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A triangle has vertices at [tex][tex]$L(2,2)$[/tex][/tex], [tex]$M(4,4)$[/tex], and [tex]$N(1,6)$[/tex]. The triangle is transformed according to the rule [tex]R_0, 180^{\circ}[/tex].

Which statements are true regarding the transformation? Select three options.

A. The rule for the transformation is [tex]$(x, y) \rightarrow (-x, -y)$[/tex].
B. The coordinates of [tex]L'[/tex] are [tex](-2, -2)$[/tex].
C. The coordinates of [tex]M'[/tex] are [tex](-4, -4)$[/tex].
D. The coordinates of [tex]N'[/tex] are [tex](6, -1)$[/tex].
E. The coordinates of [tex]N'[/tex] are [tex](-1, -6)$[/tex].

Sagot :

GinnyAnsweridn
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.
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