[/tex] are [tex]\( (6, -1) \)[/tex]:
- This statement is false. As calculated, [tex]\( N' \)[/tex] is [tex]\((-1, -6)\)[/tex].
2. The rule for the transformation is [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]:
- This statement is true. This indeed is the rule for a 180-degree rotation around the origin.
3. The coordinates of [tex]\( N' \)[/tex] are [tex]\((-1, -6)\)[/tex]:
- This statement is true. We calculated [tex]\( N' \)[/tex] to be [tex]\((-1, -6)\)[/tex].
4. The coordinates of [tex]\( L' \)[/tex] are [tex]\((-2, -2)\)[/tex]:
- This statement is true. We calculated [tex]\( L' \)[/tex] to be [tex]\((-2, -2)\)[/tex].
5. The coordinates of [tex]\( M' \)[/tex] are [tex]\((-4, 4)\)[/tex]:
- This statement is false. As calculated, [tex]\( M' \)[/tex] is [tex]\((-4, -4)\)[/tex].
Thus, the three correct statements are:
- The rule for the transformation is [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
- The coordinates of [tex]\( N' \)[/tex] are [tex]\((-1, -6)\)[/tex].
- The coordinates of [tex]\( L' \)[/tex] are [tex]\((-2, -2)\)[/tex].
Hence, there are exactly 3 correct statements out of the given options.
