Let's approach this step-by-step:
1. Understand the Transformation:
- The rule mentioned for the transformation is a [tex]$180^\circ$[/tex] rotation. In a [tex]$180^\circ$[/tex] rotation around the origin, each point [tex]\((x, y)\)[/tex] transforms to [tex]\((-x, -y)\)[/tex].
2. Apply the Transformation to Each Vertex:
- Vertex [tex]\(L\)[/tex]:
- Original coordinates: [tex]\(L(2, 2)\)[/tex]
- After transformation: [tex]\(L' = (-2, -2)\)[/tex]
- Vertex [tex]\(M\)[/tex]:
- Original coordinates: [tex]\(M(4, 4)\)[/tex]
- After transformation: [tex]\(M' = (-4, -4)\)[/tex]
- Vertex [tex]\(N\)[/tex]:
- Original coordinates: [tex]\(N(1, 6)\)[/tex]
- After transformation: [tex]\(N' = (-1, -6)\)[/tex]
3. Verify the Statements:
- Statement 1: "The rule for the transformation is [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]"
- This is true since a [tex]$180^\circ$[/tex] rotation does indeed map each [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- Statement 2: "The coordinates of [tex]\(L'\)[/tex] are [tex]\((-2, -2)\)[/tex]"
- This is true based on the transformation of vertex [tex]\(L\)[/tex].
- Statement 3: "The coordinates of [tex]\(M'\)[/tex] are [tex]\((-4, 4)\)[/tex]"
- This is false. The correct coordinates of [tex]\(M'\)[/tex] are [tex]\((-4, -4)\)[/tex].
- Statement 4: "The coordinates of [tex]\(N'\)[/tex] are [tex]\((6, -1)\)[/tex]"
- This is false. The correct coordinates of [tex]\(N'\)[/tex] are [tex]\((-1, -6)\)[/tex].
- Statement 5: "The coordinates of [tex]\(N'\)[/tex] are [tex]\((-1, -6)\)[/tex]"
- This is true based on the transformation of vertex [tex]\(N\)[/tex].
Conclusion:
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].
