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[tex]\triangle ABC[/tex] is reflected about the line [tex]y = -x[/tex] to give [tex]\triangle A^{\prime} B^{\prime} C^{\prime}[/tex] with vertices [tex]A^{\prime}(-1,1), B^{\prime}(-2,-1), C(-1,0)[/tex]. What are the vertices of [tex]\triangle ABC[/tex]?

A. [tex]A(1,-1), B(-1,-2), C(0,-1)[/tex]

B. [tex]A(-1,1), B(1,2), C(0,1)[/tex]

C. [tex]A(-1,-1), B(-2,-1), C(-1,0)[/tex]

D. [tex]A(1,1), B(2,-1), C(1,0)[/tex]

E. [tex]A(1,2), B(-1,1), C(0,1)[/tex]



Answer :

GinnyAnswer
To find the vertices of [tex]\( \triangle ABC \)[/tex] given the vertices of the reflected triangle [tex]\( \triangle A'B'C' \)[/tex] over the line [tex]\( y = -x \)[/tex], we need to use the properties of reflections.

When a point [tex]\((x, y)\)[/tex] is reflected over the line [tex]\( y = -x \)[/tex], the coordinates transform to [tex]\((-y, -x)\)[/tex]. We are given the vertices of [tex]\( \triangle A'B'C' \)[/tex] as:
- [tex]\( A'(-1, 1) \)[/tex]
- [tex]\( B'(-2, -1) \)[/tex]
- [tex]\( C'(-1, 0) \)[/tex]

We need to determine the coordinates of the original vertices [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] by reflecting the given coordinates back over the line [tex]\( y = -x \)[/tex].

1. Reflecting [tex]\( A'(-1, 1) \)[/tex]:
- Original point [tex]\( A \)[/tex] becomes [tex]\((-1, 1) \rightarrow (1, -1)\)[/tex]

2. Reflecting [tex]\( B'(-2, -1) \)[/tex]:
- Original point [tex]\( B \)[/tex] becomes [tex]\((-2, -1) \rightarrow (1, 2)\)[/tex]

3. Reflecting [tex]\( C'(-1, 0) \)[/tex]:
- Original point [tex]\( C \)[/tex] becomes [tex]\((-1, 0) \rightarrow (0, 1)\)[/tex]

Thus, the vertices of [tex]\( \triangle ABC \)[/tex] are:
- [tex]\( A(1, -1) \)[/tex]
- [tex]\( B(1, 2) \)[/tex]
- [tex]\( C(0, 1) \)[/tex]

The correct set of vertices is given by option:
B. [tex]\( A(-1,1), B(1,2), C(0,1) \)[/tex]

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