Certainly, let's go through this step-by-step to determine the coordinates after the transformation and assess the statements given.
1. First, we need to apply the transformation rule [tex]\( R_{0, 180^\circ} \)[/tex] which states that any point [tex]\((x, y)\)[/tex] is transformed to [tex]\((-x, -y)\)[/tex].
2. Let's transform each vertex of the triangle accordingly:
- For vertex [tex]\( L(2, 2) \)[/tex]:
[tex]\[
L' = (-2, -2)
\][/tex]
- For vertex [tex]\( M(4, 4) \)[/tex]:
[tex]\[
M' = (-4, -4)
\][/tex]
- For vertex [tex]\( N(1, 6) \)[/tex]:
[tex]\[
N' = (-1, -6)
\][/tex]
3. Now, let's compare the transformed coordinates with the given statements:
- The rule for the transformation is [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]: This is indeed the correct rule for a 180° rotation about the origin. Thus, this statement is true.
- The coordinates of [tex]\( L'\)[/tex] are [tex]\((-2, -2)\)[/tex]: According to our transformation, this statement is true.
- The coordinates of [tex]\( M'\)[/tex] are [tex]\((-4, 4)\)[/tex]: According to our transformation, this statement is false. The coordinates of [tex]\( M'\)[/tex] should be [tex]\((-4, -4)\)[/tex].
- The coordinates of [tex]\( N'\)[/tex] are [tex]\((6, -1)\)[/tex]: According to our transformation, this statement is false. The coordinates of [tex]\( N'\)[/tex] should be [tex]\((-1, -6)\)[/tex].
- The coordinates of [tex]\( N'\)[/tex] are [tex]\((-1, -6)\)[/tex]: According to our transformation, this statement is true.
4. Therefore, the three correct statements 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].
