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Triangle ABC with coordinates [tex]$A(3, -2), B(5, 5), C(-4, 2)$[/tex] is reflected across the x-axis. State the coordinates of the resulting triangle, [tex]A'B'C'[/tex].

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To find the reflected coordinates of triangle [tex]\(ABC\)[/tex] across the x-axis, follow these steps:

1. Understand Reflection Across the x-axis:
When reflecting a point over the x-axis, the x-coordinate remains the same while the y-coordinate changes its sign. Mathematically, if you have a point [tex]\((x, y)\)[/tex], its reflection across the x-axis will be [tex]\((x, -y)\)[/tex].

2. Apply Reflection to Each Point:

- Point A: Given coordinates are [tex]\((3, -2)\)[/tex]. The x-coordinate does not change, and the y-coordinate changes sign.
- Thus, [tex]\(A' = (3, -(-2)) = (3, 2)\)[/tex].

- Point B: Given coordinates are [tex]\((5, 5)\)[/tex]. The x-coordinate does not change, and the y-coordinate changes sign.
- Thus, [tex]\(B' = (5, -(5)) = (5, -5)\)[/tex].

- Point C: Given coordinates are [tex]\((-4, 2)\)[/tex]. The x-coordinate does not change, and the y-coordinate changes sign.
- Thus, [tex]\(C' = (-4, -(2)) = (-4, -2)\)[/tex].

3. State the New Coordinates:
- The coordinates of the reflected triangle [tex]\(A'B'C'\)[/tex] are:
- [tex]\(A' = (3, 2)\)[/tex]
- [tex]\(B' = (5, -5)\)[/tex]
- [tex]\(C' = (-4, -2)\)[/tex]

Therefore, the coordinates of triangle [tex]\(A'B'C'\)[/tex] after reflecting triangle [tex]\(ABC\)[/tex] across the x-axis are [tex]\((3, 2)\)[/tex], [tex]\((5, -5)\)[/tex], and [tex]\((-4, -2)\)[/tex].
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